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PROFESSOR: So today we're
going to keep talking
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00:00:27,370 --> 00:00:29,280
about boundary value problems.
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00:00:29,280 --> 00:00:32,080
We'll do that
again on Wednesday.
11
00:00:32,080 --> 00:00:37,195
On Friday we'll start partial
differential equations.
12
00:00:37,195 --> 00:00:41,260
And next week we'll
work with COMSOL,
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00:00:41,260 --> 00:00:44,890
which I hope you guys all have
installed on your laptops.
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00:00:44,890 --> 00:00:47,020
Please check that
you have it installed
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00:00:47,020 --> 00:00:50,590
and you can turn it on,
your licensing things
16
00:00:50,590 --> 00:00:51,850
work and stuff like that.
17
00:00:51,850 --> 00:00:56,034
We'll have a COMSOL tutorial
I think on Monday in class.
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00:00:56,034 --> 00:00:58,200
And you'll use that for one
of the homework problems
19
00:00:58,200 --> 00:01:00,430
not immediately but coming up.
20
00:01:06,790 --> 00:01:10,780
Speaking of the homework,
I received some feedback
21
00:01:10,780 --> 00:01:13,630
that the homeworks
have been taking
22
00:01:13,630 --> 00:01:16,000
an inordinate amount of time.
23
00:01:16,000 --> 00:01:19,240
And so I've discussed this
with the other people involved
24
00:01:19,240 --> 00:01:22,060
in teaching it and we
decided to drastically cut
25
00:01:22,060 --> 00:01:25,060
the amount of points
given for write-ups to try
26
00:01:25,060 --> 00:01:28,340
to encourage you not to
spend so much time on that.
27
00:01:28,340 --> 00:01:30,946
And instead I'd really
rather you spent the time
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00:01:30,946 --> 00:01:32,570
and did the reading
instead of spending
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00:01:32,570 --> 00:01:35,249
an extra hour doing write-up.
30
00:01:35,249 --> 00:01:37,040
You won't get any more
points for doing it.
31
00:01:37,040 --> 00:01:37,270
So if you want--
32
00:01:37,270 --> 00:01:38,650
I mean, I love
beautiful write-ups.
33
00:01:38,650 --> 00:01:41,020
I'm sure the graders appreciate
the beautiful write-ups.
34
00:01:41,020 --> 00:01:43,040
It's good for your
thinking to write clearly
35
00:01:43,040 --> 00:01:45,360
but it's not worth many,
many hours of time.
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00:01:48,690 --> 00:01:51,280
And in general, the instructions
at the beginning of course,
37
00:01:51,280 --> 00:01:54,420
are on average over the
course, your 14 weeks,
38
00:01:54,420 --> 00:01:57,230
it should be about nine
hours per week of homework.
39
00:01:57,230 --> 00:01:59,475
So that's maybe 13
hours per assignment,
40
00:01:59,475 --> 00:02:00,980
so you have 10 assignments.
41
00:02:00,980 --> 00:02:04,424
And so if it's getting
to be 15 hours,
42
00:02:04,424 --> 00:02:06,590
you've spent too much time
on it and just forget it.
43
00:02:06,590 --> 00:02:09,300
Just draw a line and say
I'm done, time ran out,
44
00:02:09,300 --> 00:02:12,640
and that's fine.
45
00:02:12,640 --> 00:02:16,082
This is-- getting the last
bug out of your Matlab code
46
00:02:16,082 --> 00:02:17,790
is not the main
objective of this course.
47
00:02:23,990 --> 00:02:25,640
And really, the
purpose of homework
48
00:02:25,640 --> 00:02:28,117
is to help you
learn, not that we
49
00:02:28,117 --> 00:02:30,200
need to know what the
solution to this problem is.
50
00:02:30,200 --> 00:02:30,947
This is now--
51
00:02:30,947 --> 00:02:33,280
I'm getting the solution from
somebody else in the class
52
00:02:33,280 --> 00:02:33,946
too so this is--
53
00:02:33,946 --> 00:02:36,465
I don't need it from you.
54
00:02:36,465 --> 00:02:38,090
The TAs probably
already did it already
55
00:02:38,090 --> 00:02:39,381
and they have the solution too.
56
00:02:39,381 --> 00:02:41,080
So we know the solution.
57
00:02:41,080 --> 00:02:42,320
It's not so essential.
58
00:02:42,320 --> 00:02:44,736
Its purpose is to help you
learn and how to figure it out.
59
00:02:44,736 --> 00:02:46,580
And beyond a certain
point, it's not
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00:02:46,580 --> 00:02:50,860
that instructive
in my experience.
61
00:02:50,860 --> 00:02:52,777
Though sometimes the
fifteenth and a half hour
62
00:02:52,777 --> 00:02:55,110
you suddenly have the great
insight and you learn a lot,
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00:02:55,110 --> 00:02:56,090
but most the time not.
64
00:03:01,020 --> 00:03:12,450
And by the way the reading
is in [? Beer's ?] textbook
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00:03:12,450 --> 00:03:19,490
pages 258 to 311.
66
00:03:19,490 --> 00:03:22,740
And there's also some nice
short readings by Professor
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00:03:22,740 --> 00:03:25,770
[? Brautz ?] that have been
posted, only a few pages long
68
00:03:25,770 --> 00:03:28,320
but definitely worth a look.
69
00:03:28,320 --> 00:03:38,580
Both on [? PPPs ?] and also
in ODEs and [? DAEs. ?]
70
00:03:38,580 --> 00:03:40,590
So today what we're
going to talk about
71
00:03:40,590 --> 00:03:44,510
is relaxation methods.
72
00:03:59,250 --> 00:04:02,800
And we talked last time
about the shooting method.
73
00:04:02,800 --> 00:04:04,530
So that's a good method.
74
00:04:07,999 --> 00:04:09,540
At least one famous
numerical methods
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00:04:09,540 --> 00:04:12,240
book recommends you always
shoot first then relax.
76
00:04:12,240 --> 00:04:14,130
So try the shooting method.
77
00:04:14,130 --> 00:04:15,511
If it works, you're done.
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00:04:15,511 --> 00:04:17,010
If it doesn't work
for you, then you
79
00:04:17,010 --> 00:04:20,019
may have to use a
relaxation method.
80
00:04:20,019 --> 00:04:22,920
And we'll talk a bit now
about the relaxation methods.
81
00:04:22,920 --> 00:04:28,480
And the general idea is
you're going to write your y--
82
00:04:28,480 --> 00:04:29,940
which is an
approximation, it's not
83
00:04:29,940 --> 00:04:31,750
going to be the real solution--
84
00:04:31,750 --> 00:04:36,700
and you're going to try
to write that typically
85
00:04:36,700 --> 00:04:41,200
as an expansion to
some basis functions.
86
00:04:41,200 --> 00:04:43,230
So let's say the
nth component of y.
87
00:04:50,310 --> 00:04:53,220
And then you're going to vary
these coefficients, the d's,
88
00:04:53,220 --> 00:04:55,379
to try to make your solution
as good as possible.
89
00:04:55,379 --> 00:04:57,420
And now we're talking
about different definitions
90
00:04:57,420 --> 00:05:02,850
of what good is for the
solution for a problem.
91
00:05:02,850 --> 00:05:05,040
And you have to be aware
that almost always, this
92
00:05:05,040 --> 00:05:09,190
will not exactly solve the
differential equation system.
93
00:05:09,190 --> 00:05:12,453
So it's always going to be
wrong everywhere, typically.
94
00:05:12,453 --> 00:05:15,059
And you can get to
decide sort of--
95
00:05:15,059 --> 00:05:17,100
if you wanted to be good
at some particular spot,
96
00:05:17,100 --> 00:05:18,840
you can do something about that.
97
00:05:18,840 --> 00:05:22,260
If you wanted to, on
average, be good in some way,
98
00:05:22,260 --> 00:05:23,790
you can decide that.
99
00:05:23,790 --> 00:05:28,020
And it's sort of how much
error you can tolerate.
100
00:05:28,020 --> 00:05:31,840
And in general, you have to do
a finite sum of a finite basis
101
00:05:31,840 --> 00:05:32,610
set.
102
00:05:32,610 --> 00:05:35,045
For many, many ODE,
[? PPP ?] problems,
103
00:05:35,045 --> 00:05:37,170
there's math proofs that
if-- and the limit is this
104
00:05:37,170 --> 00:05:39,240
goes to infinity if you add an
infinite number of functions
105
00:05:39,240 --> 00:05:39,900
here.
106
00:05:39,900 --> 00:05:42,844
You can always make it work
to get the true solution.
107
00:05:42,844 --> 00:05:44,760
But you can never afford
that so you're always
108
00:05:44,760 --> 00:05:48,960
finite truncated basis
set is what you're using.
109
00:05:48,960 --> 00:05:50,879
And so a lot of
the accuracy things
110
00:05:50,879 --> 00:05:52,920
have to do with exactly
what functions you choose
111
00:05:52,920 --> 00:05:56,340
and exactly how many of these
terms and sum you include.
112
00:05:56,340 --> 00:05:59,236
But typically, these problems
will start from the beginning,
113
00:05:59,236 --> 00:06:00,860
you'll say I'm only
going to include so
114
00:06:00,860 --> 00:06:05,310
many, how big my computer is.
115
00:06:05,310 --> 00:06:07,654
And so you're going to
be stuck with some error,
116
00:06:07,654 --> 00:06:08,820
so that's just the way this.
117
00:06:13,916 --> 00:06:15,290
Let's recall the
problem we have.
118
00:06:15,290 --> 00:06:17,270
If we write it in the--
119
00:06:17,270 --> 00:06:19,265
as a first order ODE
system, it's dy/dt--
120
00:06:23,350 --> 00:06:28,650
well, there's some function
of dy/dt and y and t
121
00:06:28,650 --> 00:06:33,010
that's equal to zero which
often can be written something
122
00:06:33,010 --> 00:06:40,898
like this, dy/dt is equal
to an f [INAUDIBLE]..
123
00:06:40,898 --> 00:06:42,320
Right?
124
00:06:42,320 --> 00:06:47,820
And so if you plug in
this approximation for y,
125
00:06:47,820 --> 00:06:53,460
then you'll get something here
that will be some g of t that's
126
00:06:53,460 --> 00:06:56,280
generally not going
to equal zero,
127
00:06:56,280 --> 00:06:57,680
but you would like
it to be zero.
128
00:06:57,680 --> 00:06:59,910
If you put the true solution
in, you would get zero.
129
00:06:59,910 --> 00:07:01,950
If you put your
approximate solution,
130
00:07:01,950 --> 00:07:04,450
it is not going to be zero.
131
00:07:04,450 --> 00:07:06,550
So you'll get g
of t and you want
132
00:07:06,550 --> 00:07:11,010
this equal to zero for all t.
