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PROFESSOR: OK, now
we're recording.

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00:00:29,030 --> 00:00:33,110
I think about half of my
lectures are up on Stella,

10
00:00:33,110 --> 00:00:35,850
because about a quarter,
I forgot to push record,

11
00:00:35,850 --> 00:00:38,050
and another quarter,
something went astray

12
00:00:38,050 --> 00:00:42,460
with the microphone, and all
you hear is like, blegh-egh-egh.

13
00:00:42,460 --> 00:00:44,420
But there's about
half of them up there,

14
00:00:44,420 --> 00:00:45,880
the most important ones.

15
00:00:45,880 --> 00:00:48,290
OK, so the residual
is the remainder

16
00:00:48,290 --> 00:00:53,693
when an approximate solution is
substituted into the governing

17
00:00:53,693 --> 00:00:54,192
PDE.

18
00:00:57,360 --> 00:01:01,310
And remember, what's so
powerful about the residual

19
00:01:01,310 --> 00:01:03,830
is that we can take an
approximate solution,

20
00:01:03,830 --> 00:01:07,310
we can see how close the
PDE is to being satisfied,

21
00:01:07,310 --> 00:01:10,011
which gives us some idea of
how good that solution is,

22
00:01:10,011 --> 00:01:12,510
but we don't actually need to
know the true solution, right?

23
00:01:12,510 --> 00:01:14,218
We don't need to know
the exact solution.

24
00:01:14,218 --> 00:01:16,270
So this is different
to an error.

25
00:01:16,270 --> 00:01:19,225
But it is some measure
of how far our PDE is

26
00:01:19,225 --> 00:01:20,100
from being satisfied.

27
00:01:20,100 --> 00:01:22,558
And so it turns out that this
is a really useful thing when

28
00:01:22,558 --> 00:01:28,550
we're looking for finding
good approximate solutions.

29
00:01:28,550 --> 00:01:32,590
And the example that
we worked with the most

30
00:01:32,590 --> 00:01:36,500
was the diffusion
or heat equation,

31
00:01:36,500 --> 00:01:41,160
which we wrote as K dT
/ dx, all differentiated

32
00:01:41,160 --> 00:01:44,460
with respect to x,
equal to minus q,

33
00:01:44,460 --> 00:01:46,867
where q is the heat source.

34
00:01:46,867 --> 00:01:48,450
So I can just write
the x's down here.

35
00:01:48,450 --> 00:01:51,026
This is notation for
differentiation with respect

36
00:01:51,026 --> 00:01:51,526
to x.

37
00:01:55,640 --> 00:01:58,030
So if that is our
governing equation,

38
00:01:58,030 --> 00:02:01,950
and we have an approximate
solution that we

39
00:02:01,950 --> 00:02:10,460
call T tilde of x,
then the residual

40
00:02:10,460 --> 00:02:12,920
would be [? r. ?]
It's a function

41
00:02:12,920 --> 00:02:14,530
of that approximate solution.

42
00:02:14,530 --> 00:02:17,414
It's also a function of x.

43
00:02:17,414 --> 00:02:20,200
And it would be defined
as bring everything over

44
00:02:20,200 --> 00:02:21,470
to the left-hand side.

45
00:02:21,470 --> 00:02:25,390
So K, then substitute in
the approximate solution.

46
00:02:25,390 --> 00:02:32,690
So d T tilde / dx, all
differentiated, plus q.

47
00:02:32,690 --> 00:02:35,440
So that would be the
definition of the residual.

48
00:02:35,440 --> 00:02:38,560
And if the approximate solution
were exact-- in other words,

49
00:02:38,560 --> 00:02:44,720
if we put not T tilde, but T in
here, the residual would be 0.

50
00:02:44,720 --> 00:02:46,640
And so remember, we
define the residuals.

51
00:02:46,640 --> 00:02:52,160
And then we said, let's say
that this approximate solution,

52
00:02:52,160 --> 00:02:55,910
T tilde, is discretized
and represented

53
00:02:55,910 --> 00:02:57,932
with n degrees of freedom.

54
00:02:57,932 --> 00:02:59,140
And there are different ways.

55
00:02:59,140 --> 00:03:01,460
We could use a
Fourier-type analysis

56
00:03:01,460 --> 00:03:04,600
and assume that this thing is
a linear combination of sines

57
00:03:04,600 --> 00:03:06,980
and cosines, or when we
get to finite element,

58
00:03:06,980 --> 00:03:08,910
we could assume it's
linear combinations

59
00:03:08,910 --> 00:03:10,850
of these special hat functions.

60
00:03:10,850 --> 00:03:14,140
But whatever it is, if we have n
degrees of freedom representing

61
00:03:14,140 --> 00:03:16,430
our approximate
solution, then we

62
00:03:16,430 --> 00:03:19,620
need n conditions
to be able to solve

63
00:03:19,620 --> 00:03:21,300
for that approximate solution.

64
00:03:21,300 --> 00:03:24,160
And we talked about two
different ways to get to that.

65
00:03:24,160 --> 00:03:27,480
One was the collocation
method, where, remember,

66
00:03:27,480 --> 00:03:34,590
we set the residual to be 0
at the collocation points.

67
00:03:34,590 --> 00:03:40,930
So required the
residual to be 0 at,

68
00:03:40,930 --> 00:03:46,150
let's say, capital N points,
where this approximate solution

69
00:03:46,150 --> 00:03:50,460
here is represented with
n degrees of freedom.

70
00:03:50,460 --> 00:03:52,170
So that was the
collocation method,

71
00:03:52,170 --> 00:03:55,500
pinning the residual down to
be 0 at n points, which gave us

72
00:03:55,500 --> 00:03:57,170
the n conditions
that let us solve

73
00:03:57,170 --> 00:03:58,740
for the n degrees of freedom.

74
00:03:58,740 --> 00:04:04,320
Or the method of weighted
residuals, which said,

75
00:04:04,320 --> 00:04:11,230
let's instead define the
j-th weighted residual,

76
00:04:11,230 --> 00:04:14,520
where what we do is
we take the residual,

77
00:04:14,520 --> 00:04:17,880
we weight it by
another basis function,

78
00:04:17,880 --> 00:04:21,230
and then we integrate
over the domain.

79
00:04:21,230 --> 00:04:25,750
So that means the j-th
weighted residual-- still

80
00:04:25,750 --> 00:04:28,045
a function of our
approximate solution

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00:04:28,045 --> 00:04:34,640
T-- is some weighting
function, Wj,

82
00:04:34,640 --> 00:04:41,939
times our residual, integrated
over [? our ?] 1D domain.

83
00:04:46,430 --> 00:04:49,160
So define the j-th weighted
residual in this way.

84
00:04:51,890 --> 00:04:53,105
That would be the first step.

85
00:04:53,105 --> 00:04:54,480
And then the second
step would be

86
00:04:54,480 --> 00:05:00,390
to require that n weighted
residuals now are equal to 0.

87
00:05:13,180 --> 00:05:15,680
You guys remember all of this?

88
00:05:15,680 --> 00:05:16,532
Yeah?

89
00:05:16,532 --> 00:05:18,210
And then remember,
the key is now, OK,

90
00:05:18,210 --> 00:05:20,660
what are these
weighting functions?

91
00:05:20,660 --> 00:05:22,490
And you remember, we
talked specifically

92
00:05:22,490 --> 00:05:27,590
about a kind of weighting called
Galerkin weighting, which says,

93
00:05:27,590 --> 00:05:30,666
choose the weighting
functions to be the same basis

94
00:05:30,666 --> 00:05:32,290
vectors, the same
functions that you're

95
00:05:32,290 --> 00:05:36,996
using to approximate the T tilde
and its n degrees of freedom.

96
00:05:36,996 --> 00:05:41,630
And so that's what's key
about a Galerkin method

97
00:05:41,630 --> 00:05:46,870
of weighted residuals, is
that the weighting functions

98
00:05:46,870 --> 00:05:55,410
are the same as
the functions used

99
00:05:55,410 --> 00:05:57,290
to approximate
[? as ?] a solution.

100
00:06:15,600 --> 00:06:18,115
And again, then requiring n
of those weighted residuals

101
00:06:18,115 --> 00:06:22,110
to be equal to 0 gave us,
remember, the n-by-n system,

102
00:06:22,110 --> 00:06:25,150
where each row in the
system was the equation that

103
00:06:25,150 --> 00:06:26,910
said Rj equals 0.

104
00:06:26,910 --> 00:06:28,770
And each column in
the matrix system

105
00:06:28,770 --> 00:06:31,350
corresponded to the
unknown degree of freedom

106
00:06:31,350 --> 00:06:35,220
describing our approximate
solution, T tilde.

107
00:06:35,220 --> 00:06:37,780
OK, so that's the method
of weighted residuals.

108
00:06:37,780 --> 00:06:40,310
But actually, I think once
you understand that, you have,

109
00:06:40,310 --> 00:06:41,889
more or less, all
the building blocks

110
00:06:41,889 --> 00:06:43,180
to then move to finite element.

111
00:06:45,870 --> 00:06:51,140
And so what's the jump in
terms of finite element

112
00:06:51,140 --> 00:06:54,590
is that finite element
uses a very specific kind

113
00:06:54,590 --> 00:06:57,510
of basis function
that represents

114
00:06:57,510 --> 00:07:00,690
a solution in this special
way that then turns out

115
00:07:00,690 --> 00:07:03,910
to have really nice properties.

116
00:07:03,910 --> 00:07:06,930
And there are a variety
of basis functions

117
00:07:06,930 --> 00:07:10,324
that you can use, depending on
what kind of an approximation

118
00:07:10,324 --> 00:07:12,365
you want, whether you want
a linear approximation

119
00:07:12,365 --> 00:07:15,390
in each element, or
quadratic approximation,

120
00:07:15,390 --> 00:07:17,780
or even higher order.

121
00:07:17,780 --> 00:07:21,340
And what did we talk about?

122
00:07:21,340 --> 00:07:25,850
So we talked about finite
element basis functions

123
00:07:25,850 --> 00:07:26,875
in 1D and 2D.

124
00:07:32,030 --> 00:07:37,405
And we talked mostly
about the nodal basis.

125
00:07:42,980 --> 00:07:44,932
And I think we
talked even mostly

126
00:07:44,932 --> 00:07:46,140
about the linear nodal basis.

127
00:07:46,140 --> 00:07:46,681
I'm not sure.

128
00:07:46,681 --> 00:07:50,820
Did Vikram talk about the
quadratic basis for 1D

129
00:07:50,820 --> 00:07:53,064
in the lecture that he gave?

130
00:07:53,064 --> 00:07:53,870
No.

131
00:07:53,870 --> 00:07:58,829
So I think maybe you saw
only the linear nodal basis.

132
00:08:03,050 --> 00:08:13,780
So e.g., if we look at the
linear nodal basis in 1D,

133
00:08:13,780 --> 00:08:16,340
remember that these
hat functions,

134
00:08:16,340 --> 00:08:18,940
they have the
property that they are

135
00:08:18,940 --> 00:08:24,306
0 at all nodes except their own
node, where they're equal to 1.

136
00:08:24,306 --> 00:08:31,590
So this is a plot of [? bi. ?]
And it's got a value of 1

137
00:08:31,590 --> 00:08:35,960
at node xi.

138
00:08:35,960 --> 00:08:40,390
It goes to 0 at node xi plus 1.

139
00:08:40,390 --> 00:08:46,440
And then it stays
0 for all nodes

140
00:08:46,440 --> 00:08:47,820
to the right of xi plus 1.

141
00:08:47,820 --> 00:08:53,570
It also goes to 0 at xi
minus 1 and stays to 0.

142
00:08:53,570 --> 00:08:55,320
So the basis
function is actually

143
00:08:55,320 --> 00:08:59,370
defined on the whole domain
at 0, and then it's up to 1,

144
00:08:59,370 --> 00:09:04,580
down to 0, and then
stays at 0 for the rest.

145
00:09:04,580 --> 00:09:06,880
And then, just to run by
the finite element notation,

146
00:09:06,880 --> 00:09:12,280
we talked about element i as
being the element-- the piece

147
00:09:12,280 --> 00:09:15,299
of the x-axis that lies
between xi and xi plus 1.

148
00:09:15,299 --> 00:09:16,840
The element to the
left is i minus 1.

149
00:09:16,840 --> 00:09:21,470
The element to the right
is element i plus 1.

150
00:09:21,470 --> 00:09:23,590
And what's key about
these basis functions

151
00:09:23,590 --> 00:09:25,670
is that they're 0 in
most of the domain.

152
00:09:25,670 --> 00:09:28,550
The only place
where Ci is non-zero

153
00:09:28,550 --> 00:09:31,205
is on element i minus
1 and element i.

154
00:09:31,205 --> 00:09:33,580
And we use that fact when we
get all those integrals that

155
00:09:33,580 --> 00:09:36,610
come out of the definition
of the weighted residual,

156
00:09:36,610 --> 00:09:38,730
that a whole bunch of
integrals go to 0, right?

157
00:09:38,730 --> 00:09:40,670
Because you're multiplying
things together

158
00:09:40,670 --> 00:09:44,310
that are 0 on big
parts of the domain.

159
00:09:47,831 --> 00:09:48,330
All right.

160
00:09:48,330 --> 00:09:54,410
Questions about [? your ?]
[? advisor ?] remembering this?

161
00:09:54,410 --> 00:09:54,910
Yes.

162
00:09:57,884 --> 00:09:59,550
Any questions about
either of those two?

163
00:10:04,530 --> 00:10:08,830
So I will next just remind
you about integration

164
00:10:08,830 --> 00:10:09,940
in the reference element.

165
00:10:25,630 --> 00:10:26,410
What's the idea?

166
00:10:26,410 --> 00:10:35,580
Well, first of all,
in 1D we have things

167
00:10:35,580 --> 00:10:38,620
in the physical x domain.

