WEBVTT 00:00:00.000 --> 00:00:01.950 The following content is provided 00:00:01.950 --> 00:00:06.120 by MIT OpenCourseWare under a Creative Commons license. 00:00:06.120 --> 00:00:08.230 Additional information about our license 00:00:08.230 --> 00:00:10.490 and MIT OpenCourseWare in general 00:00:10.490 --> 00:00:11.930 is available at ocw.mit.edu. 00:00:16.620 --> 00:00:18.120 PROFESSOR: This morning's SIAM News. 00:00:20.820 --> 00:00:22.460 Well, it's an article on something 00:00:22.460 --> 00:00:26.210 that's not totally up to date because it's a method that 00:00:26.210 --> 00:00:30.140 was developed 50 years ago called alternating direction 00:00:30.140 --> 00:00:32.950 and it was exactly developed-- first 00:00:32.950 --> 00:00:39.330 you did an iteration in the x direction. 00:00:39.330 --> 00:00:42.610 Of course, we're tridiagonal along every row, 00:00:42.610 --> 00:00:44.460 we're one-dimensional. 00:00:44.460 --> 00:00:49.810 And then the second half-step is an iteration 00:00:49.810 --> 00:00:51.040 in the y direction. 00:00:51.040 --> 00:00:54.360 Again, tridiagonal, so the iteration's very fast. 00:00:54.360 --> 00:00:58.200 You flip-- alternating direction describes it perfectly. 00:00:58.200 --> 00:01:00.210 And that method is still used. 00:01:00.210 --> 00:01:05.450 It got developed in the oil reservoir simulation world. 00:01:05.450 --> 00:01:07.410 That is, Houston. 00:01:07.410 --> 00:01:12.740 And the article says the evolution 00:01:12.740 --> 00:01:18.190 of techniques for oil reservoir simulation has continued. 00:01:18.190 --> 00:01:24.210 And then mentions 3-- so it went from this alternating direction 00:01:24.210 --> 00:01:32.430 to line overrelaxation, LSOR, overrelaxation, 00:01:32.430 --> 00:01:36.250 and I only spoke briefly about that. 00:01:36.250 --> 00:01:37.600 What's that word line mean? 00:01:37.600 --> 00:01:42.160 That means that we're taking whole rows at once, 00:01:42.160 --> 00:01:43.520 working with blocks. 00:01:43.520 --> 00:01:48.130 And I'll add some notes about that today. 00:01:48.130 --> 00:01:52.160 So that was the next step, line SOR. 00:01:52.160 --> 00:01:56.770 Then to the Newton-Krylov schemes, that's what's coming, 00:01:56.770 --> 00:02:03.440 Krylov, Conjugate gradients, with ILU type preconditioners. 00:02:03.440 --> 00:02:06.390 So again, we're talking about preconditioners 00:02:06.390 --> 00:02:13.120 and one important choice is this incomplete LU. 00:02:13.120 --> 00:02:19.970 So notes on Gauss-Seidel and I should maybe add incomplete LU. 00:02:19.970 --> 00:02:24.330 By the way, experiments badly needed here too. 00:02:24.330 --> 00:02:32.090 Because incomplete LU has this tolerance and as you 00:02:32.090 --> 00:02:37.470 move the tolerance you get closer or far away 00:02:37.470 --> 00:02:43.530 to the exact L and U, but you get faster or slower. 00:02:43.530 --> 00:02:45.580 So what's the balance? 00:02:45.580 --> 00:02:47.920 What's the right tolerance? 00:02:47.920 --> 00:02:51.620 How do these methods compare-- and overall, 00:02:51.620 --> 00:02:55.260 how do these iterative methods that we're speaking about 00:02:55.260 --> 00:02:58.390 compare with these direct methods? 00:02:58.390 --> 00:03:04.670 So we really are in this new chapter, solving large systems, 00:03:04.670 --> 00:03:10.080 facing a whole lot of possible new experiments. 00:03:10.080 --> 00:03:14.940 So just to continue the history or the future as described 00:03:14.940 --> 00:03:19.230 in SIAM News, so there was alternating direction, line 00:03:19.230 --> 00:03:20.460 SOR. 00:03:20.460 --> 00:03:28.700 Then to Krylov schemes with ILU preconditioners in use today. 00:03:28.700 --> 00:03:31.120 So this is a big use area. 00:03:31.120 --> 00:03:38.170 And the multiscale, multigrid solvers that are quote, 00:03:38.170 --> 00:03:41.140 now starting to come of age. 00:03:41.140 --> 00:03:47.180 So that's a third topic for today's lecture, multigrid. 00:03:47.180 --> 00:03:52.300 That-- of course, it's been around for some time, 00:03:52.300 --> 00:03:56.730 but it involved, especially at the start, 00:03:56.730 --> 00:04:01.420 more effort in programming. 