133
00:07:14,010 --> 00:07:17,940
And then in addition, you
have boundary conditions.
134
00:07:17,940 --> 00:07:21,870
And so you have boundary
conditions on the solution
135
00:07:21,870 --> 00:07:24,175
and you want those
things to be satisfied.
136
00:07:24,175 --> 00:07:25,800
And again, you can
write them in a form
137
00:07:25,800 --> 00:07:27,870
that something has to be zero.
138
00:07:27,870 --> 00:07:31,147
Often, you'll
exactly satisfy them.
139
00:07:31,147 --> 00:07:32,730
So you'll choose
your solution to make
140
00:07:32,730 --> 00:07:34,890
sure it satisfies the
boundary conditions
141
00:07:34,890 --> 00:07:36,417
and it won't satisfy
in the domain.
142
00:07:36,417 --> 00:07:38,500
It won't really satisfy
the differential equation.
143
00:07:38,500 --> 00:07:41,345
That's the most
common thing to do.
144
00:07:41,345 --> 00:07:43,720
But it could also be that it
doesn't satisfy the boundary
145
00:07:43,720 --> 00:07:47,130
conditions either, but you
want to be close to satisfy
146
00:07:47,130 --> 00:07:48,245
the boundary conditions.
147
00:07:50,910 --> 00:07:56,250
We have to think
of how are we going
148
00:07:56,250 --> 00:08:00,810
to judge if the solution
is accurate or not,
149
00:08:00,810 --> 00:08:02,310
if our approximate
solution is good?
150
00:08:08,290 --> 00:08:11,350
And from the way we wrote it,
we have our parameters, d.
151
00:08:11,350 --> 00:08:13,300
These are numbers we can adjust.
152
00:08:13,300 --> 00:08:15,940
And we're going to try to adjust
them to make the solution as
153
00:08:15,940 --> 00:08:17,850
good as possible.
154
00:08:17,850 --> 00:08:19,580
And now we just define
what a good means.
155
00:08:19,580 --> 00:08:23,610
There are several definitions
of good that are widely used.
156
00:08:23,610 --> 00:08:28,469
And one of them is
called collocation.
157
00:08:28,469 --> 00:08:29,760
This is like option number one.
158
00:08:33,919 --> 00:08:44,310
And that is you choose a set of
ts, of particular time points.
159
00:08:44,310 --> 00:08:46,800
And for those
particular type points
160
00:08:46,800 --> 00:08:52,920
you demand that g of the
time point is equal to zero.
161
00:08:52,920 --> 00:08:58,170
So you're forcing the
residuals-- this is called
162
00:08:58,170 --> 00:09:01,320
the residual, it's the error.
163
00:09:01,320 --> 00:09:03,970
And you're forcing the error to
be zero at some particular time
164
00:09:03,970 --> 00:09:05,420
points.
165
00:09:05,420 --> 00:09:07,480
And generally between
the time points
166
00:09:07,480 --> 00:09:10,887
it will not be equal to zero.
167
00:09:10,887 --> 00:09:13,220
But you can pick your time
points and depending on which
168
00:09:13,220 --> 00:09:15,470
ones you pick, you'll get a
slightly different optimal
169
00:09:15,470 --> 00:09:17,720
choice of the d's.
170
00:09:17,720 --> 00:09:19,970
Because the d's will
be adjusted to force
171
00:09:19,970 --> 00:09:23,610
the residual to be zero at
your time points you pick.
172
00:09:23,610 --> 00:09:26,200
So that's one option.
173
00:09:26,200 --> 00:09:28,381
Another one is
called Rayleigh-Ritz.
174
00:09:34,770 --> 00:09:41,450
And that one is you
minimize overall your d's.
175
00:09:45,572 --> 00:09:55,968
The integral in t zero to t
final of the norm of g of t.
176
00:10:01,720 --> 00:10:04,330
So this means you try
to make the average
177
00:10:04,330 --> 00:10:08,940
of the square of that deviation
to be as small as possible.
178
00:10:08,940 --> 00:10:10,980
So it's like sort of
like a least squares fit
179
00:10:10,980 --> 00:10:13,260
kind of thing.
180
00:10:13,260 --> 00:10:16,180
So that's another option.
181
00:10:16,180 --> 00:10:18,550
And then a third
one that people use
182
00:10:18,550 --> 00:10:20,020
a lot is called
Galerkin's Method.
183
00:10:23,470 --> 00:10:31,925
And his method is you
choose some functions--
184
00:10:31,925 --> 00:10:35,390
some of your basis
functions typically--
185
00:10:35,390 --> 00:10:38,524
and you integrate them
with each of the elements
186
00:10:38,524 --> 00:10:39,190
of the residual.
187
00:10:42,940 --> 00:10:46,310
And you demand that that
has to be equal to zero.
188
00:10:46,310 --> 00:10:46,967
Yeah?
189
00:10:46,967 --> 00:10:49,402
AUDIENCE: [INAUDIBLE]
for [? this method, ?]
190
00:10:49,402 --> 00:10:50,870
what do you [INAUDIBLE]?
191
00:10:50,870 --> 00:10:54,259
PROFESSOR: The d's,
your coefficients.
192
00:10:54,259 --> 00:10:55,800
And these ones, I
didn't write it out
193
00:10:55,800 --> 00:10:58,580
but this g depends
implicitly on the d's,
194
00:10:58,580 --> 00:11:01,166
so you can write it that way.
195
00:11:01,166 --> 00:11:04,152
A lot of d's.
196
00:11:04,152 --> 00:11:07,460
And so you optimize the
d's, you very the d's
197
00:11:07,460 --> 00:11:11,520
to force g to be
zero at certain ts.
198
00:11:11,520 --> 00:11:15,490
And this one also,
the g depends on d's.
199
00:11:15,490 --> 00:11:17,410
And so you're going
to optimize the d's
200
00:11:17,410 --> 00:11:21,395
to force this integral
equation to be satisfied.
201
00:11:21,395 --> 00:11:21,895
Yeah?
202
00:11:21,895 --> 00:11:25,109
AUDIENCE: [INAUDIBLE]
all changing d's?
203
00:11:25,109 --> 00:11:25,775
PROFESSOR: Yeah.
204
00:11:25,775 --> 00:11:26,816
They're all changing d's.
205
00:11:26,816 --> 00:11:30,590
And it's just trying to-- your
criterion, your error measure,
206
00:11:30,590 --> 00:11:34,194
how do you measure or
what do you think is good?
207
00:11:34,194 --> 00:11:36,360
You want to make something
small, some kind of error
208
00:11:36,360 --> 00:11:37,170
small, but you
have to figure out
209
00:11:37,170 --> 00:11:38,550
what are you going to
define your error to be.
210
00:11:38,550 --> 00:11:40,383
And you'll get different
solutions depending
211
00:11:40,383 --> 00:11:42,160
on which error measure you use.
212
00:11:45,170 --> 00:11:45,950
Is this OK?
213
00:11:49,950 --> 00:11:54,820
Now this one's pretty
straightforward.
214
00:11:54,820 --> 00:11:57,675
I'm just going to write it down
a bunch of algebraic equations
215
00:11:57,675 --> 00:11:59,966
that depend on my d's and
then I'm going to solve them.
216
00:11:59,966 --> 00:12:05,680
And this looks a lot like
[? an f ?] [? solve ?] problem.
217
00:12:05,680 --> 00:12:07,845
This is a Newton problem
or something like that.
218
00:12:07,845 --> 00:12:10,125
So it's just some
algebraic equations.
219
00:12:10,125 --> 00:12:12,149
Should be OK?
220
00:12:12,149 --> 00:12:14,690
So that one you should be pretty
well set up to do right now.
221
00:12:14,690 --> 00:12:17,190
Let's just think of how
many equations there are.
222
00:12:17,190 --> 00:12:27,340
So we have-- say we have
our basis set yf t dni fit.
223
00:12:30,360 --> 00:12:31,630
That's our basis set.
224
00:12:31,630 --> 00:12:35,890
And this is a sum over
i equals one to say
225
00:12:35,890 --> 00:12:37,790
the number of basis functions.
226
00:12:37,790 --> 00:12:39,040
What do you want to call that?
227
00:12:42,672 --> 00:12:44,190
N, sound good?
228
00:12:44,190 --> 00:12:46,590
So we have n basis functions.
229
00:12:46,590 --> 00:12:50,524
And so we have how
many unknowns here?
230
00:12:50,524 --> 00:12:52,190
Maybe n's not good
because n's this one.
231
00:12:52,190 --> 00:12:53,600
Let's call it something else.
232
00:12:53,600 --> 00:12:55,130
k, OK.
233
00:12:55,130 --> 00:12:56,172
All right?
234
00:12:56,172 --> 00:12:57,630
In fact, I'll even
change this to k
235
00:12:57,630 --> 00:12:58,900
too just to make life easier.
236
00:13:02,940 --> 00:13:07,140
So there are how
many dnk's are there?
237
00:13:07,140 --> 00:13:09,510
There's n where n
is the dimension,
238
00:13:09,510 --> 00:13:13,490
the number of od's or the number
of components in y times k.
239
00:13:13,490 --> 00:13:20,894
That's how many d's we got
that we're going to adjust.
240
00:13:20,894 --> 00:13:23,310
We want to have an equal number
of equations and unknowns.
241
00:13:23,310 --> 00:13:25,060
We have to have as
many equations as that.
242
00:13:28,150 --> 00:13:30,790
So what equations do we got?
243
00:13:30,790 --> 00:13:32,010
How many equations?
244
00:13:38,010 --> 00:13:43,100
So if we just have
a ODE [? BBP ?]
245
00:13:43,100 --> 00:13:47,088
we typically have n
boundary conditions.
246
00:13:51,349 --> 00:13:53,140
So that many boundary
conditions because we
247
00:13:53,140 --> 00:13:55,930
need one boundary condition
for each differential equation,
248
00:13:55,930 --> 00:13:56,430
right?
249
00:13:56,430 --> 00:13:58,330
One integration
constant for each one.
250
00:13:58,330 --> 00:14:02,360
So we took an n
boundary conditions.
251
00:14:02,360 --> 00:14:03,510
And then we have--
252
00:14:03,510 --> 00:14:06,850
in collocation, we have however
many capital M time points
253
00:14:06,850 --> 00:14:07,360
we chose.
254
00:14:10,330 --> 00:14:16,760
So we have M n equations--
255
00:14:16,760 --> 00:14:18,192
no, that's not right.
256
00:14:18,192 --> 00:14:21,345
We have an equation like that
for every component of g.
257
00:14:21,345 --> 00:14:28,722
So it's M times n equations
from the collocation.
258
00:14:34,024 --> 00:14:36,690
Is that all right?