168
00:10:38,620 --> 00:10:42,700
So we have xj,
say, and xj plus 1.

169
00:10:42,700 --> 00:10:45,230
And the idea is
that the elements

170
00:10:45,230 --> 00:10:47,160
could be all those
different [? links, ?]

171
00:10:47,160 --> 00:10:49,260
depending on how we
decided to make the bridge,

172
00:10:49,260 --> 00:10:51,009
based on what's going
on with the physics.

173
00:10:51,009 --> 00:10:53,360
So the j-th element
might be a lot bigger

174
00:10:53,360 --> 00:10:55,850
than the j minus 1
element, or smaller,

175
00:10:55,850 --> 00:10:57,340
or however we'd like to do it.

176
00:10:57,340 --> 00:11:01,800
And what we'd like to do
is map to the [? xc ?]

177
00:11:01,800 --> 00:11:09,190
space where the element
goes from minus 1 to 1.

178
00:11:09,190 --> 00:11:16,200
So every element in the x
space, element j going from xj

179
00:11:16,200 --> 00:11:19,090
to xj plus 1, could be mapped
to the reference element

180
00:11:19,090 --> 00:11:23,140
in the [? xc ?] space that
goes from minus 1 to 1.

181
00:11:23,140 --> 00:11:25,640
And the mapping
that does that is

182
00:11:25,640 --> 00:11:38,560
x-- as a function of [? xc ?]
is xj plus 1/2, 1 plus [? xc ?]

183
00:11:38,560 --> 00:11:42,170
times xj plus 1 minus xj.

184
00:11:42,170 --> 00:11:44,670
And you should be able to see
that when I set [? xc ?] equal

185
00:11:44,670 --> 00:11:46,620
to minus 1, I get xj.

186
00:11:46,620 --> 00:11:51,490
And when I set [? xc ?] equal to
plus 1, these terms cancel out

187
00:11:51,490 --> 00:11:52,680
and I get xj plus 1.

188
00:11:52,680 --> 00:11:55,910
And then everything in between
is just the linear mapping.

189
00:11:55,910 --> 00:12:00,860
The middle point here is the
middle point here, and so on.

190
00:12:00,860 --> 00:12:03,450
So that's what it
looks like in 1D.

191
00:12:03,450 --> 00:12:07,290
In 2D, we need to map to
the reference element,

192
00:12:07,290 --> 00:12:13,030
which is a standard triangle.

193
00:12:13,030 --> 00:12:18,710
So in 2D, we have
a generic element

194
00:12:18,710 --> 00:12:20,770
that might look
something like that,

195
00:12:20,770 --> 00:12:22,250
where it's got coordinates.

196
00:12:22,250 --> 00:12:29,620
Now I'll say x1,y1,
x2,y2, and x3,y3.

197
00:12:29,620 --> 00:12:33,130
Those are the xy
coordinates of the three

198
00:12:33,130 --> 00:12:36,110
nodes in this element.

199
00:12:36,110 --> 00:12:39,300
And I want to map it to the
reference element, which again,

200
00:12:39,300 --> 00:12:43,270
is in [? xc ?] space,
which, remember,

201
00:12:43,270 --> 00:12:49,840
looks like this right triangle
with this node at 0,0,

202
00:12:49,840 --> 00:12:54,600
this guy at 1,0, and
this guy here at 0,1.

203
00:12:54,600 --> 00:12:57,730
This is coordinate direction
[? xc1 ?] and this is

204
00:12:57,730 --> 00:13:03,506
coordinate direction
[? xc2. ?] And again,

205
00:13:03,506 --> 00:13:05,880
we can write down the generic
mapping just like we did it

206
00:13:05,880 --> 00:13:06,920
here.

207
00:13:06,920 --> 00:13:14,440
We now have to map x and y,
and they are given by x1--

208
00:13:14,440 --> 00:13:22,240
the mapping at x1,y1 plus a
matrix that looks like x2 minus

209
00:13:22,240 --> 00:13:30,860
x1, x3 minus x1, y2
minus y1, y3 minus y1,

210
00:13:30,860 --> 00:13:32,120
times [? xc1 ?] [? xc2. ?]

211
00:13:38,480 --> 00:13:40,540
Just in the same
way as here, we can

212
00:13:40,540 --> 00:13:43,416
see that varying xc
from minus 1 to 1

213
00:13:43,416 --> 00:13:45,297
gave back xj and xj plus 1.

214
00:13:45,297 --> 00:13:46,630
You can see the same thing here.

215
00:13:46,630 --> 00:13:50,120
If you vary [? xc1 ?] and
[? xc2 ?] from the 0,0 point,

216
00:13:50,120 --> 00:13:53,076
clearly 0,0 is going
to give you back x1,y1.

217
00:13:53,076 --> 00:13:56,760
1,0 should give you the
x2,y2, and 0,1 should give you

218
00:13:56,760 --> 00:13:57,330
the x3,y3.

219
00:14:00,110 --> 00:14:02,885
So you can define
these transformations.

220
00:14:02,885 --> 00:14:04,290
And this is generic, right?

221
00:14:04,290 --> 00:14:06,510
I can now give any
element, any triangle,

222
00:14:06,510 --> 00:14:09,605
and you can always get me
to a reference triangle.

223
00:14:09,605 --> 00:14:10,980
And what that
means is that, now,

224
00:14:10,980 --> 00:14:12,690
when you do the
integrations that show up

225
00:14:12,690 --> 00:14:19,700
in the finite element method--
so an integral over, let's say,

226
00:14:19,700 --> 00:14:27,290
a domain, over element K, some
function, function of x, these

227
00:14:27,290 --> 00:14:29,290
are the kind of integrals
that show up in the 2D

228
00:14:29,290 --> 00:14:30,810
finite element method.

229
00:14:30,810 --> 00:14:34,285
We can now do an integration
over the reference elemental,

230
00:14:34,285 --> 00:14:37,320
[INAUDIBLE] omega
[? xc. ?] We're now

231
00:14:37,320 --> 00:14:41,920
writing f as a function
of x of [? xc. ?]

232
00:14:41,920 --> 00:14:43,850
And what we need to
do is to introduce

233
00:14:43,850 --> 00:14:48,150
the Jacobian of the
mapping so that we can now

234
00:14:48,150 --> 00:14:52,480
integrate with respect
to the reference element.

235
00:14:52,480 --> 00:14:55,080
And the Jacobian
here, this is just

236
00:14:55,080 --> 00:14:56,577
the determinant of this matrix.

237
00:14:56,577 --> 00:14:58,285
Can't remember, but
I think you guys must

238
00:14:58,285 --> 00:15:00,076
have done this with
Vikram, because I think

239
00:15:00,076 --> 00:15:02,280
I was away [INAUDIBLE] yeah?

240
00:15:02,280 --> 00:15:04,320
So we end up with this
j, and this j here

241
00:15:04,320 --> 00:15:11,620
is just the determinant of this
x2 minus x1, x3 minus x1, y2

242
00:15:11,620 --> 00:15:15,840
minus y1, y3 minus y1.

243
00:15:15,840 --> 00:15:20,800
So basically, this accounts
for the change in area,

244
00:15:20,800 --> 00:15:24,110
because you're integrating
over this guy That gives you

245
00:15:24,110 --> 00:15:26,277
the reference element.

246
00:15:26,277 --> 00:15:28,360
Sometimes I think it seems
a little bit messy when

247
00:15:28,360 --> 00:15:29,180
you see it written down.

248
00:15:29,180 --> 00:15:31,555
But the reality is it makes
you code really clean, right?

249
00:15:31,555 --> 00:15:33,740
Because you just introduced
these mappings, and then

250
00:15:33,740 --> 00:15:35,239
every single
integration in the code

251
00:15:35,239 --> 00:15:36,880
takes place over the
reference element.

252
00:15:36,880 --> 00:15:39,390
You just go account for the
mappings from whichever element

253
00:15:39,390 --> 00:15:43,930
you happen to be in through
the determinant of this matrix.

254
00:15:46,495 --> 00:15:49,510
Is that good?

255
00:15:49,510 --> 00:15:54,440
OK, so somewhat related
to that, because we'll

256
00:15:54,440 --> 00:15:56,990
still talking about integration,
was Gaussian quadrature.

257
00:16:00,028 --> 00:16:03,600
A really simple idea,
but quite powerful,

258
00:16:03,600 --> 00:16:07,150
and you had a chance to practice
that on the last homework.

259
00:16:07,150 --> 00:16:15,540
So remember, for
Guassian quadrature,

260
00:16:15,540 --> 00:16:18,150
for the simple examples that
we worked through in class,

261
00:16:18,150 --> 00:16:20,800
we could do the
integrals analytically,

262
00:16:20,800 --> 00:16:24,029
because we were integrating
constants, or linear functions.

263
00:16:24,029 --> 00:16:25,945
So we were integrating
things like e to the x,

264
00:16:25,945 --> 00:16:27,280
or xc to the x.

265
00:16:27,280 --> 00:16:29,410
But in practice,
for real problems,

266
00:16:29,410 --> 00:16:32,910
you may end up with integrals
that you can't do analytically,

267
00:16:32,910 --> 00:16:36,000
either in the stiffness matrix
or in the right-hand side

268
00:16:36,000 --> 00:16:38,630
system, on the system.

269
00:16:38,630 --> 00:16:40,770
So if you can't do the
integrals analytically,

270
00:16:40,770 --> 00:16:44,530
then you resort to
Gaussian quadrature.

271
00:16:44,530 --> 00:16:46,240
And remember that
Gaussian quadrature

272
00:16:46,240 --> 00:16:49,210
in 1D looks like this.

273
00:16:49,210 --> 00:16:53,530
If we're trying to integrate
some function, g of xi,

274
00:16:53,530 --> 00:16:57,170
over the domain minus
1 to 1, we approximate

275
00:16:57,170 --> 00:17:04,410
that as the sum of the function
evaluated at quadrature points

276
00:17:04,410 --> 00:17:09,980
xi i weighted by weighting
coefficient alpha i,

277
00:17:09,980 --> 00:17:15,240
and it's the sum from
i equals 1 to nq.

278
00:17:15,240 --> 00:17:18,984
So nq here is the number
of quadrature points.

279
00:17:18,984 --> 00:17:22,383
Number of quadrature points.

280
00:17:22,383 --> 00:17:27,730
[? xci ?] here is the
i-th quadrature point,

281
00:17:27,730 --> 00:17:30,950
and alpha i here is the
i-th quadrature weighting.

282
00:17:39,210 --> 00:17:41,370
Do you remember-- where
do the rules come from?

283
00:17:41,370 --> 00:17:44,056
How do we figure out the
[? xci ?] [? alpha i ?]

284
00:17:44,056 --> 00:17:51,744
[? pairs? ?] Perfect
integration of a polynomial.

285
00:17:51,744 --> 00:17:57,180
So if, for example, we wanted to
just use one quadrature point,

286
00:17:57,180 --> 00:17:58,850
then we would say,
with one point

287
00:17:58,850 --> 00:18:01,625
we can perfectly integrate
a linear function, right?

288
00:18:01,625 --> 00:18:04,386
Because if I take
any linear function,

289
00:18:04,386 --> 00:18:07,170
and what I'm interested in is
the area under the function,

290
00:18:07,170 --> 00:18:10,030
I can get that perfectly by
just evaluating its midpoint,

291
00:18:10,030 --> 00:18:13,491
and the area of the rectangle
is the same area of the--

292
00:18:13,491 --> 00:18:14,904
what is this thing called?

293
00:18:14,904 --> 00:18:15,850
AUDIENCE: Trapezoid.

294
00:18:15,850 --> 00:18:18,818
PROFESSOR: Trapezoid, yeah.

295
00:18:18,818 --> 00:18:21,790
So with one quadrature
point, I can typically

296
00:18:21,790 --> 00:18:23,370
integrate a linear function.

297
00:18:23,370 --> 00:18:26,950
And I would do that by
setting alpha 1 equal to 2

298
00:18:26,950 --> 00:18:29,320
and xc 1 equal to 0.

299
00:18:29,320 --> 00:18:32,600
So again, midpoint and the area.

300
00:18:32,600 --> 00:18:35,010
If I am willing to have
two quadrature points,

301
00:18:35,010 --> 00:18:37,220
it turns out that I can
integrate linear functions

302
00:18:37,220 --> 00:18:41,150
exactly, I can integrate
quadratics exactly,

303
00:18:41,150 --> 00:18:44,660
and I can actually also
integrate cubics exactly.

304
00:18:44,660 --> 00:18:47,000
And it turns out
that the weights

305
00:18:47,000 --> 00:18:52,120
I should choose are alpha 2
equals 1 and quadrature points

306
00:18:52,120 --> 00:18:55,760
at plus or minus 1 over root 3.

307
00:18:55,760 --> 00:18:58,040
And again, those
points are chosen

308
00:18:58,040 --> 00:19:01,544
to give perfect
integration of polynomials.

309
00:19:04,150 --> 00:19:08,810
And of course, when we apply
those quadrature schemes

310
00:19:08,810 --> 00:19:12,470
to non-polynomial functions,
we introduce an approximation,

311
00:19:12,470 --> 00:19:15,120
but typically that's
something that we're

312
00:19:15,120 --> 00:19:18,220
willing to tolerate so that
we use not too many quadrature

313
00:19:18,220 --> 00:19:22,540
points in the scheme.

314
00:19:22,540 --> 00:19:25,260
OK, questions about
reference elements

315
00:19:25,260 --> 00:19:26,362
or Guassian quadrature?

316
00:19:29,038 --> 00:19:31,275
Is that good?

317
00:19:31,275 --> 00:19:32,050
Yeah, Kevin.

318
00:19:32,050 --> 00:19:34,515
AUDIENCE: [INAUDIBLE]
just slightly [INAUDIBLE]

319
00:19:34,515 --> 00:19:35,994
quadrature.