00:04:04.040 --> 00:04:09.460 So it didn't quickly-- I mean, it gave great results 00:04:09.460 --> 00:04:12.970 on model problems. 00:04:12.970 --> 00:04:15.290 On other problems, too, but still 00:04:15.290 --> 00:04:19.790 it kind of hasn't yet, but is penetrating 00:04:19.790 --> 00:04:23.430 into production code, where the oil reservoir 00:04:23.430 --> 00:04:26.420 industry or other big industries would use it. 00:04:26.420 --> 00:04:33.520 So here you see, really, a pretty fair picture of where we 00:04:33.520 --> 00:04:36.800 are today with large solvers. 00:04:36.800 --> 00:04:39.760 These are very important. 00:04:39.760 --> 00:04:44.830 Minimum degree-- if the size is not overwhelming, 00:04:44.830 --> 00:04:48.700 direct methods are going to be right up there. 00:04:48.700 --> 00:04:53.850 And these are the pure iterations, 00:04:53.850 --> 00:05:01.050 stationary iterations that, with this idea, this incomplete LU 00:05:01.050 --> 00:05:04.620 idea, get pretty good. 00:05:04.620 --> 00:05:07.410 But the formula's always simple there. 00:05:07.410 --> 00:05:13.730 The new x comes from the old x by the same iterations. 00:05:13.730 --> 00:05:18.450 Whereas, if we use the ideas there as 00:05:18.450 --> 00:05:25.060 preconditioners and look to see-- so the point was, 00:05:25.060 --> 00:05:28.840 and I'll repeat it again, that Gauss-Seidel and Jacobi, 00:05:28.840 --> 00:05:32.760 the standard iterations, were quite satisfactory 00:05:32.760 --> 00:05:34.810 on high frequencies. 00:05:34.810 --> 00:05:37.960 They damped those out quite well in the error, 00:05:37.960 --> 00:05:42.290 but they don't tackle the low frequency, the smooth part 00:05:42.290 --> 00:05:44.560 of the error, very efficiently. 00:05:44.560 --> 00:05:49.470 And that's what multigrid does incredibly well, so it's 00:05:49.470 --> 00:05:51.890 a nice mixture of the two. 00:05:51.890 --> 00:05:55.300 These are the smoothers, this is the change 00:05:55.300 --> 00:06:00.170 of scale, change of grid that's coming now and will come. 00:06:00.170 --> 00:06:06.190 So all this will be my subject for this week and next, 00:06:06.190 --> 00:06:07.350 prior to spring break. 00:06:07.350 --> 00:06:13.390 And I think you will have enough to go on to do experiments. 00:06:13.390 --> 00:06:15.570 It could be highly interesting. 00:06:15.570 --> 00:06:20.810 OK, so that's the SIAM News report. 00:06:20.810 --> 00:06:26.910 OK, what do I want to say about minimum degree? 00:06:26.910 --> 00:06:29.440 And more will go on the web. 00:06:29.440 --> 00:06:37.100 Just a few more notes about what we did on Monday after class. 00:06:37.100 --> 00:06:43.710 If you take a matrix like that, you take a matrix 00:06:43.710 --> 00:06:47.940 and-- how is it described in sparse format? 00:06:47.940 --> 00:06:52.290 Sparse format you can see-- normally you 00:06:52.290 --> 00:06:54.490 don't have to see it, it's usually 00:06:54.490 --> 00:06:58.230 inside the sparse MATLAB, but if you 00:06:58.230 --> 00:07:03.970 want to see what's happening, you could ask for i, j, and s. 00:07:03.970 --> 00:07:05.930 Now what are those? 00:07:05.930 --> 00:07:10.660 What will that produce? i will be 00:07:10.660 --> 00:07:14.140 the list of nonzeros, the list of the row 00:07:14.140 --> 00:07:16.420 numbers of the nonzeros. 00:07:16.420 --> 00:07:23.060 So i will be-- maybe I'll write it as a row so I'm transposing, 00:07:23.060 --> 00:07:28.820 So the row numbers would be row 1, row 2, not row 3. 00:07:28.820 --> 00:07:32.270 Row 1, 2, 3, row 2, 3. 00:07:35.320 --> 00:07:39.890 The column numbers would be-- these 00:07:39.890 --> 00:07:42.040 are going to be the pairs (i, j). 00:07:42.040 --> 00:07:48.010 So that's column 1 twice, column 2 three times. 00:07:48.010 --> 00:07:49.070 Column 3 twice. 00:07:52.520 --> 00:07:58.440 And what's s? s is the actual numbers in position (i, j), 00:07:58.440 --> 00:08:00.340 is the number 2. 00:08:00.340 --> 00:08:04.240 In position (2, 1) is the number minus 1. 00:08:04.240 --> 00:08:08.720 So those numbers, minus 1, 2, minus 1 is that column. 