259
00:14:36,690 --> 00:14:39,150
So just looking
at this, it looks
260
00:14:39,150 --> 00:14:47,220
like we have n times
M plus one equations
261
00:14:47,220 --> 00:14:52,630
and we have n times k unknowns
that we're trying to adjust.
262
00:14:52,630 --> 00:14:56,220
And so therefore, this
says we should choose k
263
00:14:56,220 --> 00:14:57,620
to be equal to M plus one.
264
00:15:01,280 --> 00:15:03,760
So if we choose we want to say
we want 100 basis functions,
265
00:15:03,760 --> 00:15:07,510
then we need 99 time points to
do collocations at it in order
266
00:15:07,510 --> 00:15:10,511
to exactly determine everything.
267
00:15:10,511 --> 00:15:11,360
Is that OK?
268
00:15:13,865 --> 00:15:15,240
How many people
think this is OK?
269
00:15:17,830 --> 00:15:18,330
OK.
270
00:15:18,330 --> 00:15:18,829
He agrees.
271
00:15:18,829 --> 00:15:20,090
It's OK.
272
00:15:20,090 --> 00:15:21,899
The rest of you, no opinion.
273
00:15:21,899 --> 00:15:23,690
This is like the American
political system.
274
00:15:23,690 --> 00:15:25,370
Only 5% people vote.
275
00:15:25,370 --> 00:15:27,920
Everybody else just
listens to Donald Trump.
276
00:15:33,530 --> 00:15:37,750
So this is how many
equations we need.
277
00:15:37,750 --> 00:15:42,030
Now sometimes people will
choose the basis functions
278
00:15:42,030 --> 00:15:44,540
so that, say, some of the
boundary condition equations
279
00:15:44,540 --> 00:15:46,782
might be satisfied
automatically.
280
00:15:46,782 --> 00:15:49,090
And I'll talk a little bit
in a minute about cleverness
281
00:15:49,090 --> 00:15:51,330
in choosing basis functions.
282
00:15:51,330 --> 00:15:53,830
So one possible thing is you
can try to cleverly choose
283
00:15:53,830 --> 00:15:56,560
basis functions so that no
matter what values of d's you
284
00:15:56,560 --> 00:15:59,320
choose, you're always going to
satisfy some of the boundary
285
00:15:59,320 --> 00:16:01,410
conditions.
286
00:16:01,410 --> 00:16:05,500
And so long as you don't
get a full value of n
287
00:16:05,500 --> 00:16:07,640
because some of these
boundary conditions
288
00:16:07,640 --> 00:16:10,250
don't help you determine
the d's, any values
289
00:16:10,250 --> 00:16:11,380
of d's will work.
290
00:16:11,380 --> 00:16:14,309
And so in those cases, you need
a few-- might need another time
291
00:16:14,309 --> 00:16:15,100
point or something.
292
00:16:15,100 --> 00:16:16,335
Get some more equations.
293
00:16:20,320 --> 00:16:21,910
So that's collocation.
294
00:16:21,910 --> 00:16:26,750
And I think this should be
perfectly straightforward.
295
00:16:26,750 --> 00:16:28,660
You just evaluate your y's.
296
00:16:28,660 --> 00:16:30,370
You need to have your dy/dts.
297
00:16:30,370 --> 00:16:33,040
You'll need them, because
they appear in g as well.
298
00:16:33,040 --> 00:16:37,040
And they're just going to be
the summation over k [? of ?]
299
00:16:37,040 --> 00:16:42,880
[? dnk ?] v prime k of t.
300
00:16:42,880 --> 00:16:45,030
And so you choose
basis functions
301
00:16:45,030 --> 00:16:47,230
that the analytical
derivatives of.
302
00:16:47,230 --> 00:16:49,640
So now you know these answers.
303
00:16:49,640 --> 00:16:55,980
And now you can evaluate
this at any time point tm.
304
00:16:55,980 --> 00:16:57,860
So you evaluate
your time points.
305
00:16:57,860 --> 00:17:00,190
You evaluate these guys
at your time points.
306
00:17:00,190 --> 00:17:04,920
You plug them all into your
g expression over here,
307
00:17:04,920 --> 00:17:10,270
and your force is equal to
zero, by varying the d's, right?
308
00:17:10,270 --> 00:17:11,534
No problem.
309
00:17:11,534 --> 00:17:12,450
So that's collocation.
310
00:17:12,450 --> 00:17:13,619
That's pretty easy.
311
00:17:13,619 --> 00:17:19,109
And because it's so easy, it's
kind of pretty widely used
312
00:17:19,109 --> 00:17:22,447
as one way to go.
313
00:17:22,447 --> 00:17:24,780
All it requires is that you
have to know the derivatives
314
00:17:24,780 --> 00:17:27,569
of your basis functions.
315
00:17:27,569 --> 00:17:30,035
In particular, doesn't
require any integrals.
316
00:17:30,035 --> 00:17:31,660
When you see the
other methods, they're
317
00:17:31,660 --> 00:17:32,784
going to involve integrals.
318
00:17:32,784 --> 00:17:35,520
So we'll have to figure out
how we're going evaluate those.
319
00:17:35,520 --> 00:17:37,580
So if you have
functions you don't
320
00:17:37,580 --> 00:17:39,800
know how to integrate then
this is definitely the way
321
00:17:39,800 --> 00:17:41,900
to go, to do collocation.
322
00:17:41,900 --> 00:17:44,990
And if you use collocation
with enough points,
323
00:17:44,990 --> 00:17:47,394
you're forcing the error
to be zero a lot of points,
324
00:17:47,394 --> 00:17:49,810
then probably it won't get
that big in between the points.
325
00:17:49,810 --> 00:17:51,620
At least you can
hope that I won't get
326
00:17:51,620 --> 00:17:53,090
that big between the points.
327
00:17:53,090 --> 00:17:55,310
And you can try it with
different numbers of points.
328
00:17:55,310 --> 00:17:56,726
And different size
numbers of base
329
00:17:56,726 --> 00:17:59,720
functions and see
what you can do.
330
00:17:59,720 --> 00:18:02,120
See if it converges to
something, if you're lucky.
331
00:18:02,120 --> 00:18:02,620
All right.
332
00:18:02,620 --> 00:18:04,680
So that's one way to go.
333
00:18:04,680 --> 00:18:07,516
And it's just f solve problem.
334
00:18:07,516 --> 00:18:09,890
And all that's happening is
it's just forming a Dracovian
335
00:18:09,890 --> 00:18:11,639
Matrix inside there.
336
00:18:11,639 --> 00:18:14,180
Now, you still have to choose
which basis functions you want.
337
00:18:14,180 --> 00:18:15,680
And there's a lot
of basis functions
338
00:18:15,680 --> 00:18:19,130
you know that you know
the derivatives of.
339
00:18:19,130 --> 00:18:25,250
So there's some issues here
about what choice is best.
340
00:18:25,250 --> 00:18:27,321
And we'll talk a little
about that in a minute.
341
00:18:27,321 --> 00:18:28,820
One thing you have
to watch out for,
342
00:18:28,820 --> 00:18:30,403
is you don't want
your basis functions
343
00:18:30,403 --> 00:18:34,370
to be linearly dependent
because then you'll end up
344
00:18:34,370 --> 00:18:37,520
with indeterminate values a d.
345
00:18:37,520 --> 00:18:44,080
Different sets of d's will
give you exactly the same y.
346
00:18:44,080 --> 00:18:48,290
If these phi's-- if one of these
phi's is a linear combination
347
00:18:48,290 --> 00:18:52,455
of other phi's in your set, then
you could have different values
348
00:18:52,455 --> 00:18:54,830
of d's that would actually
correspond to exactly the same
349
00:18:54,830 --> 00:18:55,460
y.
350
00:18:55,460 --> 00:18:58,850
Because of that, when you a
Jacobian Matrix as part of it
351
00:18:58,850 --> 00:19:00,470
your Newton solve,
the Jacobian's
352
00:19:00,470 --> 00:19:02,630
going to be singular, and
then the whole thing's
353
00:19:02,630 --> 00:19:04,570
not going to work.
354
00:19:04,570 --> 00:19:07,337
So that's one thing
to watch out for.
355
00:19:07,337 --> 00:19:08,920
And actually, it
doesn't really matter
356
00:19:08,920 --> 00:19:12,610
if the functions are orthogonal
in the sense of being functions
357
00:19:12,610 --> 00:19:18,940
orthogonal, it's really whether
the vector, vk evaluated tm,
358
00:19:18,940 --> 00:19:21,049
if those are independent
of each other or not.
359
00:19:21,049 --> 00:19:23,340
I feel like we talked about
this before, one time, yes?
360
00:19:23,340 --> 00:19:23,840
Yes.
361
00:19:23,840 --> 00:19:25,710
All right.
362
00:19:25,710 --> 00:19:28,650
So anyway, as long as
the vk evaluate of tm,
363
00:19:28,650 --> 00:19:30,210
if those things
are vectors which
364
00:19:30,210 --> 00:19:31,970
are not linerally
dependent on each other,
365
00:19:31,970 --> 00:19:33,474
this should work fine.
366
00:19:33,474 --> 00:19:35,720
OK?
367
00:19:35,720 --> 00:19:36,220
All right.
368
00:19:40,680 --> 00:19:44,310
Now, really [INAUDIBLE].
369
00:19:44,310 --> 00:19:49,890
This one here often gets
to be kind of messy.
370
00:19:49,890 --> 00:19:53,860
Part of it is that inside
this norm of this vector,
371
00:19:53,860 --> 00:19:57,035
it's square, so you end up
getting squares of your d's.
372
00:19:57,035 --> 00:19:58,410
So, the whole
thing is definitely
373
00:19:58,410 --> 00:20:01,155
non-linear in d to start with.
374
00:20:01,155 --> 00:20:03,780
And then you have to be able to
evaluate all the integrals that
375
00:20:03,780 --> 00:20:04,310
come up.
376
00:20:04,310 --> 00:20:12,840
So this is really
sum over n, of g
377
00:20:12,840 --> 00:20:23,262
and of td squared that
you're trying to integrate.
378
00:20:23,262 --> 00:20:26,800
[INAUDIBLE] Right?
379
00:20:26,800 --> 00:20:29,770
So, you have a lot of
these function squared.
380
00:20:29,770 --> 00:20:31,430
You have to add them up.
381
00:20:31,430 --> 00:20:34,370
You could do it, but the algebra
gets a little complicated.
382
00:20:34,370 --> 00:20:36,900
So, this is not so
commonly done unless the g
383
00:20:36,900 --> 00:20:40,005
has some special form that makes
the algebra a little easier.
384
00:20:40,005 --> 00:20:42,130
But there's no reason you
can't do it in principle,
385
00:20:42,130 --> 00:20:45,030
but it just is a little bit of
a mess because you get a lot
386
00:20:45,030 --> 00:20:48,260
of integrals of cross
terms between the g's.