320
00:19:35,994 --> 00:19:39,691
This is on [INAUDIBLE] assuming
there's no quadrature that

321
00:19:39,691 --> 00:19:42,400
can approximate [INAUDIBLE]

322
00:19:42,400 --> 00:19:44,190
PROFESSOR: Yeah.

323
00:19:44,190 --> 00:19:47,950
So the question is that-- so
it's not only for polynomials,

324
00:19:47,950 --> 00:19:49,370
but these are the
points that are

325
00:19:49,370 --> 00:19:53,210
derived by considering exact
integration of polynomials.

326
00:19:53,210 --> 00:19:56,610
And yes, absolutely, there
are other famous quadrature

327
00:19:56,610 --> 00:19:57,730
schemes.

328
00:19:57,730 --> 00:20:00,282
There are Chebyshev
points, and there

329
00:20:00,282 --> 00:20:02,740
are all kinds of points that
are derived in different ways.

330
00:20:02,740 --> 00:20:04,198
Some of them are
derived by looking

331
00:20:04,198 --> 00:20:06,446
at sinusoidal functions.

332
00:20:06,446 --> 00:20:08,370
Yeah, absolutely.

333
00:20:08,370 --> 00:20:12,762
Yeah, it's a whole study
of math, quadrature points

334
00:20:12,762 --> 00:20:13,720
and quadrature schemes.

335
00:20:13,720 --> 00:20:15,573
It's actually kind
of neat stuff.

336
00:20:15,573 --> 00:20:16,072
Yeah.

337
00:20:19,720 --> 00:20:25,770
All right, so let's move on and
talk about boundary conditions,

338
00:20:25,770 --> 00:20:27,830
and then we'll close
by forming the system.

339
00:20:27,830 --> 00:20:32,860
And then, that will be
finite element in a nutshell.

340
00:20:40,550 --> 00:20:43,650
Number five is
boundary conditions.

341
00:20:43,650 --> 00:20:47,870
And the main types of boundary
conditions that we discussed

342
00:20:47,870 --> 00:20:51,870
were Dirichlet, where, remember,
for a Dirichlet condition,

343
00:20:51,870 --> 00:20:59,505
you specify-- what
do we specify?

344
00:20:59,505 --> 00:21:00,850
Yeah, the value of the unknown.

345
00:21:00,850 --> 00:21:01,850
The solution at a point.

346
00:21:06,820 --> 00:21:13,060
And for a Dirichlet
condition, we

347
00:21:13,060 --> 00:21:21,470
are going to replace the
weighted residual equation.

348
00:21:21,470 --> 00:21:24,045
So we're going to
form the system,

349
00:21:24,045 --> 00:21:26,580
we're going to look at forming
the system in just a second.

350
00:21:26,580 --> 00:21:30,020
But we already said that
setting each weighted residual

351
00:21:30,020 --> 00:21:34,700
equal to 0 gives us one row
in our matrix system, right?

352
00:21:34,700 --> 00:21:38,430
One row corresponds to the
equation [? Rj ?] equals 0.

353
00:21:38,430 --> 00:21:41,110
So if there's a Dirichlet
condition in a node-- that's

354
00:21:41,110 --> 00:21:43,720
usually at the boundary,
maybe at the first one--

355
00:21:43,720 --> 00:21:44,580
we're going to come
in and we're going

356
00:21:44,580 --> 00:21:46,788
to strike out that weighted
residual equation that we

357
00:21:46,788 --> 00:21:48,955
got from multiplying
by [? c1 ?] and getting

358
00:21:48,955 --> 00:21:50,080
the weighted residual to 0.

359
00:21:50,080 --> 00:21:52,390
We're going to strike
that out and replace it

360
00:21:52,390 --> 00:21:56,830
with an equation that might look
like 10000 equals to whatever

361
00:21:56,830 --> 00:21:58,080
the Dirichlet condition is.

362
00:21:58,080 --> 00:22:00,177
Then we're going
to fit the value.

363
00:22:00,177 --> 00:22:01,760
And we have to be a
little bit careful

364
00:22:01,760 --> 00:22:04,426
and make sure that we're setting
truly the value of the unknown.

365
00:22:04,426 --> 00:22:05,940
If we're using the
nodal basis-- I

366
00:22:05,940 --> 00:22:08,439
guess I didn't state explicitly
when I drew the nodal basis,

367
00:22:08,439 --> 00:22:13,090
but what's so special
about the nodal basis?

368
00:22:13,090 --> 00:22:15,132
What does it mean
for the unknowns?

369
00:22:20,469 --> 00:22:22,510
Yeah, that's right, that
the unknown coefficients

370
00:22:22,510 --> 00:22:24,020
that you solve
for turn out to be

371
00:22:24,020 --> 00:22:26,310
the values of the approximate
solution at the node.

372
00:22:26,310 --> 00:22:29,110
And that's because, by
construction, those basis

373
00:22:29,110 --> 00:22:32,520
functions are 1 at their own
node and 0 everywhere else.

374
00:22:32,520 --> 00:22:34,300
So if you're using
the nodal basis,

375
00:22:34,300 --> 00:22:36,850
the unknown
coefficients correspond

376
00:22:36,850 --> 00:22:40,069
to the value of the approximate
solution at the node.

377
00:22:40,069 --> 00:22:41,860
Which means that for
a Dirichlet condition,

378
00:22:41,860 --> 00:22:43,960
you can directly manipulate
that coefficient.

379
00:22:47,070 --> 00:22:49,120
And then the other kind
of boundary condition

380
00:22:49,120 --> 00:22:51,960
that we talked about was
a Neumann condition, where

381
00:22:51,960 --> 00:22:53,690
now you specify the flux.

382
00:22:56,930 --> 00:23:00,095
And what we see is that when we
do the weighted residual, when

383
00:23:00,095 --> 00:23:04,610
you integrate by
parts, you get a term

384
00:23:04,610 --> 00:23:09,300
out that is a boundary term
that's where the flux shows up

385
00:23:09,300 --> 00:23:10,300
exactly.

386
00:23:10,300 --> 00:23:13,460
And so a Neumann
condition contributes

387
00:23:13,460 --> 00:23:14,550
to the weighted residual.

388
00:23:14,550 --> 00:23:16,350
You'll see its place
exactly in the derivation

389
00:23:16,350 --> 00:23:18,350
of your weighted residual,
where you can go in--

390
00:23:18,350 --> 00:23:20,445
if the Neumann condition
is flux equal to 0,

391
00:23:20,445 --> 00:23:22,720
it basically strikes
out that term.

392
00:23:22,720 --> 00:23:24,726
If it's a non-zero, a
non-homogenous Neumann

393
00:23:24,726 --> 00:23:27,100
condition, then you could go
in and put that contribution

394
00:23:27,100 --> 00:23:28,620
as the flux.

395
00:23:28,620 --> 00:23:34,140
So in this case, you leave the
weighted residual equation,

396
00:23:34,140 --> 00:23:37,580
but you add the contribution.

397
00:23:37,580 --> 00:23:40,734
So two different boundary
conditions, two different

398
00:23:40,734 --> 00:23:41,400
implementations.

399
00:23:45,975 --> 00:23:48,350
And so what this means is it's
an additional contribution

400
00:23:48,350 --> 00:23:49,440
to the stiffness matrix.

401
00:24:00,660 --> 00:24:01,940
So it's getting a little low.

402
00:24:01,940 --> 00:24:04,530
It says, add contribution to
weighted residual, additional

403
00:24:04,530 --> 00:24:06,323
contribution to
stiffness matrix.

404
00:24:17,190 --> 00:24:19,940
OK, so last thing.

405
00:24:19,940 --> 00:24:22,240
I'll just kind of put
all those pieces together

406
00:24:22,240 --> 00:24:25,310
for the 1D diffusion,
and then finally,

407
00:24:25,310 --> 00:24:30,745
end with just showing how the
matrix system gets formed.

408
00:24:34,020 --> 00:24:34,520
Yep?

409
00:24:34,520 --> 00:24:35,395
AUDIENCE: [INAUDIBLE]

410
00:24:38,010 --> 00:24:39,010
PROFESSOR: That's right.

411
00:24:39,010 --> 00:24:41,320
The Robin condition is a
combination of the two.

412
00:24:41,320 --> 00:24:45,364
So there's a Dirichlet component
and a Neumann component.

413
00:24:45,364 --> 00:24:46,780
I don't know if
you actually did--

414
00:24:46,780 --> 00:24:48,270
were they discussed
in the finite element?

415
00:24:48,270 --> 00:24:49,210
They were mostly
discussed in, I think,

416
00:24:49,210 --> 00:24:51,260
the finite volume, finite
difference section.

417
00:24:51,260 --> 00:24:53,679
But yeah, it's
just a combination.

418
00:24:53,679 --> 00:24:55,470
We've put something in
the right-hand side.

419
00:24:55,470 --> 00:24:56,206
What's that?

420
00:24:56,206 --> 00:24:56,940
AUDIENCE: We had
them in the project.

421
00:24:56,940 --> 00:24:57,920
PROFESSOR: Oh, you had
them in the project.

422
00:24:57,920 --> 00:25:00,086
So then you know how
to do them, right?

423
00:25:00,086 --> 00:25:01,580
Or you know how not to do them.

424
00:25:01,580 --> 00:25:03,062
[LAUGHTER]

425
00:25:05,038 --> 00:25:07,020
Yeah.

426
00:25:07,020 --> 00:25:13,730
OK, so just as the last
bringing it all together,

427
00:25:13,730 --> 00:25:19,860
let's think about finite
element for 1D diffusion,

428
00:25:19,860 --> 00:25:22,669
and we need to make sure
that you are clear that you

429
00:25:22,669 --> 00:25:23,960
can go through all these steps.

430
00:25:23,960 --> 00:25:29,880
So starting off with the
governing equation, K dT / dx,

431
00:25:29,880 --> 00:25:31,643
differentiated
again with respect

432
00:25:31,643 --> 00:25:37,925
to x, equals minus q on the
domain minus L over 2 to L

433
00:25:37,925 --> 00:25:39,665
over 2 [INAUDIBLE] x.

434
00:25:42,647 --> 00:25:48,330
So let's see.

435
00:25:48,330 --> 00:25:50,340
We would write the
approximate solution,

436
00:25:50,340 --> 00:25:59,430
T of x, as being the
sum from i equals 1 to n

437
00:25:59,430 --> 00:26:06,859
of sum ai times Ci of x.

438
00:26:06,859 --> 00:26:09,420
These here are our finite
element basis functions.

439
00:26:09,420 --> 00:26:13,600
We might, for example, choose
to use the linear nodal basis,

440
00:26:13,600 --> 00:26:15,100
those hat functions.

441
00:26:18,370 --> 00:26:20,930
So again, those ai, the
unknown coefficients

442
00:26:20,930 --> 00:26:23,010
that we're going
to solve for, are

443
00:26:23,010 --> 00:26:25,210
going to end up
corresponding to the value

444
00:26:25,210 --> 00:26:28,530
of our approximate solution at
the nodes in the finite element

445
00:26:28,530 --> 00:26:29,630
mesh.

446
00:26:29,630 --> 00:26:33,229
We want to write down, then,
the j-th weighted residual.

447
00:26:39,145 --> 00:26:46,400
We'll call it [? Rj. ?]
And what is it?

448
00:26:46,400 --> 00:26:51,220
It's take everything over on
the left-hand side, substitute

449
00:26:51,220 --> 00:26:53,900
in the approximate solution.

450
00:26:53,900 --> 00:26:57,810
So we're going to get a K d T
tilde / dx, all differentiated,

451
00:26:57,810 --> 00:26:59,620
plus q.

452
00:26:59,620 --> 00:27:02,880
Weight it with the
j-th spaces function.

453
00:27:02,880 --> 00:27:05,630
And because it's Galerkin,
we're going to use the same C.

454
00:27:05,630 --> 00:27:10,190
So we weight it with Cj, and
then integrate over the domain,

455
00:27:10,190 --> 00:27:13,040
integrate from minus
L over 2 to L over 2.

456
00:27:13,040 --> 00:27:14,690
I could write out
all the steps, but I

457
00:27:14,690 --> 00:27:16,940
think you've seen it
many times in the notes.

458
00:27:16,940 --> 00:27:19,148
So we get that, and then
what's the next thing we do?

459
00:27:24,620 --> 00:27:25,250
What do we do?

460
00:27:27,836 --> 00:27:30,490
What's the first thing-- what
are we going to end up with?

461
00:27:30,490 --> 00:27:34,250
We're going to end
up with [? Cxx. ?]

462
00:27:37,020 --> 00:27:38,820
We can assume K is
constant, if you want.

463
00:27:38,820 --> 00:27:41,830
We're going to end up with
[? Cxx's ?] multiplied by C,

464
00:27:41,830 --> 00:27:44,080
so we're going to end up
with [INAUDIBLE] the integral

465
00:27:44,080 --> 00:27:46,690
of [? C ?] times the second
derivative of C with respect

466
00:27:46,690 --> 00:27:47,310
to x.

467
00:27:47,310 --> 00:27:48,750
What do we do?

468
00:27:48,750 --> 00:27:50,300
Integrate by parts.

469
00:27:50,300 --> 00:27:52,660
[? Usually you ?] always
[INAUDIBLE] integrate by parts.

470
00:27:52,660 --> 00:27:55,380
That's going to move one
differential off the C,

471
00:27:55,380 --> 00:27:58,590
onto the other C. Yeah.

472
00:27:58,590 --> 00:28:04,430
So get the approximate
solution, substitute it in,

473
00:28:04,430 --> 00:28:06,810
bring everything to the
left-hand side, multiply by Cj,

474
00:28:06,810 --> 00:28:07,950
and integrate.