00:08:08.720 --> 00:08:12.810 And then minus 1, 2 is that column. 00:08:12.810 --> 00:08:16.290 So of course, we've got all the information in the matrix. 00:08:16.290 --> 00:08:23.190 We know every nonzero position and we know what that entry is. 00:08:23.190 --> 00:08:27.180 And of course, our matrices are much bigger than this one, 00:08:27.180 --> 00:08:35.370 but already we can see one useful point-- point 00:08:35.370 --> 00:08:41.170 being that key word-- that the column numbers, this-- I mean, 00:08:41.170 --> 00:08:45.940 imagine we have a large matrix, so this j is not 00:08:45.940 --> 00:08:52.610 very efficiently recorded here because what's j? 00:08:52.610 --> 00:08:55.600 We're looking at nonzeros a column at a time, 00:08:55.600 --> 00:09:00.090 so of course, we'll have a few ones for the nonzeros in column 00:09:00.090 --> 00:09:07.340 1 and then some twos and then some threes and some fours, 00:09:07.340 --> 00:09:14.650 but all this row is-- the real information in that row 00:09:14.650 --> 00:09:20.290 is a pointer to-- in other words, 00:09:20.290 --> 00:09:24.330 I don't have to repeat 2 three times. 00:09:24.330 --> 00:09:30.780 So really, if I put inside here a compressed version 00:09:30.780 --> 00:09:36.720 of j, a pointer, would be a short factor that just has a 1. 00:09:36.720 --> 00:09:42.860 It says look in position 1 for the start of column 1. 00:09:42.860 --> 00:09:48.610 Then a 3-- see, I didn't need that because that just 00:09:48.610 --> 00:09:49.830 continued column 1. 00:09:49.830 --> 00:09:52.660 That just repeated the j I already had. 00:09:52.660 --> 00:09:57.700 But this tells me that in the third position 00:09:57.700 --> 00:09:59.850 I start down column 2. 00:09:59.850 --> 00:10:04.230 And this would tell me that in position number 4, 5, 6, 00:10:04.230 --> 00:10:09.540 I guess, I start down column 3. 00:10:09.540 --> 00:10:11.030 You see the point of that 6. 00:10:11.030 --> 00:10:15.670 That 6 picks out-- yes-- 1, 2, 3, 4, 5, 00:10:15.670 --> 00:10:20.400 the sixth row number is the beginning 00:10:20.400 --> 00:10:22.670 of information on column 3. 00:10:22.670 --> 00:10:25.110 And then it's conventional to have 00:10:25.110 --> 00:10:31.010 a pointer, 8, to say finish. 00:10:31.010 --> 00:10:36.100 So that's pointing to empty space. 00:10:36.100 --> 00:10:40.080 So in other words, j got compressed to 1, 3, 6, 8 00:10:40.080 --> 00:10:44.690 and in a large problem it would get seriously compressed. 00:10:44.690 --> 00:10:54.800 OK, so that's the form in which the code keeps the matrix 00:10:54.800 --> 00:11:00.490 and does the reorderings. 00:11:00.490 --> 00:11:05.190 Let me just mention that the opposite of this 00:11:05.190 --> 00:11:09.630 would create the matrix out of these i, j, s. 00:11:09.630 --> 00:11:12.870 What command would that be? 00:11:12.870 --> 00:11:18.450 You could create the matrix A out of-- the command sparse 00:11:18.450 --> 00:11:19.960 would be good. 00:11:19.960 --> 00:11:25.110 Sparse of the inputs, now, would be i, j, s. 00:11:25.110 --> 00:11:31.670 If I input i, j, and s then that lab creates a matrix. 00:11:31.670 --> 00:11:33.900 And sometimes I may want to include 00:11:33.900 --> 00:11:41.970 the m and n, the shape of the matrix, as further parameters, 00:11:41.970 --> 00:11:45.340 but actually here I wouldn't have to. 00:11:45.340 --> 00:11:48.970 So this is the opposite command from this one 00:11:48.970 --> 00:11:52.920 that we're interested in. 00:11:52.920 --> 00:11:53.770 You can imagine. 00:11:53.770 --> 00:12:04.250 Suppose by using minimum degree or some other decision, 00:12:04.250 --> 00:12:07.010 I've eliminated up to a certain point 00:12:07.010 --> 00:12:11.420 and I want to put the remaining columns in a different order. 00:12:11.420 --> 00:12:15.950 I can do that just by playing with the pointers. 00:12:15.950 --> 00:12:21.130 So it's a very efficient structure for sparse matrices. 00:12:27.690 --> 00:12:31.110 And then comes the question, OK, what's the good order? 