387
00:20:48,260 --> 00:20:51,040
And in the d's inside there.
388
00:20:51,040 --> 00:20:55,510
And so, not so commonly done.
389
00:20:55,510 --> 00:20:57,470
Though one case where
it is done a lot
390
00:20:57,470 --> 00:20:59,990
is in actually
quantum chemistry.
391
00:20:59,990 --> 00:21:02,780
Methods called variational
methods in quantum chemistry
392
00:21:02,780 --> 00:21:05,900
use real [INAUDIBLE] to
figure out the coefficients,
393
00:21:05,900 --> 00:21:08,520
and like your orbitals,
and stuff like that.
394
00:21:08,520 --> 00:21:10,080
So, it'd be certain methods.
395
00:21:10,080 --> 00:21:12,080
But they've actually gone
out of favor recently.
396
00:21:12,080 --> 00:21:14,500
So, you probably won't
use it that often.
397
00:21:14,500 --> 00:21:17,960
But in the past,
that was a big deal.
398
00:21:17,960 --> 00:21:22,070
Galerkin is used the
most of anything,
399
00:21:22,070 --> 00:21:25,950
so let's talk about
that one for a minute.
400
00:21:25,950 --> 00:21:29,210
Now, the concept, where does
this equation come from?
401
00:21:29,210 --> 00:21:30,840
I guess that's one thing.
402
00:21:30,840 --> 00:21:32,960
So maybe we can back
up and say, well,
403
00:21:32,960 --> 00:21:35,274
where did the
collocation come from?
404
00:21:35,274 --> 00:21:37,190
Collocation is actually
like a special version
405
00:21:37,190 --> 00:21:40,340
of this where I choose, instead
of these basis functions,
406
00:21:40,340 --> 00:21:42,540
I use delta functions.
407
00:21:42,540 --> 00:21:47,460
So, if I integrated with
delta functions, t minus tm.
408
00:21:47,460 --> 00:21:49,320
Another way to look
at the collocation
409
00:21:49,320 --> 00:21:55,440
is demanding that the integral
of delta function t minus tm g
410
00:21:55,440 --> 00:22:01,100
n a t dt is equal to zero.
411
00:22:01,100 --> 00:22:03,540
OK.
412
00:22:03,540 --> 00:22:07,550
So, now instead of
using delta functions,
413
00:22:07,550 --> 00:22:09,450
I'm using these functions.
414
00:22:09,450 --> 00:22:11,250
Basis functions.
415
00:22:11,250 --> 00:22:13,300
So I don't know if it's
particularly obvious why
416
00:22:13,300 --> 00:22:16,180
you'd use one or the other,
but anyway, you have a choice.
417
00:22:16,180 --> 00:22:19,560
And any time you do this kind
of integral with some number
418
00:22:19,560 --> 00:22:22,110
of functions, you get some
number of equations that can
419
00:22:22,110 --> 00:22:25,032
help you determine the d's.
420
00:22:25,032 --> 00:22:29,250
This particular choice
of the basis functions--
421
00:22:29,250 --> 00:22:34,740
one way to look at it is,
you can say, well, suppose
422
00:22:34,740 --> 00:22:43,540
my solution g g n,
suppose I could write this
423
00:22:43,540 --> 00:22:47,770
as a sum of some different
expansion coefficients.
424
00:22:59,760 --> 00:23:22,857
So I can relate this plus
425
00:23:22,857 --> 00:23:23,940
OK, so, there's two terms.
426
00:23:23,940 --> 00:23:26,870
One is the expansion of
the residuals in the basis
427
00:23:26,870 --> 00:23:30,909
that I'm using, and
one is all the rest
428
00:23:30,909 --> 00:23:32,700
going on to infinity
of all the other basis
429
00:23:32,700 --> 00:23:35,140
functions in the universe.
430
00:23:35,140 --> 00:23:40,230
And if I think my basis set
is very complete, that it kind
431
00:23:40,230 --> 00:23:41,820
of cover all the
kinds of functions
432
00:23:41,820 --> 00:23:45,270
I'm ever going to deal
with, both in y and in g,
433
00:23:45,270 --> 00:23:48,214
in the residual, then this
would be a reasonable thing,
434
00:23:48,214 --> 00:23:50,130
and you might expect
that this term-- if you'd
435
00:23:50,130 --> 00:23:52,990
pick big K big enough,
this might be small.
436
00:23:52,990 --> 00:23:54,780
So that's sort of
where the idea is.
437
00:23:54,780 --> 00:23:58,480
So then, if I think
this is small,
438
00:23:58,480 --> 00:24:00,540
then I want to make
sure I make this part as
439
00:24:00,540 --> 00:24:03,340
close to accurate as possible.
440
00:24:03,340 --> 00:24:07,370
And this condition is
basically doing that.
441
00:24:07,370 --> 00:24:13,750
It's saying that I want the
[? error ?] to be orthogonal
442
00:24:13,750 --> 00:24:17,030
to the basis functions,
in the sense that,
443
00:24:17,030 --> 00:24:19,655
for two functions, the integral
of the two functions like this,
444
00:24:19,655 --> 00:24:21,863
is like an inner product,
just like the inner product
445
00:24:21,863 --> 00:24:23,020
between two vectors.
446
00:24:23,020 --> 00:24:25,360
So I'm saying, like,
in the vector space,
447
00:24:25,360 --> 00:24:27,310
in the function space
that I'm working in,
448
00:24:27,310 --> 00:24:28,610
I don't want to have any error.
449
00:24:28,610 --> 00:24:30,430
That's what this is saying.
450
00:24:30,430 --> 00:24:34,390
But I'm going to have is that
there's other g terms which
451
00:24:34,390 --> 00:24:36,610
are the rest over here.
452
00:24:36,610 --> 00:24:42,360
These guys will not be
orthogonal to the first part.
453
00:24:42,360 --> 00:24:44,160
Does that make sense?
454
00:24:44,160 --> 00:24:47,270
So there's still some
error left, in my g.
455
00:24:47,270 --> 00:24:52,480
But the part in here, I can
make a good [? answer. ?]
456
00:24:52,480 --> 00:24:56,331
So that's where this equation
comes from conceptually.
457
00:24:56,331 --> 00:24:58,205
All right.
458
00:24:58,205 --> 00:25:00,705
And we'll come back-- when we
do least square [? setting, ?]
459
00:25:00,705 --> 00:25:03,090
we'll come back
to that same idea.
460
00:25:06,930 --> 00:25:08,970
So this is what the
equation looks like,
461
00:25:08,970 --> 00:25:15,310
and the disadvantage of this one
is it involves some integrals.
462
00:25:15,310 --> 00:25:18,520
And so you have to be able
to evaluate the integrals.
463
00:25:18,520 --> 00:25:20,080
All right.
464
00:25:20,080 --> 00:25:22,580
So that's the downside
of this function.
465
00:25:22,580 --> 00:25:24,580
So cleverly choosing
using your basis functions
466
00:25:24,580 --> 00:25:27,790
to make it easy to evaluate the
integrals is like the key thing
467
00:25:27,790 --> 00:25:28,960
to make this a good method.
468
00:25:31,790 --> 00:25:35,210
And just like we needed
analytical expressions
469
00:25:35,210 --> 00:25:38,450
for the derivatives here, now
we need analytical expressions
470
00:25:38,450 --> 00:25:39,550
for integrals.
471
00:25:39,550 --> 00:25:41,340
That we're going to
get from this guy.
472
00:25:41,340 --> 00:25:41,840
OK.
473
00:25:45,960 --> 00:25:49,170
So, let's think about what basis
functions we can choose that
474
00:25:49,170 --> 00:25:52,260
might make it easier to
evaluate the integrals.
475
00:25:52,260 --> 00:25:54,220
Suppose we can
write g explicitly.
476
00:25:54,220 --> 00:25:54,720
Like, this.
477
00:25:54,720 --> 00:26:02,307
So, g is equal to d
gx dyn/dt minus f/n.
478
00:26:05,020 --> 00:26:06,480
Suppose.
479
00:26:06,480 --> 00:26:07,560
OK?
480
00:26:07,560 --> 00:26:10,840
Then what integrals
am I going to get?
481
00:26:10,840 --> 00:26:15,420
Well, dyn/dt is the sum of
the derivatives of the basis
482
00:26:15,420 --> 00:26:22,290
functions, and y is
just this sum up there.
483
00:26:22,290 --> 00:26:24,102
Some of the basis functions.
484
00:26:24,102 --> 00:26:25,560
And so, I'm going
to have integrals
485
00:26:25,560 --> 00:26:41,210
that look like this is like
I have an integral phi j
486
00:26:41,210 --> 00:26:46,200
summation d phi prime.
487
00:26:46,200 --> 00:26:48,878
No, k.
488
00:26:48,878 --> 00:26:49,378
t.
489
00:26:53,710 --> 00:26:54,210
dt.
490
00:26:54,210 --> 00:26:56,490
That will be the integrals
from the first term.
491
00:26:59,320 --> 00:27:05,380
And then I'll have minus some
integrals from the second term
492
00:27:05,380 --> 00:27:05,990
here.
493
00:27:05,990 --> 00:27:19,250
So, phi j fn ft y, where y
is the sum-- it's actually
494
00:27:19,250 --> 00:27:20,447
like a matrix.
495
00:27:32,640 --> 00:27:33,500
All right.
496
00:27:33,500 --> 00:27:35,510
Because this y is a vector.
497
00:27:35,510 --> 00:27:38,110
It's all the yns.
498
00:27:38,110 --> 00:27:45,400
And so, the whole y vector
is d times the phi vector.
499
00:27:51,400 --> 00:27:53,837
All right.
500
00:27:53,837 --> 00:27:54,920
Where d's that big matrix.
501
00:27:54,920 --> 00:28:02,130
The elements are
[? dnk. ?] And so,
502
00:28:02,130 --> 00:28:04,920
if you cut these
integrals, this thing
503
00:28:04,920 --> 00:28:09,400
here is a summation
of [? dnk. ?]
504
00:28:09,400 --> 00:28:11,930
The integral of phi j.
505
00:28:11,930 --> 00:28:12,780
Phi k prime.
506
00:28:18,280 --> 00:28:20,920
And so, if you choose your
basis functions cleverly,
507
00:28:20,920 --> 00:28:24,230
you might be able to know all
these integrals analytically.
508
00:28:24,230 --> 00:28:24,937
OK?
509
00:28:24,937 --> 00:28:26,520
But if you don't
choose them cleverly,
510
00:28:26,520 --> 00:28:28,853
who knows what kind of horrible
mess you'll end up here.
511
00:28:28,853 --> 00:28:30,150
All right.