475
00:28:07,950 --> 00:28:10,510
Integrate by parts, and when
you integrate by parts--

476
00:28:10,510 --> 00:28:13,175
I can write it out, if
you feel uncomfortable,

477
00:28:13,175 --> 00:28:15,050
but again, I think you've
seen it many times,

478
00:28:15,050 --> 00:28:15,966
and it's in the notes.

479
00:28:15,966 --> 00:28:19,120
I'm going to keep the T tilde,
rather than writing the sum.

480
00:28:19,120 --> 00:28:21,654
This thing is really the
sum of the [? ai's ?] times

481
00:28:21,654 --> 00:28:22,570
the [? Ci's, ?] right?

482
00:28:22,570 --> 00:28:24,320
But I'm just going to
write it as T tilde,

483
00:28:24,320 --> 00:28:26,560
because I think it's
a little bit clearer.

484
00:28:26,560 --> 00:28:28,780
You can [INAUDIBLE] if you're
working through things,

485
00:28:28,780 --> 00:28:31,600
feel free to leave
the T tilde in there

486
00:28:31,600 --> 00:28:33,240
until you get to
the bottom, so you

487
00:28:33,240 --> 00:28:34,448
don't have to write too many.

488
00:28:34,448 --> 00:28:36,789
So there's the-- we're
integrating v to u,

489
00:28:36,789 --> 00:28:38,580
so there's the [? uv ?]
term that comes out

490
00:28:38,580 --> 00:28:41,420
on the boundaries, out of
the integration by parts,

491
00:28:41,420 --> 00:28:47,110
minus now the integral
where we have moved--

492
00:28:47,110 --> 00:28:49,060
we're going to
differentiate Cj, and this

493
00:28:49,060 --> 00:28:51,200
is the Cj that's coming
from the weighting

494
00:28:51,200 --> 00:28:53,630
for the j-th weighted residual.

495
00:28:53,630 --> 00:28:56,510
And then our other one has
been integrated, so now we just

496
00:28:56,510 --> 00:29:00,712
have a K d T tilde / dx.

497
00:29:00,712 --> 00:29:03,020
dx.

498
00:29:03,020 --> 00:29:05,860
And then we have
the q term, which

499
00:29:05,860 --> 00:29:10,500
looks like an integral from
minus L over 2 to L over 2.

500
00:29:10,500 --> 00:29:13,190
And again, it's
the Cj times q dx.

501
00:29:13,190 --> 00:29:15,090
We don't have to do any
integration by parts.

502
00:29:15,090 --> 00:29:18,940
This side just
stays the way it is.

503
00:29:18,940 --> 00:29:22,360
So I jumped one step there,
but again, the [? first line ?]

504
00:29:22,360 --> 00:29:26,510
of the residual would have a Cj
times this full term with the T

505
00:29:26,510 --> 00:29:27,790
tilde there.

506
00:29:27,790 --> 00:29:31,180
One step of integration by
parts gives us the boundary term

507
00:29:31,180 --> 00:29:33,270
minus the integral
with one derivative

508
00:29:33,270 --> 00:29:35,250
on the Cj, the weighting
Cj, and one derivative

509
00:29:35,250 --> 00:29:37,630
on the approximate solution.

510
00:29:37,630 --> 00:29:40,170
Yes?

511
00:29:40,170 --> 00:29:42,450
No?

512
00:29:42,450 --> 00:29:43,710
Yes?

513
00:29:43,710 --> 00:29:45,710
OK.

514
00:29:45,710 --> 00:29:49,780
So where do all the bits go?

515
00:29:49,780 --> 00:29:52,850
Here is if we had a
Neumann condition.

516
00:29:52,850 --> 00:29:54,990
This is where this
contribution would go.

517
00:29:54,990 --> 00:29:55,490
Yeah?

518
00:29:59,860 --> 00:30:06,480
This term here is going to
give us the stiffness matrix,

519
00:30:06,480 --> 00:30:09,386
and this term here is going to
give us the right-hand side.

520
00:30:09,386 --> 00:30:09,886
Yep.

521
00:30:15,190 --> 00:30:22,020
Let's just remember how all that
goes together, conceptually.

522
00:30:25,961 --> 00:30:28,464
This guy here is going to
go to our stiffness matrix.

523
00:30:33,870 --> 00:30:36,220
This guy here is going to
go the right-hand side.

524
00:30:40,450 --> 00:30:52,390
And if we think about just
integral xj minus 1 to xj

525
00:30:52,390 --> 00:31:01,256
plus 1 of Cj x K d T tilde / dx.

526
00:31:04,672 --> 00:31:08,088
So why did I change
the limits here?

527
00:31:08,088 --> 00:31:09,552
AUDIENCE: [INAUDIBLE]

528
00:31:11,710 --> 00:31:13,008
PROFESSOR: What is it?

529
00:31:13,008 --> 00:31:13,960
Mumble a bit louder.

530
00:31:13,960 --> 00:31:14,460
Yeah?

531
00:31:14,460 --> 00:31:15,756
AUDIENCE: [INAUDIBLE]

532
00:31:15,997 --> 00:31:17,580
PROFESSOR: Yeah,
because we're looking

533
00:31:17,580 --> 00:31:18,470
at the j-th weighted residual.

534
00:31:18,470 --> 00:31:20,470
We've got the Cj x
sitting out in front here,

535
00:31:20,470 --> 00:31:24,020
and we know-- again,
if you sketch the Cj x,

536
00:31:24,020 --> 00:31:27,200
you know that it's only
non-zero over elements

537
00:31:27,200 --> 00:31:30,570
that go from xj minus 1
to xj and xj to xj plus 1.

538
00:31:30,570 --> 00:31:34,100
So all the other parts
of the integral go to 0.

539
00:31:34,100 --> 00:31:36,230
So that's the only
bit that's left.

540
00:31:36,230 --> 00:31:40,840
And again, I'll skip
over a few steps,

541
00:31:40,840 --> 00:31:42,640
but it's all derived
in the notes.

542
00:31:42,640 --> 00:31:47,870
If you then substitute in the
T tilde [? has been ?] the sum

543
00:31:47,870 --> 00:31:51,070
of the [? ai ?] times
the [? Ci's, ?] again,

544
00:31:51,070 --> 00:31:52,960
a lot of the [? Ci's ?]
are going to be 0.

545
00:31:52,960 --> 00:31:56,820
The only ones that are going
to be non-zero on this part

546
00:31:56,820 --> 00:32:04,330
are going to be this one,
so Ci equal to Cj, and then

547
00:32:04,330 --> 00:32:05,220
this one, right?

548
00:32:05,220 --> 00:32:08,320
Ci equal to Cj minus 1.

549
00:32:08,320 --> 00:32:10,180
So itself and its
left neighbor are

550
00:32:10,180 --> 00:32:14,400
the only ones that are going
to be non-zero over this.

551
00:32:14,400 --> 00:32:17,730
Because in the next one--
oh, and this guy here.

552
00:32:17,730 --> 00:32:18,580
And this guy here.

553
00:32:18,580 --> 00:32:19,746
That's right, there you are.

554
00:32:19,746 --> 00:32:20,529
Three of them.

555
00:32:20,529 --> 00:32:22,570
Itself, its left neighbor,
and its right neighbor

556
00:32:22,570 --> 00:32:25,525
are all going to contribute
to the way I've written it,

557
00:32:25,525 --> 00:32:27,470
this full integral.

558
00:32:27,470 --> 00:32:29,390
And so what does it
end up looking like?

559
00:32:29,390 --> 00:32:31,098
It ends up looking
something like there's

560
00:32:31,098 --> 00:32:39,550
an a j minus an a j minus 1
over delta x j minus 1 squared,

561
00:32:39,550 --> 00:32:44,670
and here are I've left in
the K. If K were a constant,

562
00:32:44,670 --> 00:32:46,434
it would just be coming out.

563
00:32:46,434 --> 00:32:47,850
And then there's
another term that

564
00:32:47,850 --> 00:32:53,225
looks like a j plus 1
minus a j over delta x j

565
00:32:53,225 --> 00:32:59,760
squared integral
x j to xj plus 1.

566
00:32:59,760 --> 00:33:04,220
OK, so I mean you could
work through all of these,

567
00:33:04,220 --> 00:33:05,980
but what's the
important concept?

568
00:33:05,980 --> 00:33:09,000
The important concept is that--
let me draw this a little bit

569
00:33:09,000 --> 00:33:11,530
more clearly.

570
00:33:11,530 --> 00:33:16,294
The important concept is that
when you're an-- x j-- always

571
00:33:16,294 --> 00:33:17,960
draw the picture for
yourself, because I

572
00:33:17,960 --> 00:33:20,220
think it really helps.

573
00:33:20,220 --> 00:33:22,270
Don't get too embroiled
in trying to integrate

574
00:33:22,270 --> 00:33:23,547
things and plug things in.

575
00:33:23,547 --> 00:33:24,630
It's easy to make mistake.

576
00:33:24,630 --> 00:33:27,083
Just think about where
things are going to be going.

577
00:33:27,083 --> 00:33:29,730
So there's Cj.

578
00:33:33,950 --> 00:33:35,515
There Cj plus 1.

579
00:33:40,320 --> 00:33:44,891
And there's Cj minus 1.

580
00:33:47,540 --> 00:33:50,820
So when we think
about this element,

581
00:33:50,820 --> 00:33:52,700
the j minus 1
element, which goes

582
00:33:52,700 --> 00:33:56,640
from x j minus 1 to x j,
which two get activated?

583
00:33:56,640 --> 00:33:58,426
It's j minus 1 and j.

584
00:33:58,426 --> 00:34:01,469
So we expect the coefficients
aj and aj minus 1

585
00:34:01,469 --> 00:34:03,510
to be the ones that are
going to get coefficients

586
00:34:03,510 --> 00:34:05,676
put in front of them in the
stiffness matrix, right?

587
00:34:05,676 --> 00:34:06,510
Nothing else.

588
00:34:06,510 --> 00:34:10,671
And when we think about this
element, xj and xj-- from xj

589
00:34:10,671 --> 00:34:12,650
to xj plus 1, this
is the j-th element,

590
00:34:12,650 --> 00:34:15,480
it's Cj and Cj plus 1 that
are going to get activated.

591
00:34:15,480 --> 00:34:19,780
So we expect to see
coefficients aj and aj plus 1,

592
00:34:19,780 --> 00:34:23,150
with something
[? multiplying ?] them.

593
00:34:23,150 --> 00:34:29,010
And so before we went through
we worked out all the integrals,

594
00:34:29,010 --> 00:34:33,310
again, you could do it by just
plugging and substituting,

595
00:34:33,310 --> 00:34:36,000
but you could also just
think, conceptually,

596
00:34:36,000 --> 00:34:39,196
what is the system
going to look like?

597
00:34:39,196 --> 00:34:40,654
At the end of the
day, you're going

598
00:34:40,654 --> 00:34:45,949
to end up with a big matrix, the
stiffness matrix, multiplying

599
00:34:45,949 --> 00:34:48,960
these unknown coefficients,
which correspond

600
00:34:48,960 --> 00:34:53,350
to the approximate solution at
the nodes in the finite element

601
00:34:53,350 --> 00:34:56,510
[? match, ?] and then
the right-hand side.

602
00:35:01,280 --> 00:35:07,840
And again, remember
that the j-th row

603
00:35:07,840 --> 00:35:13,780
corresponds to setting the j-th
weighted residual equal to 0.

604
00:35:13,780 --> 00:35:16,590
So each row in the matrix
comes from setting a weighted

605
00:35:16,590 --> 00:35:21,160
residual equal to 0,
and then each column--

606
00:35:21,160 --> 00:35:32,710
so i-th column multiplies
[? an ?] [? ai. ?] So how does

607
00:35:32,710 --> 00:35:34,910
this system start looking?

608
00:35:34,910 --> 00:35:37,249
Well, we know if this
were the first element,

609
00:35:37,249 --> 00:35:38,290
we would be getting what?

610
00:35:41,860 --> 00:35:44,350
[? c1 ?] and a [? c2, ?] so
we might get entries here

611
00:35:44,350 --> 00:35:45,290
and here.

612
00:35:45,290 --> 00:35:47,150
Although, if we had a
Dirichlet condition,

613
00:35:47,150 --> 00:35:54,460
we'd see 0 [AUDIO OUT] entries.

614
00:35:59,579 --> 00:36:00,620
First, second, and third.

615
00:36:03,540 --> 00:36:06,890
The matrix starts
to look like this.

616
00:36:06,890 --> 00:36:09,880
And again, it's because when we
do the j-th weighted residual,

617
00:36:09,880 --> 00:36:12,340
which is here, we're going
to trigger contributions to j

618
00:36:12,340 --> 00:36:16,470
minus 1, to j, and to j plus 1.

619
00:36:16,470 --> 00:36:18,750
And that's because the
j-th weighting function,

620
00:36:18,750 --> 00:36:20,870
again, triggers
two elements, the j

621
00:36:20,870 --> 00:36:25,440
minus 1 element and
the j-th element.

622
00:36:25,440 --> 00:36:28,920
OK, so I think
[INAUDIBLE] two sets

623
00:36:28,920 --> 00:36:30,050
of skills you have to have.

624
00:36:30,050 --> 00:36:31,190
One is that you have
to be comfortable

625
00:36:31,190 --> 00:36:33,565
doing the integration, writing
down the weighted residual

626
00:36:33,565 --> 00:36:36,080
and doing integration by parts.

627
00:36:36,080 --> 00:36:37,603
And then the other
is not to lose

628
00:36:37,603 --> 00:36:40,980
sight of the big picture of
forming this matrix system.

629
00:36:40,980 --> 00:36:43,309
Columns correspond
to the unknowns,

630
00:36:43,309 --> 00:36:45,100
rows correspond to j-th
weighted residuals,

631
00:36:45,100 --> 00:36:47,391
and it has a very specific
structure because of the way

632
00:36:47,391 --> 00:36:48,773
we choose the nodal basis.