00:12:34.270 --> 00:12:39.150 I was surprised to realize how open a problem that 00:12:39.150 --> 00:12:47.720 still is, even from the expert who's developing the key code. 00:12:52.470 --> 00:12:56.340 So approximate minimum degree, of course, by that word 00:12:56.340 --> 00:13:00.370 you see that it allows freedom. 00:13:00.370 --> 00:13:02.140 There are also decisions to be made 00:13:02.140 --> 00:13:06.340 when minimum degree is a tie between several nodes, 00:13:06.340 --> 00:13:08.080 as it commonly is. 00:13:08.080 --> 00:13:12.430 And it's rather nice to get a MATLAB movie that 00:13:12.430 --> 00:13:18.260 shows the order in which nodes are 00:13:18.260 --> 00:13:24.670 eliminated, edges are removed from the graph, from the mesh. 00:13:24.670 --> 00:13:27.040 You'll see that. 00:13:27.040 --> 00:13:30.900 All right, I'm ready to go ahead now to some comment 00:13:30.900 --> 00:13:32.190 on Gauss-Seidel. 00:13:32.190 --> 00:13:35.110 Maybe I'll put that here. 00:13:35.110 --> 00:13:37.780 In fact, maybe I'll just take the same matrix. 00:13:37.780 --> 00:13:43.880 So now I want to remember, what is the Gauss-Seidel method 00:13:43.880 --> 00:13:45.830 for A*x equal b. 00:13:45.830 --> 00:13:47.430 What's the iteration now? 00:13:47.430 --> 00:13:52.230 And again, A will be the same: 2, minus 1, zero; minus 1, 00:13:52.230 --> 00:13:53.890 2, minus 1. 00:13:53.890 --> 00:13:57.620 My favorite, OK. 00:13:57.620 --> 00:14:00.850 So all these iterations, you remember, 00:14:00.850 --> 00:14:04.720 are splittings of the matrix. 00:14:04.720 --> 00:14:07.470 Some of the matrix goes on the left-hand side. 00:14:07.470 --> 00:14:09.700 That's the preconditioner. 00:14:09.700 --> 00:14:11.940 So it's P, the preconditioner that's 00:14:11.940 --> 00:14:16.970 sort of close to A in some sense, but easy to work with, 00:14:16.970 --> 00:14:21.550 multiplies the new thing and on the other side 00:14:21.550 --> 00:14:25.890 is A minus A-- the rest of the matrix-- that 00:14:25.890 --> 00:14:31.900 multiplies the old plus the right-hand side b. 00:14:31.900 --> 00:14:36.750 Let's just remember again, that if we converge to the point 00:14:36.750 --> 00:14:42.300 where this is the same as this, then we have the right answer. 00:14:42.300 --> 00:14:45.510 We have the right answer because if that's the same as that, 00:14:45.510 --> 00:14:49.780 P*x is the same as P*x, this comes over on the other side, 00:14:49.780 --> 00:14:51.830 A*x equals b. 00:14:51.830 --> 00:14:56.870 So when it converges, it converges to the right answer, 00:14:56.870 --> 00:14:59.300 but you remember that the key equation was 00:14:59.300 --> 00:15:06.920 that the new error is the old error multiplied 00:15:06.920 --> 00:15:14.240 by this matrix I minus P inverse A. That's the iteration matrix, 00:15:14.240 --> 00:15:17.190 you could say. 00:15:17.190 --> 00:15:21.400 I just get it when I multiply both sides by P inverse. 00:15:21.400 --> 00:15:30.240 And so the problem is choose P so that this is easy to do 00:15:30.240 --> 00:15:34.310 and at the same time, this has small eigenvalues 00:15:34.310 --> 00:15:35.820 or as small as you can get. 00:15:35.820 --> 00:15:38.810 So the Gauss-Seidel is a particular choice, 00:15:38.810 --> 00:15:43.430 which takes the lower-- P is the lower triangular part here. 00:15:43.430 --> 00:15:48.030 So P is, so I just thought I'd better write down, explicitly: 00:15:48.030 --> 00:15:55.020 Gauss-Seidel takes that, with zeros there, 00:15:55.020 --> 00:16:00.200 as multiplying-- that's P. And what's P minus A? 00:16:00.200 --> 00:16:02.730 It's the rest. 00:16:02.730 --> 00:16:05.640 And it's been moved to the other side 00:16:05.640 --> 00:16:10.740 so the rest is going to be the strictly upper triangular part. 00:16:10.740 --> 00:16:12.840 And because it's moved over to the other side 00:16:12.840 --> 00:16:14.430 it'll have a plus sign. 00:16:14.430 --> 00:16:15.750 It'll be 1 and 1. 00:16:20.600 --> 00:16:24.740 x_k plus b. 00:16:24.740 --> 00:16:31.810 I just write it out so that you see, totally explicitly, 00:16:31.810 --> 00:16:41.420 how being triangular makes the solution step immediate. 