512
00:28:30,150 --> 00:28:32,400
Ideally, you really want to
get these all analytically
513
00:28:32,400 --> 00:28:35,029
so you don't have to do
numerical integration.
514
00:28:35,029 --> 00:28:36,820
As a loop inside, all
the rest of the work,
515
00:28:36,820 --> 00:28:38,893
you're going to do
in this problem.
516
00:28:38,893 --> 00:28:40,160
OK?
517
00:28:40,160 --> 00:28:43,310
And then, these ones also,
now knowing something about
518
00:28:43,310 --> 00:28:45,350
this is really
important or you may
519
00:28:45,350 --> 00:28:48,830
have to do the numerical
integration here, because this
520
00:28:48,830 --> 00:28:50,200
could be really a mess.
521
00:28:50,200 --> 00:28:52,490
You have a lot of phis
inside some function, which
522
00:28:52,490 --> 00:28:54,466
could be a non-linear
function of these guys.
523
00:28:54,466 --> 00:28:56,590
And so, in principle this
could be really horrible.
524
00:29:00,350 --> 00:29:05,384
So you may have to do numerical
quadrature for those guys.
525
00:29:05,384 --> 00:29:06,348
All right.
526
00:29:13,110 --> 00:29:17,770
OK, and so, if you do Galerkin's
method, a big part of that
527
00:29:17,770 --> 00:29:20,080
is thinking ahead of time,
oh, what basis function am I
528
00:29:20,080 --> 00:29:20,720
going to use?
529
00:29:20,720 --> 00:29:22,219
How can I make a
basis function so I
530
00:29:22,219 --> 00:29:24,550
can evaluate the
integrals easily,
531
00:29:24,550 --> 00:29:27,870
and then I might have
a chance to do it?
532
00:29:27,870 --> 00:29:29,322
All right, questions so far?
533
00:29:32,268 --> 00:29:34,240
AUDIENCE: Phi j
534
00:29:34,240 --> 00:29:36,990
PROFESSOR: Phi j.
535
00:29:36,990 --> 00:29:38,180
Sorry, phi j of t.
536
00:29:38,180 --> 00:29:39,940
Here, I'll get rid of the t's.
537
00:29:39,940 --> 00:29:41,800
These are both functions of t.
538
00:29:41,800 --> 00:29:44,640
You're integrating over t.
539
00:29:44,640 --> 00:29:47,770
And these integrals
from t0 to to t phi.
540
00:29:47,770 --> 00:29:48,870
Your domain.
541
00:29:52,550 --> 00:29:53,420
All right.
542
00:29:53,420 --> 00:29:55,010
Now in Galerkin's
method, in addition
543
00:29:55,010 --> 00:29:56,630
to these integral
equations, I still
544
00:29:56,630 --> 00:29:58,046
have the integrals,
the equations,
545
00:29:58,046 --> 00:29:59,570
that the boundary conditions.
546
00:29:59,570 --> 00:30:01,028
So I still have
some equations that
547
00:30:01,028 --> 00:30:06,134
look like collocation equations
that are evaluated at tm.
548
00:30:06,134 --> 00:30:08,990
Do I have that anywhere?
549
00:30:08,990 --> 00:30:10,550
Nowhere.
550
00:30:10,550 --> 00:30:12,960
I have an equation like this,
except it's not g, it's q,
551
00:30:12,960 --> 00:30:14,819
it's the boundary
condition equations
552
00:30:14,819 --> 00:30:16,610
have to be true at the
boundary conditions.
553
00:30:16,610 --> 00:30:17,110
Right?
554
00:30:17,110 --> 00:30:21,110
So they're the same as
before, q [? or just ?]
555
00:30:21,110 --> 00:30:33,540
before qn of dy/dt
evaluated at tn y of tn.
556
00:30:36,396 --> 00:30:37,570
tn.
557
00:30:37,570 --> 00:30:38,530
This is equal to 0.
558
00:30:38,530 --> 00:30:42,320
This is sort of the general way
to write a boundary condition.
559
00:30:42,320 --> 00:30:45,240
And so there'll be some special
ends, the boundaries, where I
560
00:30:45,240 --> 00:30:47,404
want to have extra conditions.
561
00:30:47,404 --> 00:30:48,820
I'll get some
equations like this.
562
00:30:48,820 --> 00:30:52,650
These have to be satisfied
in Galerkin's method as well
563
00:30:52,650 --> 00:30:55,650
and they will not be integrals.
564
00:30:55,650 --> 00:30:57,640
They are just that, tn.
565
00:30:57,640 --> 00:31:02,610
And so in addition to the
integral equations, that you
566
00:31:02,610 --> 00:31:05,110
have to solve over here, you
want these things to be zero
567
00:31:05,110 --> 00:31:07,818
and you also want to satisfy
the boundary conditions.
568
00:31:12,950 --> 00:31:15,930
So then, all these
methods, a big part of it
569
00:31:15,930 --> 00:31:20,890
gets to be cleverness of a basis
function, a choice of basis
570
00:31:20,890 --> 00:31:22,270
functions.
571
00:31:22,270 --> 00:31:26,690
And there's kind of two
families of approaches.
572
00:31:26,690 --> 00:31:33,280
So, one family is
global basis functions.
573
00:31:33,280 --> 00:31:37,240
So basis functions that are
defined on the whole domain.
574
00:31:37,240 --> 00:31:42,280
And there are special
ones, sines and cosines,
575
00:31:42,280 --> 00:31:45,630
Bessel functions,
all these functions
576
00:31:45,630 --> 00:31:48,210
inferred from your classes,
all the special functions.
577
00:31:48,210 --> 00:31:52,605
And a lot of them, the integrals
are known for a lot of cases.
578
00:31:52,605 --> 00:31:54,230
There might be special
tricks that make
579
00:31:54,230 --> 00:31:56,590
the integrals easy to evaluate.
580
00:31:56,590 --> 00:31:59,740
They can satisfy the boundary
conditions automatically.
581
00:31:59,740 --> 00:32:02,590
And some of them, for
example, many of the problems
582
00:32:02,590 --> 00:32:06,820
we have with like the heat
flow equation, so you have--
583
00:32:10,070 --> 00:32:12,704
actually it won't be d.
584
00:32:12,704 --> 00:32:14,245
I'm probably going
to get this wrong.
585
00:32:14,245 --> 00:32:16,372
It's, like, kappa or alpha.
586
00:32:16,372 --> 00:32:17,600
Which one is it?
587
00:32:17,600 --> 00:32:18,920
Alpha.
588
00:32:18,920 --> 00:32:25,810
Alpha d square of td, x
squared minus, there's
589
00:32:25,810 --> 00:32:30,016
some source of heat that might
depend on the temperature,
590
00:32:30,016 --> 00:32:31,640
and this has to be,
say, equal to zero.
591
00:32:34,980 --> 00:32:38,970
Right, does this seem what
you've seen in classes before?
592
00:32:38,970 --> 00:32:40,230
So this is a common one.
593
00:32:40,230 --> 00:32:47,040
So, in this case, you might try
t to be a sum of, say, sines.
594
00:32:47,040 --> 00:32:56,440
So [? dnk ?] sine of k
something something something.
595
00:32:56,440 --> 00:33:00,600
In there, and the
cleverness of this
596
00:33:00,600 --> 00:33:10,740
is that d squared
phi k dx squared is
597
00:33:10,740 --> 00:33:14,550
going to be equal to
some number times phi k.
598
00:33:14,550 --> 00:33:19,822
Because the second derivatives
of sines are also sines.
599
00:33:19,822 --> 00:33:20,760
Right?
600
00:33:20,760 --> 00:33:23,130
So all your
differentials will solve
601
00:33:23,130 --> 00:33:27,520
and in fact, the derivatives
will be really simple.
602
00:33:27,520 --> 00:33:30,272
Now, the q terms could
still be a horrible mess.
603
00:33:30,272 --> 00:33:32,480
For example, this could be
an Arrhenius thing where's
604
00:33:32,480 --> 00:33:33,999
the t's up in the exponent.
605
00:33:33,999 --> 00:33:35,790
And then the whole
thing might be horrible.
606
00:33:35,790 --> 00:33:36,700
Anyway.
607
00:33:36,700 --> 00:33:39,120
But at least the differential
part is, like, super easy.
608
00:33:39,120 --> 00:33:40,320
OK.
609
00:33:40,320 --> 00:33:43,910
And you might have a boundary
condition say, at one end,
610
00:33:43,910 --> 00:33:52,600
that dt/dx at someplace like d
final x final is equal to zero.
611
00:33:52,600 --> 00:33:53,100
Right?
612
00:33:53,100 --> 00:33:55,060
That might be like, you're
up against a x insulator.
613
00:33:55,060 --> 00:33:55,840
Or something like that.
614
00:33:55,840 --> 00:33:57,110
So, you don't have a heat flow.
615
00:33:57,110 --> 00:33:59,230
So that would be a boundary
condition you might have.
616
00:33:59,230 --> 00:34:01,730
And then, by cleverly choosing
your definition of the sines,
617
00:34:01,730 --> 00:34:04,020
you could force that
all the sines satisfy
618
00:34:04,020 --> 00:34:06,082
this derivative condition.
619
00:34:06,082 --> 00:34:12,021
OK, so, rescale your coordinates
so that ends up with pi over 2,
620
00:34:12,021 --> 00:34:13,770
an all sides of
everything and pi over two
621
00:34:13,770 --> 00:34:17,130
is always zero, the
derivative, so you're good.
622
00:34:17,130 --> 00:34:20,019
So, you can do some
clever trickiness
623
00:34:20,019 --> 00:34:21,810
to try to make the
problem easier to solve.
624
00:34:21,810 --> 00:34:25,050
And so, that's one whole
branch of these basis function
625
00:34:25,050 --> 00:34:27,570
methods, is clever
basis functions
626
00:34:27,570 --> 00:34:30,239
to match the special
problem you have.
627
00:34:30,239 --> 00:34:30,820
OK?
628
00:34:30,820 --> 00:34:33,510
So that's one option.
629
00:34:33,510 --> 00:34:36,844
Then, the other option is
the non-clever approach,
630
00:34:36,844 --> 00:34:38,760
where you just say, well,
I've got a computer.
631
00:34:38,760 --> 00:34:39,968
Who cares about being clever?
632
00:34:39,968 --> 00:34:41,460
Let's just brute force it.
633
00:34:41,460 --> 00:34:41,961
All right.
634
00:34:41,961 --> 00:34:44,168
And instead, you just want
to write a general method,
635
00:34:44,168 --> 00:34:45,270
any problem you can solve.
636
00:34:45,270 --> 00:34:47,020
And this is more or
less what COMSOL does.
637
00:34:47,020 --> 00:34:49,100
And you're going to do it.