633
00:36:54,395 --> 00:36:54,895
Questions?

634
00:36:58,652 --> 00:37:00,360
You guys feel like
you have a good handle

635
00:37:00,360 --> 00:37:01,380
on the finite element?

636
00:37:08,310 --> 00:37:10,990
Is that a no, or
a yes, or you will

637
00:37:10,990 --> 00:37:13,746
have by Monday and Tuesday?

638
00:37:13,746 --> 00:37:19,810
So the last thing I
want to do is go back,

639
00:37:19,810 --> 00:37:26,170
and I want to remind
you that I've given you

640
00:37:26,170 --> 00:37:30,690
a very specific list of
measurable outcomes, which

641
00:37:30,690 --> 00:37:33,630
represent the things that
I want you to take away

642
00:37:33,630 --> 00:37:34,542
from the class.

643
00:37:34,542 --> 00:37:36,000
And so when I
construct assessments

644
00:37:36,000 --> 00:37:37,791
like the final exam,
that's what I'm doing,

645
00:37:37,791 --> 00:37:39,750
is I'm trying to
measure those outcomes.

646
00:37:39,750 --> 00:37:44,030
So if-- when you are
studying for the final exam,

647
00:37:44,030 --> 00:37:47,370
going back to those measurable
outcomes and going through them

648
00:37:47,370 --> 00:37:52,400
I think is, hopefully, a
really helpful guide for you.

649
00:37:52,400 --> 00:37:58,840
Because it really, again--
let me get plugged in here--

650
00:37:58,840 --> 00:38:01,270
really, again, tells
you exactly what

651
00:38:01,270 --> 00:38:04,150
it is that I want
you to be able to do.

652
00:38:04,150 --> 00:38:06,370
So let's go through them.

653
00:38:06,370 --> 00:38:08,910
And I just-- so
I've pulled them.

654
00:38:08,910 --> 00:38:13,270
And I'll go back to the
[INAUDIBLE] site, or the MITx

655
00:38:13,270 --> 00:38:14,465
site where they are.

656
00:38:14,465 --> 00:38:18,740
But starting at Measurable
Outcome 212, this one is,

657
00:38:18,740 --> 00:38:23,434
"Describe"-- and I also want
to point out that the verbs are

658
00:38:23,434 --> 00:38:28,229
also important-- "Describe
how the method of weighted

659
00:38:28,229 --> 00:38:30,270
residuals can be used to
calculate an approximate

660
00:38:30,270 --> 00:38:31,015
solution to a PDE."

661
00:38:31,015 --> 00:38:32,470
Hopefully, you feel
comfortable with that.

662
00:38:32,470 --> 00:38:34,110
"Describe the differences
between method

663
00:38:34,110 --> 00:38:36,026
of weighted residuals
and collocation method."

664
00:38:36,026 --> 00:38:38,810
We didn't actually talk
about least squares, so just

665
00:38:38,810 --> 00:38:40,150
the first two.

666
00:38:40,150 --> 00:38:42,560
"Describe the Galerkin method
of weighted residuals."

667
00:38:42,560 --> 00:38:44,930
And we just went
over all of that.

668
00:38:44,930 --> 00:38:47,490
Then under finite element,
describe the choice

669
00:38:47,490 --> 00:38:50,290
of approximate solutions,
which are sometimes

670
00:38:50,290 --> 00:38:51,420
called the test functions.

671
00:38:51,420 --> 00:38:53,544
So that's the choice of
the [? Ci's ?] that you use

672
00:38:53,544 --> 00:38:55,220
to approximate the solution.

673
00:38:55,220 --> 00:38:57,900
Give examples of a
basis, in particular,

674
00:38:57,900 --> 00:38:58,980
including a nodal basis.

675
00:38:58,980 --> 00:39:00,200
And I've crossed
out the quadratic,

676
00:39:00,200 --> 00:39:02,325
because I don't think
[? Vikrum ?] actually covered

677
00:39:02,325 --> 00:39:03,380
it in that lecture.

678
00:39:03,380 --> 00:39:05,680
But if you're comfortable
with the linear nodal basis,

679
00:39:05,680 --> 00:39:07,196
that will be good.

680
00:39:07,196 --> 00:39:08,820
To describe how
integrals are performed

681
00:39:08,820 --> 00:39:10,310
using a reference element.

682
00:39:10,310 --> 00:39:13,140
Explain how Guassian
quadrature rules are derived,

683
00:39:13,140 --> 00:39:16,010
and also how they're used, in
terms of doing integrations

684
00:39:16,010 --> 00:39:18,530
in the reference element
for the finite element.

685
00:39:18,530 --> 00:39:20,730
We went through all of these.

686
00:39:20,730 --> 00:39:23,020
Explain how Dirichlet and
Neumann boundary conditions

687
00:39:23,020 --> 00:39:25,770
are implemented for
the diffusion equation,

688
00:39:25,770 --> 00:39:27,784
or Laplace's equation.

689
00:39:27,784 --> 00:39:33,230
So Robin not actually necessary
for the finite element.

690
00:39:33,230 --> 00:39:35,930
And then describe how that all
goes together and gives you

691
00:39:35,930 --> 00:39:38,470
this system of
discrete equations.

692
00:39:38,470 --> 00:39:40,810
Describe the meaning of the
entries, rows and columns.

693
00:39:40,810 --> 00:39:42,500
So again, this is
what I was meaning

694
00:39:42,500 --> 00:39:44,570
by, this is kind of
the big picture, right?

695
00:39:44,570 --> 00:39:45,710
Now, to do the
integrals and come up

696
00:39:45,710 --> 00:39:47,160
with numbers is one
thing, but explaining

697
00:39:47,160 --> 00:39:49,284
what the system means, and
what all the pieces are,

698
00:39:49,284 --> 00:39:52,340
and how they go
together, is important.

699
00:39:52,340 --> 00:39:56,450
So I think that's essentially
what we just covered.

700
00:39:56,450 --> 00:40:02,080
And I also want to remind you
that if you go to the MITx web

701
00:40:02,080 --> 00:40:04,700
page, so there's the
Courseware where the notes are,

702
00:40:04,700 --> 00:40:06,574
but remember, there's
this Measurable Outcome

703
00:40:06,574 --> 00:40:10,540
Index that has all the
measurable outcomes.

704
00:40:10,540 --> 00:40:13,539
And they're also
linked to the notes.

705
00:40:13,539 --> 00:40:15,330
And I know one piece
of feedback I've heard

706
00:40:15,330 --> 00:40:17,860
is that it would be really
helpful to be able to search

707
00:40:17,860 --> 00:40:19,115
on this web page.

708
00:40:19,115 --> 00:40:20,990
I know you can't search,
but this is actually

709
00:40:20,990 --> 00:40:24,420
one way to navigate, that
we could come in here

710
00:40:24,420 --> 00:40:26,840
and click on one of these.

711
00:40:26,840 --> 00:40:29,480
And I think our tagging
is pretty thorough,

712
00:40:29,480 --> 00:40:30,990
but as you click
on that, here are

713
00:40:30,990 --> 00:40:33,882
all the places in the notes
where-- in particular one,

714
00:40:33,882 --> 00:40:36,090
describe how integrals are
performed in the reference

715
00:40:36,090 --> 00:40:39,740
element-- these are the sections
in the notes that relate

716
00:40:39,740 --> 00:40:41,770
to this measurable
outcome, and then these

717
00:40:41,770 --> 00:40:44,350
are the sections in the notes
that have embedded questions

718
00:40:44,350 --> 00:40:45,210
in them.

719
00:40:45,210 --> 00:40:47,360
So again, if you're
looking for a study guide,

720
00:40:47,360 --> 00:40:49,230
going through these
measurable outcomes,

721
00:40:49,230 --> 00:40:50,896
and going back and
looking at the notes,

722
00:40:50,896 --> 00:40:57,810
and making sure that you can
execute whatever it says here.

723
00:40:57,810 --> 00:40:58,310
Yep.

724
00:41:01,370 --> 00:41:05,350
OK, any questions
about finite element?

725
00:41:11,000 --> 00:41:13,012
You guys look worried or tired.

726
00:41:15,930 --> 00:41:17,156
OK.

727
00:41:17,156 --> 00:41:19,280
All right, so I'm going to
do the same thing, then,

728
00:41:19,280 --> 00:41:20,840
for probabilistic
analysis and optimization.

729
00:41:20,840 --> 00:41:22,298
I'll go through a
bit more quickly,

730
00:41:22,298 --> 00:41:25,920
just because hopefully that
stuff is a little bit fresher.

731
00:41:25,920 --> 00:41:29,955
But again, I just want to
touch on the main topics.

732
00:41:29,955 --> 00:41:31,580
We'll hit the high
points on the board,

733
00:41:31,580 --> 00:41:34,038
and then we'll just take a look
at the measurable outcomes.

734
00:41:34,038 --> 00:41:35,690
And particularly,
I just want to make

735
00:41:35,690 --> 00:41:37,680
sure you're clear on a
lot the last bits, which

736
00:41:37,680 --> 00:41:39,780
are a little bit introductory,
what it is that I expect

737
00:41:39,780 --> 00:41:40,655
you to be able to do.

738
00:41:47,984 --> 00:41:50,400
Does it feel like you've learned
a lot since spring break?

739
00:41:50,400 --> 00:41:52,120
Remember on spring break,
wherever you guys were,

740
00:41:52,120 --> 00:41:54,560
in Florida or whatever, you
didn't know finite element.

741
00:41:54,560 --> 00:41:56,537
You didn't know Monte Carlo.

742
00:41:56,537 --> 00:41:58,620
You maybe didn't know
anything about optimization.

743
00:41:58,620 --> 00:42:00,560
It seems like a
long time ago, no?

744
00:42:03,560 --> 00:42:06,648
AUDIENCE: It's a long
time, but a short time.

745
00:42:06,648 --> 00:42:07,606
PROFESSOR: It's a what?

746
00:42:07,606 --> 00:42:08,689
AUDIENCE: Long short time.

747
00:42:08,689 --> 00:42:10,292
PROFESSOR: A long short time?

748
00:42:10,292 --> 00:42:11,284
AUDIENCE: [INAUDIBLE]

749
00:42:14,377 --> 00:42:15,252
PROFESSOR: All right.

750
00:42:22,196 --> 00:42:25,417
So over there are
the six main topics

751
00:42:25,417 --> 00:42:27,250
that we've covered in
probabilistic analysis

752
00:42:27,250 --> 00:42:30,920
and optimization, the
basics of Monte Carlo,

753
00:42:30,920 --> 00:42:32,390
the Monte Carlo estimators.

754
00:42:32,390 --> 00:42:34,160
We talked about various
reduction methods.

755
00:42:34,160 --> 00:42:36,040
We talked about design
of experiment methods

756
00:42:36,040 --> 00:42:37,490
for sampling.

757
00:42:37,490 --> 00:42:40,270
We did a very basic intro
to design optimization,

758
00:42:40,270 --> 00:42:42,680
and then we talked about
responsiveness models

759
00:42:42,680 --> 00:42:44,038
in the last lecture.

760
00:42:44,038 --> 00:42:50,750
So let's just go through
and-- where are my notes?

761
00:42:56,260 --> 00:42:59,355
So this is probabilistic
analysis and optimization,

762
00:42:59,355 --> 00:43:03,036
so I'm going to start
numbering from 1 again.

763
00:43:03,036 --> 00:43:07,710
So again, Monte Carlo
simulation, I hope by now,

764
00:43:07,710 --> 00:43:13,290
after the project, you
feel pretty comfortable

765
00:43:13,290 --> 00:43:15,210
with the idea.

766
00:43:15,210 --> 00:43:17,920
I think the little block
diagram says a lot.

767
00:43:17,920 --> 00:43:20,890
If we have a model,
we have inputs.

768
00:43:20,890 --> 00:43:25,180
Maybe we have x1, x2, and
x3, and maybe the model

769
00:43:25,180 --> 00:43:28,400
produces one or more outputs.

770
00:43:28,400 --> 00:43:32,400
So inputs, in this case,
maybe just a single output

771
00:43:32,400 --> 00:43:34,150
is always considered.

772
00:43:34,150 --> 00:43:39,870
That we want to represent our
inputs as random variables,

773
00:43:39,870 --> 00:43:44,360
and so if we propagate random
variables through the model,

774
00:43:44,360 --> 00:43:46,935
then our output will also
be a random variable.

775
00:43:52,220 --> 00:43:56,470
And the idea of Monte
Carlo simulation

776
00:43:56,470 --> 00:43:58,595
is really quite simple.

777
00:43:58,595 --> 00:44:02,300
Remember, we broke it
into the three steps.

778
00:44:02,300 --> 00:44:08,176
The first step is to
define the input PDEs.

779
00:44:08,176 --> 00:44:09,880
So once you make the
decision that you

780
00:44:09,880 --> 00:44:11,840
want to-- not PDEs, PDFs.

781
00:44:16,250 --> 00:44:17,760
Input PDFs.

782
00:44:17,760 --> 00:44:21,540
Once you decide you want
to model these things

783
00:44:21,540 --> 00:44:23,839
as random variables, you
have to come up with some way

784
00:44:23,839 --> 00:44:25,380
where you can
probabilistic analysis,

785
00:44:25,380 --> 00:44:28,330
some way of describing
what kind of distribution

786
00:44:28,330 --> 00:44:29,720
these guys follow.

787
00:44:29,720 --> 00:44:31,870
And for pretty much
everything that we did,

788
00:44:31,870 --> 00:44:33,030
those were given to you.

789
00:44:33,030 --> 00:44:34,780
Remember I said that,
in practice, this is

790
00:44:34,780 --> 00:44:36,520
a really difficult thing to do.

791
00:44:36,520 --> 00:44:38,460
You can use data that
you might have collected

792
00:44:38,460 --> 00:44:40,750
from manufacturing, or
you might query experts,

793
00:44:40,750 --> 00:44:44,780
but you somehow have to come
up and define the input PDFs.