00:16:41.420 --> 00:16:45.060 Because the first equation will tell us the first component, 00:16:45.060 --> 00:16:46.680 right? 00:16:46.680 --> 00:16:49.340 The first equation, because it's triangular 00:16:49.340 --> 00:16:51.150 there's only an entry here. 00:16:51.150 --> 00:16:57.860 And by the way, all these codes, including Tim Davis's 00:16:57.860 --> 00:17:00.870 minimum degree codes, their very first step 00:17:00.870 --> 00:17:05.100 is reorder the equations if necessary 00:17:05.100 --> 00:17:09.870 to guarantee that the main diagonal has no zeros. 00:17:09.870 --> 00:17:13.250 We want to know that in advance. 00:17:13.250 --> 00:17:16.320 So that you just do, let's assume it. 00:17:16.320 --> 00:17:18.690 Of course, for us it happens automatically. 00:17:18.690 --> 00:17:21.320 OK, so that's not zero. 00:17:21.320 --> 00:17:24.580 And that first equation gives us the first component, 00:17:24.580 --> 00:17:28.690 then we know the first component so we use it here, 00:17:28.690 --> 00:17:32.470 and the second equation tells us what the second component is. 00:17:32.470 --> 00:17:35.940 We use that second component in the third equation 00:17:35.940 --> 00:17:40.590 to find the third component. 00:17:40.590 --> 00:17:45.700 So there's no loss of speed compared to diagonal 00:17:45.700 --> 00:17:50.030 and actually, it's faster because the storage-- 00:17:50.030 --> 00:17:55.210 we're changing-- we're using the new first component to find 00:17:55.210 --> 00:17:59.000 the second and the new second component to find the third, 00:17:59.000 --> 00:18:04.920 so we can overwrite x_k by x_(k+1). 00:18:08.330 --> 00:18:08.900 Let's see. 00:18:08.900 --> 00:18:13.490 I guess if I had space and had prepared I would figure out 00:18:13.490 --> 00:18:16.700 what this matrix is. 00:18:16.700 --> 00:18:19.000 Maybe you could do that. 00:18:19.000 --> 00:18:23.080 Figure out what-- here's P. It's invertible. 00:18:23.080 --> 00:18:26.170 P inverse is going to be quite simple. 00:18:26.170 --> 00:18:30.080 Actually, it would be extremely simple. 00:18:30.080 --> 00:18:32.800 You'll be able to see its eigenvalues immediately. 00:18:35.590 --> 00:18:38.940 We could stop to do it, but I think if you do it, 00:18:38.940 --> 00:18:41.960 it's more valuable. 00:18:41.960 --> 00:18:45.510 So we would find that its eigenvalues were below 1 00:18:45.510 --> 00:18:52.360 and that they were the squares of the Jacobi eigenvalues, 00:18:52.360 --> 00:18:55.890 so the method is twice as fast as Jacobi. 00:18:55.890 --> 00:19:02.180 But it has the same good feature of damping the high frequencies 00:19:02.180 --> 00:19:07.420 and the same bad feature of not very much damping 00:19:07.420 --> 00:19:12.080 the low, smooth part of the error. 00:19:12.080 --> 00:19:16.180 Now we're ready for multigrid, which is intended to solve it. 00:19:16.180 --> 00:19:28.570 So multigrid is going to be-- the point is, by changing grid, 00:19:28.570 --> 00:19:38.270 by changing from the fine grid to a coarse grid, 00:19:38.270 --> 00:19:42.060 the smooth oscillation doesn't look so smooth. 00:19:42.060 --> 00:19:48.620 On the coarse grid, its effective frequency 00:19:48.620 --> 00:19:55.060 is effectively doubled and it's moving over in the direction 00:19:55.060 --> 00:19:58.750 where the smoother can attack it. 00:19:58.750 --> 00:20:01.100 We'll see that happen. 00:20:01.100 --> 00:20:04.450 So now, I have to, really-- this begins the lecture 00:20:04.450 --> 00:20:07.470 on multigrid, which is going to take, 00:20:07.470 --> 00:20:09.270 certainly, today and Friday. 00:20:12.700 --> 00:20:17.010 There's a very good book by Briggs and others. 00:20:24.110 --> 00:20:29.910 He wrote the first edition of the book 00:20:29.910 --> 00:20:33.470 and coauthored the second one. 00:20:33.470 --> 00:20:36.560 And it's beautifully short and clear and simple 00:20:36.560 --> 00:20:45.750 and this presentation will follow the same exposition 00:20:45.750 --> 00:20:47.282 that the book does. 00:20:51.400 --> 00:21:00.380 So the key idea is to see what-- how 00:21:00.380 --> 00:21:03.260 to go from a fine grid problem, so this 00:21:03.260 --> 00:21:07.670 is on the fine grid with step size h. 00:21:07.670 --> 00:21:09.810 So this A_h is our problem. 