638
00:34:49,100 --> 00:34:52,750
And so the distinction
here is that this kind
639
00:34:52,750 --> 00:34:56,620
of thing, this sine function,
has a value everywhere, all
640
00:34:56,620 --> 00:34:57,781
across the domain.
641
00:34:57,781 --> 00:34:59,530
So this is kind of
like a global function.
642
00:35:05,571 --> 00:35:06,070
OK?
643
00:35:08,690 --> 00:35:11,570
And the alternative
is to do local basis.
644
00:35:11,570 --> 00:35:13,480
Try to have basis
functions that are only
645
00:35:13,480 --> 00:35:15,547
defined in little tiny areas.
646
00:35:15,547 --> 00:35:17,630
And then, at least when I
integrate the integrals,
647
00:35:17,630 --> 00:35:19,140
I don't have to integrate
over the whole range.
648
00:35:19,140 --> 00:35:19,820
I don't have to have
to integrate right
649
00:35:19,820 --> 00:35:21,180
around my little basis function.
650
00:35:24,314 --> 00:35:26,980
So, this global basis function--
this is sort of similar to what
651
00:35:26,980 --> 00:35:28,104
we're doing, interpolation?
652
00:35:28,104 --> 00:35:30,520
Do you remember you could
use high order polynomial
653
00:35:30,520 --> 00:35:34,030
to interpolate, or you
could alternatively
654
00:35:34,030 --> 00:35:36,960
do a little piecewise
interpolations,
655
00:35:36,960 --> 00:35:39,820
say, with straight lines
between your points.
656
00:35:39,820 --> 00:35:43,150
And, you know, it's
not so clear, actually,
657
00:35:43,150 --> 00:35:44,770
which would be the
best way to do it.
658
00:35:44,770 --> 00:35:45,820
Or maybe you want to
do some combination.
659
00:35:45,820 --> 00:35:47,569
Do little parabolas
between little triples
660
00:35:47,569 --> 00:35:48,610
of points, or something.
661
00:35:48,610 --> 00:35:52,060
That might be a good
interpolation procedure.
662
00:35:52,060 --> 00:35:53,530
It's the same thing here.
663
00:35:53,530 --> 00:35:56,770
Some problems, you can find
a global basis set that works
664
00:35:56,770 --> 00:35:59,110
great and you should use it.
665
00:35:59,110 --> 00:36:00,880
In other problems
you might do better
666
00:36:00,880 --> 00:36:03,130
to just break up the domain
in little tiny pieces
667
00:36:03,130 --> 00:36:06,290
and then do simple little
polynomials or something
668
00:36:06,290 --> 00:36:08,036
in those little domains.
669
00:36:08,036 --> 00:36:09,684
All right.
670
00:36:09,684 --> 00:36:10,850
So this is the global basis.
671
00:36:10,850 --> 00:36:12,730
Let's see a local basis.
672
00:36:17,795 --> 00:36:19,290
It's OK to delete this?
673
00:36:30,650 --> 00:36:33,655
So, local basis functions.
674
00:36:39,672 --> 00:36:44,272
A really common choice for
these guys are the b-splines.
675
00:36:47,510 --> 00:36:49,690
And, in particular,
first order of b-splines
676
00:36:49,690 --> 00:36:56,085
and these functions have
the shape phi phi k,
677
00:36:56,085 --> 00:36:57,090
it looks like this.
678
00:37:14,560 --> 00:37:20,190
So, at zero, all the
way up to ti minus 1.
679
00:37:20,190 --> 00:37:23,860
Then it goes up to
one at ti, and then it
680
00:37:23,860 --> 00:37:27,930
goes down to zero
again, and goes out.
681
00:37:27,930 --> 00:37:28,430
OK.
682
00:37:28,430 --> 00:37:30,110
So this function is a b-spline.
683
00:37:30,110 --> 00:37:33,110
It's also called a tent function
because it looks like a tent.
684
00:37:33,110 --> 00:37:34,700
Some people call
it a hat function.
685
00:37:34,700 --> 00:37:36,920
I don't have a pointy head so it
doesn't look like a hat to me,
686
00:37:36,920 --> 00:37:38,670
but they call it a hat
function so I guess
687
00:37:38,670 --> 00:37:42,260
they must have hats like that.
688
00:37:42,260 --> 00:37:45,260
And this is a very
common basis set.
689
00:37:45,260 --> 00:37:48,560
And the nice thing about
this basis function
690
00:37:48,560 --> 00:37:54,198
is, it's zero except in this
little tiny domain around it.
691
00:37:54,198 --> 00:37:55,540
OK?
692
00:37:55,540 --> 00:37:59,310
And it's the only basis
function in the whole set that
693
00:37:59,310 --> 00:38:01,960
has a non-zero value of ti.
694
00:38:01,960 --> 00:38:04,820
And so if you want the function
to equal something in ti,
695
00:38:04,820 --> 00:38:07,190
it's going to be equal to
the coefficient of this basis
696
00:38:07,190 --> 00:38:07,856
function, right?
697
00:38:07,856 --> 00:38:11,630
Because we're going
to write y n of t
698
00:38:11,630 --> 00:38:17,960
is equal to summation
[? dnk ?] phi k of t.
699
00:38:17,960 --> 00:38:23,500
And so, if I really care
about yn evaluated at ti,
700
00:38:23,500 --> 00:38:27,610
the only one basis function
this whole sum is going
701
00:38:27,610 --> 00:38:29,240
to have a non-zero value there.
702
00:38:29,240 --> 00:38:37,070
So that's going to
be equal to dn dnk, k
703
00:38:37,070 --> 00:38:40,031
where this is the special k.
704
00:38:40,031 --> 00:38:42,625
k prime matches up to ti.
705
00:38:44,880 --> 00:38:47,130
OK, so there's one base
function that looks like this.
706
00:38:47,130 --> 00:38:50,870
There's another one
over here that's
707
00:38:50,870 --> 00:38:52,760
the one base center
on this point,
708
00:38:52,760 --> 00:38:54,630
and there's another
one over here.
709
00:38:54,630 --> 00:38:56,354
So on this point,
it's [INAUDIBLE]..
710
00:38:56,354 --> 00:38:58,520
And I have as many basis
functions as I have points.
711
00:39:01,070 --> 00:39:04,610
So this is like a way I
discretize the problem,
712
00:39:04,610 --> 00:39:09,030
but I've kept my solution
as continuous function
713
00:39:09,030 --> 00:39:10,820
because my y--
714
00:39:10,820 --> 00:39:12,650
that's the sum of these guys--
715
00:39:12,650 --> 00:39:14,490
has a value everywhere.
716
00:39:14,490 --> 00:39:16,487
It's a continuous function.
717
00:39:16,487 --> 00:39:18,570
The way this is written
it looks like it might not
718
00:39:18,570 --> 00:39:20,850
be differentiable
at all the points
719
00:39:20,850 --> 00:39:23,640
because it has all
these kinks, but there's
720
00:39:23,640 --> 00:39:27,660
a clever trick you can do
to deal with the kinks.
721
00:39:27,660 --> 00:39:30,264
So, actually, it's
not a problem.
722
00:39:30,264 --> 00:39:32,122
AUDIENCE: So, for
these basis functions,
723
00:39:32,122 --> 00:39:34,080
how would you define if
you had the boundaries?
724
00:39:34,080 --> 00:39:35,821
Like t0?
725
00:39:35,821 --> 00:39:38,320
PROFESSOR: So, you'll have a
basis function at the very end.
726
00:39:38,320 --> 00:39:40,460
Suppose this is t0 here.
727
00:39:40,460 --> 00:39:43,940
You have one like that.
728
00:39:43,940 --> 00:39:46,830
OK, so it's just
a half of a tent.
729
00:39:46,830 --> 00:39:49,250
Can you get a
[? sparse ?] d-matrix?
730
00:39:49,250 --> 00:39:49,920
Yes.
731
00:39:49,920 --> 00:39:51,710
So locality is really
good because it
732
00:39:51,710 --> 00:39:56,690
makes the Jacobian matrix
sparse, the overall problem.
733
00:39:56,690 --> 00:40:02,510
So, when I compute the
integrals of this thing,
734
00:40:02,510 --> 00:40:14,160
for example, the integral of phi
i of t, phi i minus 1 of t dt,
735
00:40:14,160 --> 00:40:19,880
this turns out to be equal
to 1 or 2 times ti minus ti
736
00:40:19,880 --> 00:40:20,880
[INAUDIBLE].
737
00:40:23,863 --> 00:40:26,280
No.
738
00:40:26,280 --> 00:40:27,100
I take that back.
739
00:40:27,100 --> 00:40:29,150
Just one half.
740
00:40:29,150 --> 00:40:31,920
Just half.
741
00:40:31,920 --> 00:40:35,750
So it is like a brilliant thing.
742
00:40:35,750 --> 00:40:36,820
Is that right?
743
00:40:36,820 --> 00:40:39,550
It does have a ti [INAUDIBLE],,
not a [INAUDIBLE]..
744
00:40:39,550 --> 00:40:40,960
I'll have to double check.
745
00:40:40,960 --> 00:40:43,190
Possibly including delta t.
746
00:40:43,190 --> 00:40:45,050
I can't remember.
747
00:40:45,050 --> 00:40:47,800
All right.
748
00:40:47,800 --> 00:40:51,760
But the integrals
are very analytical,
749
00:40:51,760 --> 00:40:54,490
and only certain
ones are non-zero.
750
00:40:54,490 --> 00:41:00,790
So only one that the two is when
the two is differ by one unit,
751
00:41:00,790 --> 00:41:02,260
do they have a
non-zero integral?
752
00:41:02,260 --> 00:41:03,718
All the rest of
them are non-zeros.
753
00:41:03,718 --> 00:41:05,350
So, when I write
down these equations,
754
00:41:05,350 --> 00:41:07,414
I mean, many, many,
many, many zeros,
755
00:41:07,414 --> 00:41:09,580
so it looks horrible when
I write Galerkin's method,
756
00:41:09,580 --> 00:41:11,871
I get so many integrals, but
actually a zillion of them
757
00:41:11,871 --> 00:41:15,550
are zero, and then there's
a bunch of special tricks
758
00:41:15,550 --> 00:41:19,900
I can do that make it
even better than that.
759
00:41:19,900 --> 00:41:22,210
OK?
760
00:41:22,210 --> 00:41:23,810
So then, the Jacobian sparse.
761
00:41:23,810 --> 00:41:27,130
So, that'll save you a lot of
time in linear algebra, which
762
00:41:27,130 --> 00:41:28,575
allows you use a lot of points.
763
00:41:28,575 --> 00:41:31,585
So, you can use a very large
basis set because you end up
764
00:41:31,585 --> 00:41:33,210
with sparse Jacobians,
I mean, it's not
765
00:41:33,210 --> 00:41:35,350
going to fill your memory
storing all the elements
766
00:41:35,350 --> 00:41:38,270
in the Jacobian, even if the
number of points is very large.