794
00:44:44,780 --> 00:44:48,700
Then, once you have
those PDFs defined,

795
00:44:48,700 --> 00:44:50,125
you sample your inputs randomly.

796
00:44:59,410 --> 00:45:03,280
Each random [? draw ?] you solve
by running through the model,

797
00:45:03,280 --> 00:45:08,250
and so what this means is that
if we make capital N samples,

798
00:45:08,250 --> 00:45:09,890
then we have to do n solves.

799
00:45:09,890 --> 00:45:13,650
So each solve is going to
be one run of our model,

800
00:45:13,650 --> 00:45:15,540
which might be a
finite element model

801
00:45:15,540 --> 00:45:20,590
or whatever kind of
stimulation it would be.

802
00:45:20,590 --> 00:45:24,790
So now we've taken N samples,
we've run the model n times,

803
00:45:24,790 --> 00:45:29,020
we have n samples of the
output, and the third step

804
00:45:29,020 --> 00:45:32,526
is to analyze those
resulting samples.

805
00:45:32,526 --> 00:45:40,303
So analyze the resulting
samples of the output,

806
00:45:40,303 --> 00:45:43,270
or outputs, if
there's more than one.

807
00:45:43,270 --> 00:45:45,277
And in particular,
we might be-- well,

808
00:45:45,277 --> 00:45:47,547
we're usually
interested in estimating

809
00:45:47,547 --> 00:45:48,505
statistics of interest.

810
00:45:52,650 --> 00:45:55,070
Statistics of interest.

811
00:45:55,070 --> 00:45:59,620
And those statistics might
include means, variances,

812
00:45:59,620 --> 00:46:08,510
probabilities of
failure, or whatever it

813
00:46:08,510 --> 00:46:11,410
is that we want to calculate.

814
00:46:15,210 --> 00:46:20,270
And if these inputs are
uniform random variables,

815
00:46:20,270 --> 00:46:23,080
then drawing a sample from
them is pretty straightforward.

816
00:46:23,080 --> 00:46:26,445
You can draw them just randomly
[? under ?] [? 01. ?] If these

817
00:46:26,445 --> 00:46:29,100
things are not uniform random
variables, then, remember,

818
00:46:29,100 --> 00:46:32,370
we talked about the inversion
method, where you use the CDF,

819
00:46:32,370 --> 00:46:34,640
you generate a uniform
random variable,

820
00:46:34,640 --> 00:46:37,430
which is like picking where
you're going to be on the CDF,

821
00:46:37,430 --> 00:46:38,980
go along, invert
through the CDF,

822
00:46:38,980 --> 00:46:40,760
and that gives
you a sample of x.

823
00:46:40,760 --> 00:46:43,440
And in doing that, by sampling
uniformly on this axis,

824
00:46:43,440 --> 00:46:45,410
remember, we graphically
saw that that

825
00:46:45,410 --> 00:46:48,032
puts the right [? entity ?]
of points on the x-axis.

826
00:46:48,032 --> 00:46:50,640
And again, you implemented
that in the project

827
00:46:50,640 --> 00:46:53,400
with the triangular
distributions.

828
00:46:53,400 --> 00:46:54,380
What?

829
00:46:54,380 --> 00:46:55,850
AUDIENCE: [INAUDIBLE]

830
00:47:02,240 --> 00:47:03,532
PROFESSOR: Nothing awful.

831
00:47:03,532 --> 00:47:04,860
Yeah.

832
00:47:04,860 --> 00:47:07,110
I think the triangular
is the only one that's

833
00:47:07,110 --> 00:47:10,380
really analytically tractable.

834
00:47:10,380 --> 00:47:11,730
AUDIENCE: [INAUDIBLE]

835
00:47:14,900 --> 00:47:15,810
PROFESSOR: Yeah.

836
00:47:15,810 --> 00:47:18,773
I'm not going to have you do
Monte Carlo simulation by hand.

837
00:47:18,773 --> 00:47:19,460
[LAUGHTER]

838
00:47:19,460 --> 00:47:22,775
Although it could
be interesting.

839
00:47:22,775 --> 00:47:24,900
We could correlate your
grade with how many samples

840
00:47:24,900 --> 00:47:28,040
you could execute in a
fixed amount of time.

841
00:47:28,040 --> 00:47:31,880
See what the
[? process ?] of power is.

842
00:47:31,880 --> 00:47:32,970
But yeah, no.

843
00:47:32,970 --> 00:47:33,910
Triangular is fine.

844
00:47:33,910 --> 00:47:36,368
Triangular is fine, as long as
you understand it generally.

845
00:47:36,368 --> 00:47:38,660
AUDIENCE: Can we
build [INAUDIBLE]

846
00:47:39,454 --> 00:47:41,370
PROFESSOR: If you can
do that in the half-hour

847
00:47:41,370 --> 00:47:43,370
you have to prepare with
the pieces of the paper

848
00:47:43,370 --> 00:47:45,410
that you have with you.

849
00:47:45,410 --> 00:47:46,820
That would be impressive.

850
00:47:46,820 --> 00:47:52,190
OK, so I think the more
conceptually difficult part

851
00:47:52,190 --> 00:47:55,260
of Monte Carlo are
the estimators.

852
00:47:55,260 --> 00:47:58,020
And let me just-- I
know, I feel like I've

853
00:47:58,020 --> 00:48:00,380
been harping on about
these for a while,

854
00:48:00,380 --> 00:48:02,712
but let me just make sure
that they're really clear,

855
00:48:02,712 --> 00:48:05,136
because I think it
can get confusing.

856
00:48:05,136 --> 00:48:08,810
So we talked about three
main estimators, the mean,

857
00:48:08,810 --> 00:48:11,250
the variance, and
probability estimators.

858
00:48:11,250 --> 00:48:20,650
So if we want to estimate the
mean, and it's the mean of y,

859
00:48:20,650 --> 00:48:28,500
y being our output of interest--
so let's call it mu sub y.

860
00:48:28,500 --> 00:48:29,470
Then what do we do?

861
00:48:29,470 --> 00:48:33,930
We use the sample
mean as the estimator,

862
00:48:33,930 --> 00:48:37,340
and we denote the
sample mean as y bar.

863
00:48:37,340 --> 00:48:43,870
And it's equal to 1 over n, the
sum from i equals 1 n of y i,

864
00:48:43,870 --> 00:48:48,795
where this guy is the
i-th Monte Carlo sample.

865
00:48:52,060 --> 00:48:54,180
The y that we get when we
run the i-th Monte Carlo

866
00:48:54,180 --> 00:48:57,380
sample through our model.

867
00:48:57,380 --> 00:49:00,460
So we're trying to estimate mu
of y, which is the real mean.

868
00:49:00,460 --> 00:49:02,240
To do that, we use
the sample mean,

869
00:49:02,240 --> 00:49:05,720
y bar, which is just the
standard sample mean.

870
00:49:05,720 --> 00:49:08,522
And what we know is
that for n, as long

871
00:49:08,522 --> 00:49:12,970
as we take a large
enough n-- large, 30

872
00:49:12,970 --> 00:49:19,130
or 40-- that this estimator--
so let me back up a second.

873
00:49:19,130 --> 00:49:23,120
This estimator-- this
thing is the true mean.

874
00:49:23,120 --> 00:49:24,480
It's a number.

875
00:49:24,480 --> 00:49:26,644
The estimator is
a random variable.

876
00:49:26,644 --> 00:49:28,310
And it's a random
variable because we're

877
00:49:28,310 --> 00:49:30,870
sampling randomly, so
any time we do this,

878
00:49:30,870 --> 00:49:33,230
we could get a slightly
different answer.

879
00:49:33,230 --> 00:49:35,040
So this is a
deterministic quantity.

880
00:49:35,040 --> 00:49:36,762
This is a random variable.

881
00:49:36,762 --> 00:49:38,220
We need to be able
to say something

882
00:49:38,220 --> 00:49:40,053
about the properties
of this random variable

883
00:49:40,053 --> 00:49:42,220
to be able to say
how accurate it is.

884
00:49:42,220 --> 00:49:44,030
What we know is that
for large n, where

885
00:49:44,030 --> 00:49:46,680
large means of the
order of 30 or 40,

886
00:49:46,680 --> 00:49:51,770
this random variable, y bar,
will be normally distributed.

887
00:49:51,770 --> 00:49:56,100
And the mean of--
let's call it mu

888
00:49:56,100 --> 00:50:00,952
sub y bar, the mean of
x-- mean of mu sub y bar,

889
00:50:00,952 --> 00:50:04,640
and variant sigma y bar squared.

890
00:50:04,640 --> 00:50:09,960
OK, so this is the mean of y
bar and a variant of y bar.

891
00:50:09,960 --> 00:50:14,460
And what we can show
is that the mean of y

892
00:50:14,460 --> 00:50:18,050
bar-- the expected value
of our estimator, y bar,

893
00:50:18,050 --> 00:50:20,180
is in fact equal to mu of y.

894
00:50:20,180 --> 00:50:22,900
It's equal to the thing that
we're trying to estimate.

895
00:50:22,900 --> 00:50:25,660
And so this is what's called
an unbiased estimator,

896
00:50:25,660 --> 00:50:31,240
because on average, it
gives us the correct result.

897
00:50:31,240 --> 00:50:32,620
So that's great, on average.

898
00:50:32,620 --> 00:50:34,330
But now, the variance
of that estimator

899
00:50:34,330 --> 00:50:37,150
is important, because it
tells us how far off could we

900
00:50:37,150 --> 00:50:39,940
be if we just run it one time.

901
00:50:39,940 --> 00:50:44,970
And we could also derive-- that
variance of the mean estimator

902
00:50:44,970 --> 00:50:50,730
is given by the variance
of y itself divided

903
00:50:50,730 --> 00:50:54,940
by the number of
samples that we drew.

904
00:50:54,940 --> 00:50:56,960
And the square root of
this thing, sigma y bar,

905
00:50:56,960 --> 00:50:59,622
is sometimes called the
standard error of the estimator.

906
00:50:59,622 --> 00:51:01,580
I tend to not use that
term so much, because it

907
00:51:01,580 --> 00:51:02,860
confuses me a little bit.

908
00:51:02,860 --> 00:51:06,370
I just like to think of this as
the variance of the estimator.

909
00:51:06,370 --> 00:51:08,900
So I really recommend that
when you write these things,

910
00:51:08,900 --> 00:51:14,580
use the subscripts to denote
mean of y bar, mean of y,

911
00:51:14,580 --> 00:51:17,570
variance of y bar,
variance of y.

912
00:51:17,570 --> 00:51:19,920
Keep it straight.

913
00:51:19,920 --> 00:51:21,890
Yes, Kevin.

914
00:51:21,890 --> 00:51:23,315
AUDIENCE: [INAUDIBLE]

915
00:51:31,490 --> 00:51:33,690
PROFESSOR: This is
not an estimator.

916
00:51:33,690 --> 00:51:35,700
The estimator is y bar.

917
00:51:35,700 --> 00:51:38,890
This is the mean and the
variance of that estimator.

918
00:51:38,890 --> 00:51:41,920
We haven't yet talked about
the estimator for the variance,

919
00:51:41,920 --> 00:51:43,874
which could biased.

920
00:51:43,874 --> 00:51:45,290
Yeah, so this is
not an estimator.

921
00:51:45,290 --> 00:51:50,150
This is just the variance
of this estimator.

922
00:51:50,150 --> 00:51:50,650
OK?

923
00:51:50,650 --> 00:51:51,150
Yep.

924
00:51:54,660 --> 00:51:58,440
OK, so that's the
mean estimator.

925
00:51:58,440 --> 00:52:01,650
And the other ones
we talked about

926
00:52:01,650 --> 00:52:07,726
were variance and probability.

927
00:52:07,726 --> 00:52:14,710
So let's say we want to
estimate the variance of y.

928
00:52:14,710 --> 00:52:17,454
And again, y is the
output of our code.

929
00:52:17,454 --> 00:52:21,210
And so this is sigma y squared.

930
00:52:21,210 --> 00:52:23,410
Then, here we have a variety
of different options.

931
00:52:23,410 --> 00:52:25,474
I think Alex presented
three to you.

932
00:52:28,920 --> 00:52:34,960
Maybe the one of
choice is f y squared,

933
00:52:34,960 --> 00:52:41,350
which is given by 1 over n minus
1, the sum from i equal 1 to n

934
00:52:41,350 --> 00:52:45,915
of the [? yi's ?] minus
the y bar squared.

935
00:52:49,520 --> 00:52:54,510
OK, so again-- and I guess
the notation is a little bit

936
00:52:54,510 --> 00:52:59,100
unfortunate, because this
isn't what people use.

937
00:52:59,100 --> 00:53:04,230
The f y squared is the
estimator for the variance sigma

938
00:53:04,230 --> 00:53:05,720
y squared.

939
00:53:05,720 --> 00:53:07,650
And again, this is
a random variable,

940
00:53:07,650 --> 00:53:10,470
and we want to know what its
expectation and its variance

941
00:53:10,470 --> 00:53:12,980
are, because that tells us
how good of an estimator

942
00:53:12,980 --> 00:53:14,380
it might be.

943
00:53:14,380 --> 00:53:17,800
And what we can show is
that the expected value

944
00:53:17,800 --> 00:53:21,082
of this particular
estimator is actually

945
00:53:21,082 --> 00:53:23,040
equal to the variance
we're trying to estimate.

946
00:53:23,040 --> 00:53:24,940
So this one is unbiased.

947
00:53:24,940 --> 00:53:28,070
And Kevin, to your question,
if we had 1 over n there,

948
00:53:28,070 --> 00:53:29,330
that would be biased.

949
00:53:29,330 --> 00:53:31,034
And I think
[INAUDIBLE] Alex also

950
00:53:31,034 --> 00:53:34,502
showed you 1 over n minus 1/2?