00:21:09.810 --> 00:21:16.490 A_h is A. A_h, b sub h, we're looking 00:21:16.490 --> 00:21:20.430 for u_h on the fine grid. 00:21:20.430 --> 00:21:24.040 Fine grid means lots of mesh points, 00:21:24.040 --> 00:21:26.840 lots of unknowns, big problem. 00:21:26.840 --> 00:21:30.760 OK, so the idea is going to be, somehow, 00:21:30.760 --> 00:21:35.220 to-- let me just start with two grids. 00:21:39.310 --> 00:21:46.820 Well, you can see what you do on the fine grid at the start. 00:21:46.820 --> 00:21:47.700 You iterate. 00:21:47.700 --> 00:21:51.210 You use Gauss-Seidel or Jacobi, whatever. 00:21:51.210 --> 00:21:52.940 Maybe three times. 00:21:52.940 --> 00:21:54.930 Maybe three iterations, but don't 00:21:54.930 --> 00:21:56.650 continue to a thousand iterations, 00:21:56.650 --> 00:21:59.450 it's a waste of time. 00:21:59.450 --> 00:22:03.530 Then, here is the real-- this is the multigrid part. 00:22:07.460 --> 00:22:09.580 The multigrid part is going-- you 00:22:09.580 --> 00:22:14.070 get an answer after a few iterations, 00:22:14.070 --> 00:22:16.040 after you've smoothed it. 00:22:16.040 --> 00:22:21.390 You compute this residual, which is 00:22:21.390 --> 00:22:27.410 the amount you're wrong in the equation, the difference 00:22:27.410 --> 00:22:32.950 between the right side and the left side. 00:22:32.950 --> 00:22:35.710 I can put h's here too, to emphasize. 00:22:35.710 --> 00:22:39.430 This is residuals being computed with what we have, 00:22:39.430 --> 00:22:42.920 which is on the fine grid. 00:22:42.920 --> 00:22:45.620 Now here's the change of grid. 00:22:45.620 --> 00:22:50.160 Number two says, restrict to a coarse grid. 00:22:50.160 --> 00:22:57.000 Take that r_h, that residual which is probably a bit smooth, 00:22:57.000 --> 00:23:01.310 and move it to the coarse grid. 00:23:01.310 --> 00:23:05.490 So the coarse grid we identify by the fact 00:23:05.490 --> 00:23:11.630 that its mesh width is twice as big, 2h. 00:23:11.630 --> 00:23:14.730 And so we're going to need some restriction, 00:23:14.730 --> 00:23:18.550 so the input to this multigrid is 00:23:18.550 --> 00:23:24.880 to decide on a restriction matrix, how shall we take 00:23:24.880 --> 00:23:27.800 values that are given on a fine grid 00:23:27.800 --> 00:23:33.340 and restrict them to a coarse grid? 00:23:33.340 --> 00:23:37.270 Of course, one restriction is-- and it's not 00:23:37.270 --> 00:23:42.670 that bad-- is just to take the values of r 00:23:42.670 --> 00:23:46.300 that are on the coarse grid at every other grid point, 00:23:46.300 --> 00:23:48.910 use those. 00:23:48.910 --> 00:23:54.020 Or we could take into account the neighbors. 00:23:54.020 --> 00:23:56.090 You see the question. 00:23:56.090 --> 00:24:00.130 So this is coming now in this restriction. 00:24:00.130 --> 00:24:04.070 So the restriction is, restriction R, 00:24:04.070 --> 00:24:11.110 this is fine to coarse, and we have this question 00:24:11.110 --> 00:24:18.310 of use neighbors or not? 00:24:18.310 --> 00:24:21.760 That will enter. 00:24:21.760 --> 00:24:25.680 Well, the decision on what the matrix R is. 00:24:25.680 --> 00:24:27.130 That's the key. 00:24:27.130 --> 00:24:30.840 OK, so we choose an R, we restrict, 00:24:30.840 --> 00:24:33.630 and we have now a half-size problem. 00:24:33.630 --> 00:24:37.570 Well, half-size in 1D, quarter-size in 2D, 00:24:37.570 --> 00:24:44.130 eighth-size in 3D because the mesh width has 00:24:44.130 --> 00:24:46.290 doubled in every coordinate. 00:24:48.880 --> 00:24:51.030 Now I've put solve. 00:24:51.030 --> 00:24:54.080 I better put quotes around that. 00:24:54.080 --> 00:24:58.500 Well, solve the problem on the coarse mesh. 00:24:58.500 --> 00:25:02.420 So the idea is that that's much faster. 00:25:02.420 --> 00:25:04.540 Well, you might say, what problem? 