767
00:41:38,270 --> 00:41:42,930
And also, there's vast numerical
solution methods for those.
768
00:41:42,930 --> 00:41:46,650
So that's the idea of this.
769
00:41:46,650 --> 00:41:49,800
I guess should we try to
carry one of these out?
770
00:41:49,800 --> 00:41:52,050
You guys are [INAUDIBLE]
trying to do a lot of algebra
771
00:41:52,050 --> 00:41:53,124
on the board?
772
00:41:53,124 --> 00:41:55,520
Do you think I can do
the algebra on the board?
773
00:41:55,520 --> 00:41:57,710
That's the real question.
774
00:41:57,710 --> 00:41:58,210
All right.
775
00:42:22,907 --> 00:42:24,490
Should I do it for
collocation, or you
776
00:42:24,490 --> 00:42:27,390
guys confident you
can do collocation?
777
00:42:27,390 --> 00:42:30,292
You're all right
with collocation?
778
00:42:30,292 --> 00:42:31,500
You're fine with collocation.
779
00:42:31,500 --> 00:42:32,290
Great.
780
00:42:32,290 --> 00:42:33,200
OK.
781
00:42:33,200 --> 00:42:36,320
So we'll just go
right into Galerkin.
782
00:42:36,320 --> 00:42:38,076
So, Galerkin.
783
00:42:38,076 --> 00:42:39,605
That way, we use a local basis.
784
00:42:45,860 --> 00:42:50,450
So, most of the equations we
have to solve are this type.
785
00:42:50,450 --> 00:43:05,720
Phi j t times the residual
function of summation [? dnk ?]
786
00:43:05,720 --> 00:43:06,833
phi k prime.
787
00:43:12,990 --> 00:43:21,720
Summation [? dnk ?]
of phi k and t.
788
00:43:21,720 --> 00:43:22,540
Is equal to zero.
789
00:43:22,540 --> 00:43:24,540
All right, we have a lot
of equations like that.
790
00:43:24,540 --> 00:43:26,610
We're trying to
find the d's that
791
00:43:26,610 --> 00:43:30,760
are going to force all
these integrals to be zero.
792
00:43:30,760 --> 00:43:31,260
OK?
793
00:43:31,260 --> 00:43:35,290
So we have a lot of different
j's we're going to try.
794
00:43:35,290 --> 00:43:38,430
We want this to be
true for all the n's.
795
00:43:38,430 --> 00:43:41,580
And then we're going to adjust
these [? dnk's ?] to try
796
00:43:41,580 --> 00:43:44,260
to force this to be zero.
797
00:43:44,260 --> 00:43:44,760
All right?
798
00:43:44,760 --> 00:43:45,920
That's the main problem.
799
00:43:45,920 --> 00:43:47,294
And then, on top
of this, there's
800
00:43:47,294 --> 00:43:49,054
some equations with
boundary conditions.
801
00:43:52,330 --> 00:43:56,600
So, now we have to look
at what the form is.
802
00:43:56,600 --> 00:44:01,380
If the form is, oh, I erased
it, if I can make this explicit
803
00:44:01,380 --> 00:44:05,640
in the derivatives, which
I can do very often,
804
00:44:05,640 --> 00:44:16,210
for example, this could
be dyn/dt minus [? fm. ?]
805
00:44:16,210 --> 00:44:20,800
So then, dyn/dt is just this.
806
00:44:20,800 --> 00:44:23,300
And then I'll have another term,
[? fm, ?] which will depend
807
00:44:23,300 --> 00:44:25,890
on that [INAUDIBLE].
808
00:44:25,890 --> 00:44:31,380
And so the integral
phi j, and I'll just
809
00:44:31,380 --> 00:44:43,170
remind you that phi j
is not equal to zero.
810
00:44:43,170 --> 00:44:52,821
If tj minus 1 is
less than tj plus 1.
811
00:44:52,821 --> 00:44:53,320
All right.
812
00:44:53,320 --> 00:44:56,290
That's the only places
where my local basis
813
00:44:56,290 --> 00:44:58,040
function is non-zero.
814
00:44:58,040 --> 00:44:59,660
That's where the tent is.
815
00:44:59,660 --> 00:45:01,790
And all the rest of it's zero.
816
00:45:01,790 --> 00:45:03,920
So when I have this
integral, originally have it
817
00:45:03,920 --> 00:45:07,680
t0 to t-final, but
actually I could replace
818
00:45:07,680 --> 00:45:12,466
this with tj minus 1 to tj.
819
00:45:12,466 --> 00:45:13,840
And it's just the same.
820
00:45:13,840 --> 00:45:16,330
Because of all the integral
outside that domain is zero.
821
00:45:16,330 --> 00:45:18,460
Because this function is zero.
822
00:45:18,460 --> 00:45:18,970
All right?
823
00:45:18,970 --> 00:45:20,636
So at least I have a
small little domain
824
00:45:20,636 --> 00:45:22,577
to do the integral over.
825
00:45:22,577 --> 00:45:25,815
AUDIENCE: [INAUDIBLE] tj?
826
00:45:25,815 --> 00:45:26,690
PROFESSOR: tj plus 1.
827
00:45:26,690 --> 00:45:28,370
Thank you.
828
00:45:28,370 --> 00:45:28,955
Yes.
829
00:45:28,955 --> 00:45:29,460
Yes.
830
00:45:29,460 --> 00:45:29,960
Plus 1.
831
00:45:29,960 --> 00:45:31,696
Yes.
832
00:45:31,696 --> 00:45:34,160
Is that all right?
833
00:45:34,160 --> 00:45:34,870
OK.
834
00:45:34,870 --> 00:45:36,550
So, I have this
integral, and then
835
00:45:36,550 --> 00:45:43,005
I have this times the
derivative term, so it's dnk
836
00:45:43,005 --> 00:45:50,792
dk prime minus the same
integral, j minus 1.
837
00:45:50,792 --> 00:45:52,480
J plus 1.
838
00:45:52,480 --> 00:46:01,348
Phi j sorry [? fm ?]
[INAUDIBLE] summation of dn.
839
00:46:01,348 --> 00:46:03,828
k phi.
840
00:46:03,828 --> 00:46:05,316
k [INAUDIBLE] t.
841
00:46:11,264 --> 00:46:11,764
OK.
842
00:46:14,740 --> 00:46:21,395
Now, this derivative
of phi k, well, I just
843
00:46:21,395 --> 00:46:22,270
told you what it was.
844
00:46:22,270 --> 00:46:25,330
It's just like the
[INAUDIBLE] set functions.
845
00:46:25,330 --> 00:46:28,600
So, this integral,
you'll know analytically.
846
00:46:28,600 --> 00:46:32,590
Like, it's either 0 or it's 1/2.
847
00:46:32,590 --> 00:46:34,090
Maybe minus 1/2.
848
00:46:34,090 --> 00:46:35,920
Whatever, but it's
nothing complicated,
849
00:46:35,920 --> 00:46:37,670
and you'll just know
it right off the bat.
850
00:46:37,670 --> 00:46:40,300
So this is all
that can be known.
851
00:46:40,300 --> 00:46:41,260
And sparse.
852
00:46:41,260 --> 00:46:45,517
It's only going to be when k is
equal to j minus 1 or j plus 1
853
00:46:45,517 --> 00:46:47,600
that this will be non-zero,
and all the rest of it
854
00:46:47,600 --> 00:46:49,424
would be zero.
855
00:46:49,424 --> 00:46:50,380
OK?
856
00:46:50,380 --> 00:46:52,600
So, that's mostly zeros.
857
00:46:52,600 --> 00:46:56,850
This one over
here, in principle,
858
00:46:56,850 --> 00:47:00,511
I have quite a huge sum
here, k equals 1 to k,
859
00:47:00,511 --> 00:47:02,010
where I have all
my basis functions,
860
00:47:02,010 --> 00:47:05,610
which is all my points where
I put my little tents down.
861
00:47:05,610 --> 00:47:09,010
So I have a domain and
I've parked a lot of tents
862
00:47:09,010 --> 00:47:11,310
all on the domain, that's
my basis functions.
863
00:47:11,310 --> 00:47:12,060
And I'm trying to
figure out, sort of,
864
00:47:12,060 --> 00:47:13,635
how high the tent poles are.
865
00:47:13,635 --> 00:47:15,350
On all those tents.
866
00:47:15,350 --> 00:47:17,920
And my tunnel functions, the
sum of all the heights of those
867
00:47:17,920 --> 00:47:18,669
tents.
868
00:47:18,669 --> 00:47:19,168
OK?
869
00:47:21,720 --> 00:47:25,770
But this guy is only non-zero
in this little domain,
870
00:47:25,770 --> 00:47:30,766
and these guys, most of them
are zero in that domain,
871
00:47:30,766 --> 00:47:33,140
because they're mostly tents
that are far away from where
872
00:47:33,140 --> 00:47:35,440
my special tent phi j is.
873
00:47:35,440 --> 00:47:37,990
OK, so I'm going
to draw a picture.
874
00:47:37,990 --> 00:47:43,730
Here's the domain
from t0 to t-final.
875
00:47:43,730 --> 00:47:47,060
Over here is my tent
corresponding to phi j,
876
00:47:47,060 --> 00:47:50,590
which is centered around tj.
877
00:47:50,590 --> 00:47:52,780
Hence, this function.
878
00:47:52,780 --> 00:47:55,280
And then, these guys
are all the other tents.
879
00:47:55,280 --> 00:47:57,350
There's a tent here,
there's a tent like this,
880
00:47:57,350 --> 00:47:59,641
there's another tent like
this, another tent like this.
881
00:47:59,641 --> 00:48:00,556
All these tents.
882
00:48:00,556 --> 00:48:01,930
Those are all the
basis functions
883
00:48:01,930 --> 00:48:05,020
starting from k
equals 1 and going up.
884
00:48:05,020 --> 00:48:08,400
All of these guys are zero in
this domain because they all
885
00:48:08,400 --> 00:48:10,440
have zero tail.
886
00:48:10,440 --> 00:48:14,070
So, almost all of these,
the f's doesn't really
887
00:48:14,070 --> 00:48:15,840
pick up anything from
any of those guys
888
00:48:15,840 --> 00:48:17,080
in the domain I care about.
889
00:48:17,080 --> 00:48:19,830
The only domain I care about
is this domain right here.
890
00:48:19,830 --> 00:48:24,080
And this whole sum is all zero
except for a few special k's
891
00:48:24,080 --> 00:48:28,830
when k is equal to j
minus 1, j, or j plus 1.
892
00:48:28,830 --> 00:48:31,182
So I only have to worry
about three terms.