951
00:53:34,502 --> 00:53:39,880
Oh, 1 over n minus 1.5.

952
00:53:39,880 --> 00:53:42,120
That one, I think, turns
out to be unbiased estimate

953
00:53:42,120 --> 00:53:43,042
of the standard deviation.

954
00:53:43,042 --> 00:53:43,542
Yep, yep.

955
00:53:46,490 --> 00:53:50,210
OK, and so that's
the mean of it.

956
00:53:50,210 --> 00:53:55,180
It turns out that the variance,
or the standard deviation

957
00:53:55,180 --> 00:54:00,450
of this estimator is
generally not known.

958
00:54:00,450 --> 00:54:03,160
So the mean estimate, we know
by central limit theorem,

959
00:54:03,160 --> 00:54:06,960
follows the normal distribution,
and we know its variance.

960
00:54:06,960 --> 00:54:10,760
For the variance, typically
it's generally not known.

961
00:54:10,760 --> 00:54:12,252
How could we get
a handle on this?

962
00:54:15,411 --> 00:54:16,660
We could do it multiple times.

963
00:54:16,660 --> 00:54:18,576
Maybe we can't afford
to do it multiple times.

964
00:54:18,576 --> 00:54:21,090
How else could we
get [INAUDIBLE]

965
00:54:21,090 --> 00:54:23,910
what's the cheating way of
doing it multiple times?

966
00:54:23,910 --> 00:54:24,547
Bootstrapping.

967
00:54:24,547 --> 00:54:26,630
Bootstrapping, which is
redrawing from the samples

968
00:54:26,630 --> 00:54:28,140
we've already evaluated.

969
00:54:28,140 --> 00:54:31,570
That could be a way to get a
handle on how good that is.

970
00:54:31,570 --> 00:54:34,960
Turns out there's a case that
if y is normally distributed,

971
00:54:34,960 --> 00:54:38,160
then you do know what this
is, and this estimator follows

972
00:54:38,160 --> 00:54:42,190
a chi squared distribution
that's mentioned in the notes.

973
00:54:42,190 --> 00:54:43,630
That's a very restrictive case.

974
00:54:43,630 --> 00:54:45,260
So in general, we
don't know what

975
00:54:45,260 --> 00:54:52,390
this variance is, the variance
of the variance estimator.

976
00:54:52,390 --> 00:54:54,580
Yep.

977
00:54:54,580 --> 00:54:59,450
And then, the third kind of
estimator that we talked about

978
00:54:59,450 --> 00:55:00,905
was for probability.

979
00:55:00,905 --> 00:55:04,214
So estimate mean,
estimate variance,

980
00:55:04,214 --> 00:55:05,463
and then estimate probability.

981
00:55:12,221 --> 00:55:13,970
So let's say that we're
trying to estimate

982
00:55:13,970 --> 00:55:16,930
the probability of some
event A, which I'm just going

983
00:55:16,930 --> 00:55:20,740
to call p, little p for short.

984
00:55:20,740 --> 00:55:25,725
And so we would use
as an estimator,

985
00:55:25,725 --> 00:55:34,110
let's say, p hat of a,
which is na over a-- over n.

986
00:55:34,110 --> 00:55:36,560
The number of times that
a occurred in our sample

987
00:55:36,560 --> 00:55:39,010
divided by the total
number of samples.

988
00:55:39,010 --> 00:55:40,600
Now, again, for
this one, we know

989
00:55:40,600 --> 00:55:44,660
that by central limit theorem,
for large n-- so again,

990
00:55:44,660 --> 00:55:47,260
this estimator, t hat
a, is a random variable.

991
00:55:47,260 --> 00:55:53,830
For large n, it follows
a normal distribution,

992
00:55:53,830 --> 00:55:56,430
with the mean equal
to the probability

993
00:55:56,430 --> 00:55:59,170
that we're actually
trying to estimate,

994
00:55:59,170 --> 00:56:01,230
and the variance equal
to the probability

995
00:56:01,230 --> 00:56:03,730
we're trying to estimate times
1 minus the probability we're

996
00:56:03,730 --> 00:56:06,670
trying to estimate divided by n.

997
00:56:06,670 --> 00:56:09,000
So this guy here means
that it's unbiased.

998
00:56:14,752 --> 00:56:16,720
AUDIENCE: So you cannot
say that in general,

999
00:56:16,720 --> 00:56:24,100
an estimator of
[INAUDIBLE] variance,

1000
00:56:24,100 --> 00:56:28,774
true population variance, you
have to include for each type

1001
00:56:28,774 --> 00:56:29,520
of estimator--

1002
00:56:29,520 --> 00:56:31,660
PROFESSOR: That's right.

1003
00:56:31,660 --> 00:56:33,400
Yeah, and I think
that's a place where

1004
00:56:33,400 --> 00:56:36,470
some people were a little
confused in the project.

1005
00:56:36,470 --> 00:56:41,540
It turns out that this
is a variance of n.

1006
00:56:41,540 --> 00:56:44,170
But this is the variance of
the Bernoulli random variable

1007
00:56:44,170 --> 00:56:48,222
that's the [? 01 ?] indicator
that can come out of the code.

1008
00:56:48,222 --> 00:56:50,680
And so that's a way-- if you
feel more comfortable thinking

1009
00:56:50,680 --> 00:56:52,804
about it that way, you can
think about it that way.

1010
00:56:52,804 --> 00:56:56,130
But I much prefer to think
that this is the mean

1011
00:56:56,130 --> 00:56:58,530
and this is a variance--
the mean of the estimator

1012
00:56:58,530 --> 00:57:00,146
and the variance
of the estimator.

1013
00:57:00,146 --> 00:57:01,520
And the variance
of the estimator

1014
00:57:01,520 --> 00:57:05,350
is this whole thing, just like
the variance of this estimator

1015
00:57:05,350 --> 00:57:08,100
is this whole thing here.

1016
00:57:08,100 --> 00:57:10,860
And there do turn out to be
other variances divided by n,

1017
00:57:10,860 --> 00:57:13,170
but that relationship
is a little bit murky,

1018
00:57:13,170 --> 00:57:15,128
and I think you could
get yourself into trouble

1019
00:57:15,128 --> 00:57:17,260
if you think of it that way.

1020
00:57:17,260 --> 00:57:18,385
It doesn't work over there.

1021
00:57:18,385 --> 00:57:19,600
Yeah.

1022
00:57:19,600 --> 00:57:25,140
So it's definitely-- it's
not a general result. Yeah.

1023
00:57:25,140 --> 00:57:27,480
Maybe the key is that here
you can write a probability

1024
00:57:27,480 --> 00:57:31,090
as a mean, because it's the
average with that indicator

1025
00:57:31,090 --> 00:57:32,650
function.

1026
00:57:32,650 --> 00:57:35,020
And so you can use
the same result

1027
00:57:35,020 --> 00:57:36,880
that you have used here
to get to this one.

1028
00:57:36,880 --> 00:57:40,184
But again, that seems a
little-- if the ideas are

1029
00:57:40,184 --> 00:57:42,350
clear in your mind, you can
think about it that way,

1030
00:57:42,350 --> 00:57:43,550
if you want to.

1031
00:57:47,960 --> 00:57:54,315
OK, so once you have this, then
specifying an accuracy level

1032
00:57:54,315 --> 00:57:56,690
with a given confidence is
pretty straightforward, right?

1033
00:57:56,690 --> 00:57:58,148
If we say that we
want our estimate

1034
00:57:58,148 --> 00:58:02,850
to be plus or minus 0.1 with
confidence 99 percentile,

1035
00:58:02,850 --> 00:58:05,430
then what we're saying is
that this distribution's

1036
00:58:05,430 --> 00:58:08,110
standard deviation--
square root of this guy--

1037
00:58:08,110 --> 00:58:12,060
has to be within plus
or minus, if it's

1038
00:58:12,060 --> 00:58:18,282
99 percentile, 3-ish times that
0.1, or whatever I said it was.

1039
00:58:18,282 --> 00:58:20,130
Yep.

1040
00:58:20,130 --> 00:58:22,890
And so that will tell you
how many-- should make this

1041
00:58:22,890 --> 00:58:25,950
a bit of a curly
n-- how many samples

1042
00:58:25,950 --> 00:58:29,010
you need to run in order to
drive-- so the more samples you

1043
00:58:29,010 --> 00:58:32,720
run, the more you're shrinking
this distribution of p hat

1044
00:58:32,720 --> 00:58:37,450
a, and making sure that the one
run you do-- shrink, shrink,

1045
00:58:37,450 --> 00:58:39,215
shrink-- falls within
the given tolerance

1046
00:58:39,215 --> 00:58:43,039
that you've been given, with
a given amount of confidence.

1047
00:58:43,039 --> 00:58:44,580
Now, of course, the
trick is that you

1048
00:58:44,580 --> 00:58:51,370
don't know p, in order to figure
out what this variance is.

1049
00:58:51,370 --> 00:58:53,780
But like in the project,
worst-case, variance

1050
00:58:53,780 --> 00:58:56,280
is maximized when p is
1/2, so you could use that.

1051
00:58:56,280 --> 00:58:59,630
Or you could do some kind of an
on-the-fly checking as you go,

1052
00:58:59,630 --> 00:59:02,800
get an estimate for p hat,
and then use that in here,

1053
00:59:02,800 --> 00:59:03,869
and keep checking.

1054
00:59:03,869 --> 00:59:05,660
And so then again, it
would be approximate,

1055
00:59:05,660 --> 00:59:07,970
but if you built in a
little bit of conservatism,

1056
00:59:07,970 --> 00:59:10,084
then you would be OK.

1057
00:59:10,084 --> 00:59:11,500
I posted solutions
for the project

1058
00:59:11,500 --> 00:59:13,255
last night, where
that's explained.

1059
00:59:16,770 --> 00:59:23,602
OK, questions about
Monte Carlo estimators?

1060
00:59:23,602 --> 00:59:25,060
Three different
kinds of estimators

1061
00:59:25,060 --> 00:59:26,768
that I expect you to
be comfortable with.

1062
00:59:29,450 --> 00:59:32,085
OK?

1063
00:59:32,085 --> 00:59:35,070
You guys are very
talkative today.

1064
00:59:35,070 --> 00:59:37,300
Very talkative.

1065
00:59:37,300 --> 00:59:43,300
All right, so moving on,
variance reduction methods.

1066
00:59:46,110 --> 00:59:48,160
The main one that
we talked about

1067
00:59:48,160 --> 00:59:56,840
was importance
sampling, where again,

1068
00:59:56,840 --> 00:59:59,150
it's kind of a trick
to help control

1069
00:59:59,150 --> 01:00:01,992
the error in our estimates.

1070
01:00:01,992 --> 01:00:04,610
So in particular, we
looked at estimating

1071
01:00:04,610 --> 01:00:10,140
a mean, mu of x, which
is the expectation

1072
01:00:10,140 --> 01:00:11,730
of some random variable x.

1073
01:00:11,730 --> 01:00:14,610
And remember, I was putting
the little x down there

1074
01:00:14,610 --> 01:00:16,530
to denote that we're
taking expectation

1075
01:00:16,530 --> 01:00:22,060
over x, because we're going to
play around with the sampling.

1076
01:00:22,060 --> 01:00:24,560
And remember, we said
that we could write this

1077
01:00:24,560 --> 01:00:30,010
as the integral of x times
f of x dx, where this f of x

1078
01:00:30,010 --> 01:00:33,090
is the PDF of x.

1079
01:00:33,090 --> 01:00:36,380
And then, remember, we just play
this trick where we divide by z

1080
01:00:36,380 --> 01:00:40,630
and multiply by z, which
means that we don't actually

1081
01:00:40,630 --> 01:00:43,840
change anything.

1082
01:00:43,840 --> 01:00:48,090
But then we define z
such that z times f of x

1083
01:00:48,090 --> 01:00:51,010
is actually another
density, f of z.

1084
01:00:53,580 --> 01:00:55,450
Which means that
we can interpret

1085
01:00:55,450 --> 01:00:59,740
the mean of x as being the
expectation over x with respect

1086
01:00:59,740 --> 01:01:02,640
the density f x, or we
can also interpret it

1087
01:01:02,640 --> 01:01:07,110
as being the
expectation of x over z

1088
01:01:07,110 --> 01:01:09,436
with respect to the density z.

1089
01:01:13,910 --> 01:01:16,450
And why do we do that?

1090
01:01:22,510 --> 01:01:24,630
We do that because
now we have two ways

1091
01:01:24,630 --> 01:01:27,645
to think about this mean that
we're trying to estimate.

1092
01:01:30,240 --> 01:01:33,910
We can think about
it as the expectation

1093
01:01:33,910 --> 01:01:37,060
of x under the
density f of x, which

1094
01:01:37,060 --> 01:01:44,521
means that we would sample x,
again, under this density, f

1095
01:01:44,521 --> 01:01:45,020
of x.

1096
01:01:47,640 --> 01:01:50,950
And then we would get
our mean estimate.

1097
01:01:50,950 --> 01:01:53,450
And more importantly, we would
get our estimator variance

1098
01:01:53,450 --> 01:01:56,130
that we just saw a
couple minutes ago

1099
01:01:56,130 --> 01:02:04,671
as being the variance, again,
under f of x of x over root n.

1100
01:02:04,671 --> 01:02:07,436
So that would be the variance
of-- the mean [INAUDIBLE]

1101
01:02:07,436 --> 01:02:09,060
so that's just the
standard Monte Carlo

1102
01:02:09,060 --> 01:02:10,980
that we've been talking about.

1103
01:02:10,980 --> 01:02:12,650
But what we just
saw over there was

1104
01:02:12,650 --> 01:02:14,600
that we can also
think about this thing

1105
01:02:14,600 --> 01:02:19,330
as being the expectation
of x over z taken

1106
01:02:19,330 --> 01:02:23,050
with respect to the pdfz.