00:25:04.540 --> 00:25:10.080 Well, we have to create, somehow, 00:25:10.080 --> 00:25:12.730 the matrix on the coarse mesh. 00:25:12.730 --> 00:25:17.400 That's not necessarily given to us. 00:25:17.400 --> 00:25:20.070 We start the problem with A, A_h, 00:25:20.070 --> 00:25:23.030 the matrix on the fine grid. 00:25:23.030 --> 00:25:28.450 Can I just mention that this is probably, 00:25:28.450 --> 00:25:33.740 this v-cycle-- that letter v is supposed 00:25:33.740 --> 00:25:38.300 to suggest going down to the coarse mesh and back up 00:25:38.300 --> 00:25:41.360 to the fine mesh. 00:25:41.360 --> 00:25:47.540 And the standard notation in all the 00:25:47.540 --> 00:25:54.400 multigrid books is a capital V. And why do I use a small v? 00:25:54.400 --> 00:26:00.010 This is my educational contribution. 00:26:00.010 --> 00:26:06.410 Capital V is appropriate when you have several meshes. 00:26:06.410 --> 00:26:11.560 You go to 2h, you go to 4h, you go to 8h, back to 4h, back 00:26:11.560 --> 00:26:13.390 to 2h, back to h. 00:26:13.390 --> 00:26:16.130 So a deeper multigrid. 00:26:16.130 --> 00:26:19.250 So it would look more complicated. 00:26:19.250 --> 00:26:24.370 I would repeat this idea instead of solve here. 00:26:24.370 --> 00:26:27.420 Instead of solve at the 2h level, which of course, I'm 00:26:27.420 --> 00:26:29.560 not going to do exactly anyway. 00:26:29.560 --> 00:26:36.530 But if I was in a big V-cycle I would 00:26:36.530 --> 00:26:42.000 iterate a few times to smooth here, just as I did here. 00:26:42.000 --> 00:26:47.620 I would iterate using weighted Jacobi or something 00:26:47.620 --> 00:26:51.490 and then go down to the 4h mesh. 00:26:51.490 --> 00:26:53.540 So restrict to the 4h mesh. 00:26:53.540 --> 00:26:57.840 So you can see that I would stay in this little loop 00:26:57.840 --> 00:27:03.570 to get to the bottom, 8h, and then I would start back up. 00:27:03.570 --> 00:27:07.780 So I think it's a good idea to use a small letter 00:27:07.780 --> 00:27:14.010 v to tell us-- rather than saying two-grid capital 00:27:14.010 --> 00:27:19.000 V-cycle, I'm just going to use a small v to signal 00:27:19.000 --> 00:27:21.870 right away that it's two grids. 00:27:21.870 --> 00:27:23.690 Fine, coarse, fine. 00:27:23.690 --> 00:27:31.330 OK, so we have some solution, not necessarily exact 00:27:31.330 --> 00:27:34.550 and actually, there isn't that much point 00:27:34.550 --> 00:27:39.450 in getting it exact because what we want 00:27:39.450 --> 00:27:42.380 is to move toward the right answer. 00:27:44.890 --> 00:27:48.440 And notice that I'm looking here at the error, 00:27:48.440 --> 00:27:54.170 rather than looking at A_(2h), u_(2h), the actual solution. 00:27:54.170 --> 00:27:58.180 What I'm computing with is the correction term. 00:27:58.180 --> 00:28:03.150 So this is the correction term that I find. 00:28:03.150 --> 00:28:07.650 So this is the residual over here. 00:28:07.650 --> 00:28:11.220 this is the A_(2h) that I still have to choose, 00:28:11.220 --> 00:28:15.740 and then solving that will give me a correction, 00:28:15.740 --> 00:28:21.370 but that correction is on coarse mesh. 00:28:21.370 --> 00:28:29.210 I've only had that correction E_(2h) defined on the 2h mesh. 00:28:29.210 --> 00:28:35.000 So now comes the other part of multigrid. 00:28:35.000 --> 00:28:40.210 Get back to the fine mesh, climb back up. 00:28:40.210 --> 00:28:44.820 I take that E_(2h) and apply an interpolation. 00:28:44.820 --> 00:28:49.210 I need to create an interpolation. 00:28:49.210 --> 00:28:51.860 This is going to be coarse to fine. 00:28:56.420 --> 00:29:00.560 And I have to decide how to do that. 00:29:00.560 --> 00:29:05.280 Once I've decided it that gives me 00:29:05.280 --> 00:29:08.170 a correction at the fine mesh level, which 00:29:08.170 --> 00:29:13.470 is where I'm really working, so I make the correction 00:29:13.470 --> 00:29:16.100 and I have a better answer. 00:29:16.100 --> 00:29:21.720 And probably, I iterate a few times-- 00:29:21.720 --> 00:29:25.590 that would be called a post-smoother-- 00:29:25.590 --> 00:29:26.510 on that correction. 00:29:26.510 --> 00:29:32.790 And maybe that's where I stop or maybe I repeat little v-cycles, 00:29:32.790 --> 00:29:36.380 but that's not brilliant. 