893
00:48:31,182 --> 00:48:34,900
The inside contributing to
the y in the region of t
894
00:48:34,900 --> 00:48:36,893
that I care about.
895
00:48:36,893 --> 00:48:38,820
All right?
896
00:48:38,820 --> 00:48:44,850
So, when I want to compute
the Jacobian with respect
897
00:48:44,850 --> 00:48:49,540
to d, the only terms here
where the Jacobian's going
898
00:48:49,540 --> 00:48:59,350
to be non-zero are for d
I have a Jacobian, which
899
00:48:59,350 --> 00:49:05,190
is the derivative of
this whole thing, ddd nk.
900
00:49:05,190 --> 00:49:06,140
Right?
901
00:49:06,140 --> 00:49:17,955
And that's only non-zero if
k is equal to j minus 1 j,
902
00:49:17,955 --> 00:49:19,440
or j plus 1.
903
00:49:23,400 --> 00:49:27,017
You guys see that?
904
00:49:27,017 --> 00:49:28,350
How many people do not see this?
905
00:49:31,722 --> 00:49:32,805
How many people are lying?
906
00:49:36,300 --> 00:49:40,130
This is a really important
concept for the locality,
907
00:49:40,130 --> 00:49:41,880
so this is the advantage
of a local basis,
908
00:49:41,880 --> 00:49:43,421
that the Jacobian
will turn out to be
909
00:49:43,421 --> 00:49:46,830
really sparse because it's only
non-zero in this special case.
910
00:49:46,830 --> 00:49:50,660
And you might have
1,000 points, 1,000
911
00:49:50,660 --> 00:49:53,740
of these little tent, 1,000
basis functions in the sum.
912
00:49:53,740 --> 00:49:55,782
But only three of
them are non-zero.
913
00:49:55,782 --> 00:49:58,610
For four each n.
914
00:49:58,610 --> 00:50:00,490
So, actually, it's 3 times n.
915
00:50:00,490 --> 00:50:03,180
[INAUDIBLE] non-zero.
916
00:50:03,180 --> 00:50:04,720
Is that all right?
917
00:50:04,720 --> 00:50:07,210
Because it just
depends on the k.
918
00:50:07,210 --> 00:50:09,050
Three of the ks are non-zero.
919
00:50:09,050 --> 00:50:11,954
There's 1,000 of these
guys [INAUDIBLE]..
920
00:50:11,954 --> 00:50:13,790
Yep.
921
00:50:13,790 --> 00:50:17,000
So that's the big
trick of locality.
922
00:50:17,000 --> 00:50:19,250
And then, also, because
these integration ranges
923
00:50:19,250 --> 00:50:22,220
are so small, because you
choose your points really finely
924
00:50:22,220 --> 00:50:25,670
spaced, then you might get
away with simple polynomial
925
00:50:25,670 --> 00:50:29,176
expansions of f, for example.
926
00:50:29,176 --> 00:50:31,050
Around the points, or
[INAUDIBLE] expansions,
927
00:50:31,050 --> 00:50:32,610
there are all kinds
of little tricks.
928
00:50:32,610 --> 00:50:35,910
Or you could even do quadrature
and determine some points,
929
00:50:35,910 --> 00:50:38,682
do a [INAUDIBLE] quadrature
to evaluate the integral.
930
00:50:38,682 --> 00:50:40,682
But you only need a few
points, because you know
931
00:50:40,682 --> 00:50:43,570
it's a little tiny range of dt.
932
00:50:43,570 --> 00:50:46,870
And you would hope that
you've chosen so many fees, so
933
00:50:46,870 --> 00:50:49,690
many time points, that
your function doesn't
934
00:50:49,690 --> 00:50:52,450
change much from one
time point to the next.
935
00:50:52,450 --> 00:50:55,409
So, more or less, first
order is constant.
936
00:50:55,409 --> 00:50:56,950
Your function's
constant with respect
937
00:50:56,950 --> 00:50:59,710
to t in this little tiny
domain, and then maybe
938
00:50:59,710 --> 00:51:00,850
has a little slope.
939
00:51:00,850 --> 00:51:02,520
And then, if you're
really being fancy,
940
00:51:02,520 --> 00:51:04,270
you might be able to
put a parabola on it.
941
00:51:04,270 --> 00:51:05,440
But it's not going
to change that much.
942
00:51:05,440 --> 00:51:07,330
If it's changing a
lot, that's telling you
943
00:51:07,330 --> 00:51:08,705
you don't have
enough time points
944
00:51:08,705 --> 00:51:11,274
and you should go back and put
some more basis functions in.
945
00:51:11,274 --> 00:51:12,550
And then you can get
a good [INAUDIBLE]..
946
00:51:12,550 --> 00:51:14,091
Because what we're,
really doing here
947
00:51:14,091 --> 00:51:16,240
is we're using this basis set.
948
00:51:16,240 --> 00:51:18,970
If I add up these guys,
what I'm really doing
949
00:51:18,970 --> 00:51:21,040
is piecewise linear
interpolation
950
00:51:21,040 --> 00:51:22,900
between all the points.
951
00:51:22,900 --> 00:51:28,512
So I'm approximating
my y versus time.
952
00:51:28,512 --> 00:51:29,720
It's going to look like this.
953
00:51:37,390 --> 00:51:37,890
All right.
954
00:51:37,890 --> 00:51:39,910
It's piecewise
linear, because that's
955
00:51:39,910 --> 00:51:41,660
the only thing you can
make from adding up
956
00:51:41,660 --> 00:51:43,100
a bunch of straight
line segments.
957
00:51:43,100 --> 00:51:45,080
A bunch of tents is a
bunch of straight lines.
958
00:51:45,080 --> 00:51:48,410
And so this is what the
approximate function is.
959
00:51:48,410 --> 00:51:50,390
Well, you can see,
this is really bad
960
00:51:50,390 --> 00:51:52,050
if these functions are too
different from each other.
961
00:51:52,050 --> 00:51:53,300
But if they're all like this--
962
00:52:01,292 --> 00:52:02,750
and you think,
well, maybe it's not
963
00:52:02,750 --> 00:52:05,310
so bad to use piecewise linear.
964
00:52:05,310 --> 00:52:06,024
Yeah.
965
00:52:06,024 --> 00:52:09,190
AUDIENCE: [INAUDIBLE]
confused at where
966
00:52:09,190 --> 00:52:11,900
we're using [INAUDIBLE].
967
00:52:12,440 --> 00:52:13,981
PROFESSOR: Inside
[? f ?] solve, what
968
00:52:13,981 --> 00:52:16,760
is trying to solve
for the d's, it's
969
00:52:16,760 --> 00:52:21,270
solving each of this giant set
of the equations, a whole lot
970
00:52:21,270 --> 00:52:25,840
of equations like this, they
all come in to a gigantic f.
971
00:52:25,840 --> 00:52:28,914
That's the f of d that need
to make equal to to zero.
972
00:52:28,914 --> 00:52:30,330
And so, I'm
[? burying ?] the d's,
973
00:52:30,330 --> 00:52:32,780
I need the Jacobian
of that gigantic f.
974
00:52:32,780 --> 00:52:35,390
Now, the problem,
because it's gigantic,
975
00:52:35,390 --> 00:52:38,630
I need a lot of points, a
lot of closely spaced points
976
00:52:38,630 --> 00:52:40,700
to make my function look smooth.
977
00:52:40,700 --> 00:52:43,500
That means the number of base
functions is really huge.
978
00:52:43,500 --> 00:52:47,180
So the Jacobian principle
is huge number squared.
979
00:52:47,180 --> 00:52:50,510
I have 1,000 points, say,
discretizing my domain.
980
00:52:50,510 --> 00:52:52,176
And then it's 1,000
by 1,000 Jacobian.
981
00:52:52,176 --> 00:52:54,800
It might be really hard to solve
it, or cost a lot of CPU time,
982
00:52:54,800 --> 00:52:56,180
or use a lot of memory.
983
00:52:56,180 --> 00:52:58,940
But, fortunately, almost
all the elements are zero.
984
00:52:58,940 --> 00:53:03,130
So, it's like, has a
sparsity of 0.3% or something
985
00:53:03,130 --> 00:53:05,030
is the occupancy.
986
00:53:05,030 --> 00:53:06,770
So, it's almost all
completely sparse,
987
00:53:06,770 --> 00:53:10,560
and therefore you can solve
it even though it's gigantic.
988
00:53:10,560 --> 00:53:11,750
Yes.
989
00:53:11,750 --> 00:53:16,170
AUDIENCE: [INAUDIBLE]
basis function [INAUDIBLE]
990
00:53:16,170 --> 00:53:24,104
the Jacobian [INAUDIBLE]
should be [INAUDIBLE]
991
00:53:24,104 --> 00:53:25,520
PROFESSOR: We're
trying to find d,
992
00:53:25,520 --> 00:53:29,800
so we're really trying
to solve a problem that's
993
00:53:29,800 --> 00:53:34,440
f of d is equal to zero.
994
00:53:34,440 --> 00:53:37,060
That's our fundamental
problem we're trying to solve.
995
00:53:37,060 --> 00:53:38,490
We're doing Galerkin.
996
00:53:38,490 --> 00:53:40,745
We have something equal to
zero, and it depends on d.
997
00:53:40,745 --> 00:53:42,120
And we're trying
to find the d's.
998
00:53:42,120 --> 00:53:44,660
So this is really the function
we're trying to solve.
999
00:53:44,660 --> 00:53:46,410
In order to evaluate
the elements in this,
1000
00:53:46,410 --> 00:53:48,000
we have to compute a
whole bunch of integrals
1001
00:53:48,000 --> 00:53:49,041
with the Galerkin method.
1002
00:53:49,041 --> 00:53:50,452
So that's the complexity.
1003
00:53:50,452 --> 00:53:52,410
But the basis functions,
we know what they are.
1004
00:53:52,410 --> 00:53:54,360
We've pre-specified them.
1005
00:53:54,360 --> 00:53:57,560
So, the whole question
is just what the d's are.
1006
00:53:57,560 --> 00:54:01,830
And the d's get multiplied
by a whole lot of integrals.
1007
00:54:01,830 --> 00:54:03,200
Is this all right?
1008
00:54:03,200 --> 00:54:06,112
I think I may have
misunderstood the question.
1009
00:54:06,112 --> 00:54:07,570
So you have f of
d is [INAUDIBLE],,
1010
00:54:07,570 --> 00:54:10,400
so therefore we probably want
to know j with respect to d.
1011
00:54:15,080 --> 00:54:16,980
OK.
1012
00:54:16,980 --> 00:54:19,980
Time ran out before
clarity was achieved.
1013
00:54:19,980 --> 00:54:22,890
We'll try to achieve clarity
on Wednesday morning.