1107
01:02:23,050 --> 01:02:28,095
So this would mean,
conceptually, sample now x

1108
01:02:28,095 --> 01:02:33,440
over z, under now f of z.

1109
01:02:36,670 --> 01:02:39,340
And then, the real
reason that we do that

1110
01:02:39,340 --> 01:02:42,260
is that we can then play around
with the estimator variance.

1111
01:02:44,830 --> 01:02:49,810
And it now is the
variance of x over z

1112
01:02:49,810 --> 01:02:52,304
with respect to z over n.

1113
01:02:52,304 --> 01:02:53,720
And while these
guys are the same,

1114
01:02:53,720 --> 01:02:56,300
because the mean is
the linear calculation,

1115
01:02:56,300 --> 01:02:59,299
these two are not
necessarily the same.

1116
01:02:59,299 --> 01:03:01,340
So then, remember, we
talked about different ways

1117
01:03:01,340 --> 01:03:04,560
that we could come up
with a clever f of z that

1118
01:03:04,560 --> 01:03:06,080
would put more
samples, for example,

1119
01:03:06,080 --> 01:03:09,740
in a probability region that
we're trying to explore,

1120
01:03:09,740 --> 01:03:11,660
that could have the
effect of reducing

1121
01:03:11,660 --> 01:03:14,280
the variance of the estimator,
meaning less samples

1122
01:03:14,280 --> 01:03:16,810
for the same level of
accuracy, while giving still

1123
01:03:16,810 --> 01:03:19,480
the correct estimate
of the probability

1124
01:03:19,480 --> 01:03:21,260
in that particular case.

1125
01:03:21,260 --> 01:03:23,710
And we also talked about how
you would modify the Monte

1126
01:03:23,710 --> 01:03:25,730
Carlo-- remember, the
Monte Carlo estimator had

1127
01:03:25,730 --> 01:03:35,465
that x over z, and then the z
scaling back [? inside it. ?]

1128
01:03:35,465 --> 01:03:37,840
I think, unfortunately, that
is one of the lectures where

1129
01:03:37,840 --> 01:03:39,820
the audio got scrambled.

1130
01:03:39,820 --> 01:03:43,800
But I scanned in my notes,
and they're posted online.

1131
01:03:49,870 --> 01:03:53,646
OK, how are we doing on time?

1132
01:03:53,646 --> 01:04:00,816
[? 46. ?] So we talked
briefly about variance-based

1133
01:04:00,816 --> 01:04:02,190
sensitivity
analysis, and I think

1134
01:04:02,190 --> 01:04:03,356
I'll put it up on the board.

1135
01:04:03,356 --> 01:04:07,202
But I really just
introduced you to the idea

1136
01:04:07,202 --> 01:04:08,660
of doing variance-based
sensitivity

1137
01:04:08,660 --> 01:04:12,390
analysis of defining those
main effects sensitivity

1138
01:04:12,390 --> 01:04:16,600
indices that gave us a sense of
how the variance in the output

1139
01:04:16,600 --> 01:04:20,890
changes if you learn something
about one of the inputs.

1140
01:04:20,890 --> 01:04:26,250
We talked about different
DOE methods, the idea

1141
01:04:26,250 --> 01:04:38,030
being that what we're trying to
do is sample the design space,

1142
01:04:38,030 --> 01:04:41,810
but not do full random
Monte Carlo sampling,

1143
01:04:41,810 --> 01:04:43,576
maybe because we
can't afford it.

1144
01:04:43,576 --> 01:04:51,960
So in DOE [INAUDIBLE]
that the idea

1145
01:04:51,960 --> 01:04:54,740
is that we would define
factors, which are inputs.

1146
01:04:54,740 --> 01:04:57,500
And we would define discrete
levels at which we might

1147
01:04:57,500 --> 01:04:59,790
want to test those factors.

1148
01:04:59,790 --> 01:05:04,500
And that we're going to run
some combination of factors

1149
01:05:04,500 --> 01:05:07,420
and levels through
the model, and collect

1150
01:05:07,420 --> 01:05:12,300
outputs, which are sometimes
called observations

1151
01:05:12,300 --> 01:05:15,750
in DOE language.

1152
01:05:15,750 --> 01:05:17,950
And that the
different DOE methods

1153
01:05:17,950 --> 01:05:20,940
are different ways to
choose combinations

1154
01:05:20,940 --> 01:05:22,164
of factors and levels.

1155
01:05:22,164 --> 01:05:23,830
And we talked about
the parameter study,

1156
01:05:23,830 --> 01:05:27,600
the one-at-a-time, orthogonal
arrays, Latin hypercubes.

1157
01:05:27,600 --> 01:05:30,240
So again, different DOE
methods are different ways

1158
01:05:30,240 --> 01:05:33,940
to choose which combinations
of factors and levels.

1159
01:05:33,940 --> 01:05:36,860
And then we also talked
about the main effects,

1160
01:05:36,860 --> 01:05:39,370
averaging over all
the experiments

1161
01:05:39,370 --> 01:05:41,580
with a factor at a
particular level,

1162
01:05:41,580 --> 01:05:44,030
looking at that contingent
average of all experiments.

1163
01:05:44,030 --> 01:05:45,863
And remember, the paper
airplane experiment,

1164
01:05:45,863 --> 01:05:48,700
where we looked at the effects
of the different design

1165
01:05:48,700 --> 01:05:50,260
variable settings.

1166
01:05:50,260 --> 01:05:55,760
That gives us some insight
to our design space.

1167
01:05:55,760 --> 01:06:03,990
OK, and then lastly, in
optimization, not too

1168
01:06:03,990 --> 01:06:05,830
much to say here.

1169
01:06:05,830 --> 01:06:09,730
Just a basic idea of
unconstrained optimization,

1170
01:06:09,730 --> 01:06:11,220
what an optimization
problem looks

1171
01:06:11,220 --> 01:06:15,560
like, the basic idea of
how a simple algorithm

1172
01:06:15,560 --> 01:06:18,290
like steepest descent, or
conjugate gradient, or Newton's

1173
01:06:18,290 --> 01:06:21,030
method might work.

1174
01:06:21,030 --> 01:06:25,870
We talked about the different
methods to compute gradients.

1175
01:06:25,870 --> 01:06:30,339
And we saw particularly
that finite difference

1176
01:06:30,339 --> 01:06:32,880
approximations, which you saw
when you were approximating PDE

1177
01:06:32,880 --> 01:06:36,160
terms, can also be used
to compute gradients,

1178
01:06:36,160 --> 01:06:39,210
or to estimate gradients,
in the design space.

1179
01:06:39,210 --> 01:06:41,360
And then, we also
talked about how

1180
01:06:41,360 --> 01:06:45,280
we could use samples to fit
a polynomial response surface

1181
01:06:45,280 --> 01:06:50,990
model, if we needed to
approximate an expensive model,

1182
01:06:50,990 --> 01:06:52,375
and use it in an optimization.

1183
01:06:56,010 --> 01:06:59,300
So that was the last--
really, the last two lectures.

1184
01:07:05,270 --> 01:07:12,930
All right, so let me put up--
quickly look at the outcomes.

1185
01:07:12,930 --> 01:07:14,935
Everybody got-- what's that?

1186
01:07:14,935 --> 01:07:16,360
AUDIENCE: [INAUDIBLE]

1187
01:07:20,160 --> 01:07:21,950
PROFESSOR: We'll go
look at the outcomes

1188
01:07:21,950 --> 01:07:24,200
for that last section,
the probabilistic analysis

1189
01:07:24,200 --> 01:07:24,910
and optimization.

1190
01:07:24,910 --> 01:07:27,932
And again, these
are specifically

1191
01:07:27,932 --> 01:07:29,390
what I expect you
to be able to do.

1192
01:07:29,390 --> 01:07:30,500
I'm not going to ask
you to do anything

1193
01:07:30,500 --> 01:07:31,320
that's not in these outcomes.

1194
01:07:31,320 --> 01:07:33,480
If you can do every single
one of these outcomes,

1195
01:07:33,480 --> 01:07:36,510
you will be 100% fine.

1196
01:07:36,510 --> 01:07:40,910
So 3.1 and 3.2 related to
some basic probability stuff.

1197
01:07:40,910 --> 01:07:43,310
3.2 described the process
of Monte Carlo sampling

1198
01:07:43,310 --> 01:07:45,125
from uniform distributions.

1199
01:07:45,125 --> 01:07:47,200
3.4 described how
to generalize it

1200
01:07:47,200 --> 01:07:51,580
to arbitrary univariate
distributions, like triangular.

1201
01:07:54,100 --> 01:07:55,720
Use it to propagate uncertainty.

1202
01:07:55,720 --> 01:08:00,890
You've [INAUDIBLE] this one
in the project-- [? horray. ?]

1203
01:08:00,890 --> 01:08:04,970
Then, a whole bunch of outcomes
that relate to the estimators.

1204
01:08:04,970 --> 01:08:06,740
Describe what an estimator is.

1205
01:08:06,740 --> 01:08:11,470
Define its bias and variance.

1206
01:08:11,470 --> 01:08:14,830
State unbiased estimators
for the mean and variance

1207
01:08:14,830 --> 01:08:18,420
of a random variable and for
the probability of an event.

1208
01:08:18,420 --> 01:08:20,229
Describe the typical
convergence rate.

1209
01:08:20,229 --> 01:08:23,910
So this is how
does the variance,

1210
01:08:23,910 --> 01:08:26,500
how do the standard
deviations of these estimators

1211
01:08:26,500 --> 01:08:28,455
behave as n increases?

1212
01:08:32,512 --> 01:08:36,039
For the estimators, define the
standard error and the sampling

1213
01:08:36,039 --> 01:08:36,580
distribution.

1214
01:08:36,580 --> 01:08:38,038
So this is what
we're talking about

1215
01:08:38,038 --> 01:08:40,689
with the normal distributions
that we can derive for the mean

1216
01:08:40,689 --> 01:08:42,090
and for the probability.

1217
01:08:42,090 --> 01:08:43,550
In the variance,
we typically don't

1218
01:08:43,550 --> 01:08:45,800
know what it is, unless we
have that very special case

1219
01:08:45,800 --> 01:08:49,316
that the outputs
themselves are normal.

1220
01:08:49,316 --> 01:08:51,352
Give standard errors for
the sample estimators

1221
01:08:51,352 --> 01:08:53,060
of mean, variance,
and event probability.

1222
01:08:53,060 --> 01:08:56,319
This one should really only
be in that very special case.

1223
01:08:56,319 --> 01:08:58,430
And then, obtain
confidence intervals,

1224
01:08:58,430 --> 01:09:02,310
and be able to use those to
determine how many samples you

1225
01:09:02,310 --> 01:09:05,470
need in a Monte
Carlo simulation.

1226
01:09:05,470 --> 01:09:07,986
Then on Design of Experiments
and Response Surfaces,

1227
01:09:07,986 --> 01:09:10,069
so just describe how to
apply the different design

1228
01:09:10,069 --> 01:09:12,652
of experiments methods that we
talked about-- parameter study,

1229
01:09:12,652 --> 01:09:15,420
one-at-a-time, Latin
hypercube sampling,

1230
01:09:15,420 --> 01:09:16,899
and orthogonal arrays.

1231
01:09:16,899 --> 01:09:18,550
Describe the Response
Surface Method,

1232
01:09:18,550 --> 01:09:21,520
and describe how you
could construct it

1233
01:09:21,520 --> 01:09:23,970
using least squares regression.

1234
01:09:23,970 --> 01:09:25,279
Remember, we did the points.

1235
01:09:25,279 --> 01:09:27,069
We construct the
least square system.

1236
01:09:27,069 --> 01:09:29,359
Use it to get the
unknown coefficients

1237
01:09:29,359 --> 01:09:33,060
of either a linear or a
quadratic response surface.

1238
01:09:33,060 --> 01:09:35,689
And then, we didn't talk
about this in too much detail,

1239
01:09:35,689 --> 01:09:37,930
but I know you've seen
the R2-metric before,

1240
01:09:37,930 --> 01:09:40,160
and you understand
that that measures

1241
01:09:40,160 --> 01:09:42,759
the quality of the
[? search, ?] of the Response

1242
01:09:42,759 --> 01:09:45,500
Surface to the data points.

1243
01:09:45,500 --> 01:09:48,109
Then lastly, on the Introduction
to Design Optimization,

1244
01:09:48,109 --> 01:09:50,650
be able to describe the steepest
descent, conjugate gradient,

1245
01:09:50,650 --> 01:09:52,910
and the Newton method,
and to apply it

1246
01:09:52,910 --> 01:09:56,190
to simple unconstrained
design problems.

1247
01:09:56,190 --> 01:09:58,750
Describe the different methods
to estimate the gradients,

1248
01:09:58,750 --> 01:10:01,240
and be able to use finite
difference approximations

1249
01:10:01,240 --> 01:10:03,290
to actually estimate them.

1250
01:10:03,290 --> 01:10:05,150
And then, interpret
sensitivity information,

1251
01:10:05,150 --> 01:10:07,030
so things like the main
effect sensitivities

1252
01:10:07,030 --> 01:10:08,690
that we talk about,
and explain why

1253
01:10:08,690 --> 01:10:11,320
those are relevant to
aerospace design examples.

1254
01:10:11,320 --> 01:10:13,780
Remember, we talked about
how you could use sensitivity

1255
01:10:13,780 --> 01:10:16,900
information to decide
where to reduce uncertainty

1256
01:10:16,900 --> 01:10:19,350
in the system, or
where-- there was called

1257
01:10:19,350 --> 01:10:23,250
factor prioritization, or factor
fixing, where there are factors

1258
01:10:23,250 --> 01:10:25,880
that don't really matter, and
you could fix them at a value

1259
01:10:25,880 --> 01:10:28,550
and not worry about
their uncertainty.