00:29:36.380 --> 00:29:40.930 If you're going to do multiple-- if you're 00:29:40.930 --> 00:29:44.740 going to repeat multigrid, it's much more efficient 00:29:44.740 --> 00:29:52.430 to move to more and more coarse meshes 00:29:52.430 --> 00:29:56.710 because the calculations on these 00:29:56.710 --> 00:30:02.310 are way faster than on the fine mesh. 00:30:02.310 --> 00:30:07.370 You see, rather than-- I much prefer 00:30:07.370 --> 00:30:10.770 to go way down and back up than a lot of v-cycles. 00:30:10.770 --> 00:30:16.780 I mean, that would be a small w-cycle. 00:30:16.780 --> 00:30:19.370 And not brilliant. 00:30:19.370 --> 00:30:25.610 Much better to use a big V-cycle or a big W-cycle. 00:30:25.610 --> 00:30:28.830 There's a place for W-cycles, but they're capital W-cycles, 00:30:28.830 --> 00:30:31.450 because you want to get down where 00:30:31.450 --> 00:30:35.400 it is very fast, very inexpensive and in fact, 00:30:35.400 --> 00:30:40.060 so efficient that multigrid achieves 00:30:40.060 --> 00:30:48.940 this holy grail of giving you an answer with whatever accuracy 00:30:48.940 --> 00:30:54.560 you want-- giving you an answer in O of n operations. 00:30:54.560 --> 00:30:57.290 n being the matrix size. 00:30:57.290 --> 00:31:00.430 Capital N squared in our example. 00:31:00.430 --> 00:31:07.340 So that's the fantastic result from multigrid. 00:31:07.340 --> 00:31:17.450 So I'll come back to this, but just say if the size of A 00:31:17.450 --> 00:31:22.770 is n, which is capital N in 1D, capital N squared, capital N 00:31:22.770 --> 00:31:35.260 cubed, then the multigrid works in O of n flops, 00:31:35.260 --> 00:31:36.390 floating points operations. 00:31:39.480 --> 00:31:44.985 Not even log n, which we think of for the-- n log 00:31:44.985 --> 00:31:52.470 n we think of for the FFT, for the Fast Fourier Transform. 00:31:52.470 --> 00:31:58.730 But here's it's actually order of n. 00:31:58.730 --> 00:32:01.650 That really is a goal worth achieving. 00:32:06.680 --> 00:32:09.870 So we know what the smoothers might be. 00:32:09.870 --> 00:32:14.580 It's the I and the R that are new. 00:32:14.580 --> 00:32:16.890 And then the analysis of convergence, 00:32:16.890 --> 00:32:19.130 why does it converge? 00:32:19.130 --> 00:32:22.220 But to use the method, and-- so by the way, 00:32:22.220 --> 00:32:25.030 projects using multigrid are totally welcome also. 00:32:28.860 --> 00:32:33.720 So let me take the interpolation I. So what's that? 00:32:33.720 --> 00:32:35.110 Coarse to fine. 00:32:35.110 --> 00:32:39.850 Let me imagine I'm in 1D, so there's the x direction. 00:32:39.850 --> 00:32:42.650 Here's one end of the interval, here's the other end 00:32:42.650 --> 00:32:45.380 of the interval and suppose my boundary conditions are zero. 00:32:50.370 --> 00:32:58.630 So this is the coarse mesh and let me put in here, 00:32:58.630 --> 00:33:02.670 make it the fine mesh. 00:33:02.670 --> 00:33:04.400 So at this point, I have an answer 00:33:04.400 --> 00:33:08.150 on the coarse mesh, which satisfies the boundary 00:33:08.150 --> 00:33:14.890 condition, and let's say it's there, there, and there. 00:33:17.580 --> 00:33:23.170 OK, and I need values on the fine mesh at the other points. 00:33:23.170 --> 00:33:26.730 Of course, I'm going to use these values 00:33:26.730 --> 00:33:27.930 when I interpolate them. 00:33:27.930 --> 00:33:31.200 That word interpolation means-- implies, 00:33:31.200 --> 00:33:34.440 keep what you have on the given points. 00:33:34.440 --> 00:33:38.030 And I'm just going to do linear interpolation. 00:33:38.030 --> 00:33:46.270 So halfway there, halfway there, and halfway there. 00:33:46.270 --> 00:33:51.810 So I'm inputting-- ignoring the boundary conditions, the zeros, 00:33:51.810 --> 00:33:59.410 I'm-- it's going to be 5 by 2 I think, 00:33:59.410 --> 00:34:07.990 this matrix I in this example will be 5 by 2 because it has 2 00:34:07.990 --> 00:34:11.320 inputs, these, and it has 5 outputs. 00:34:11.320 --> 00:34:12.660 And what's the matrix? 00:34:16.590 --> 00:34:21.050 Well, for the second and fourth outputs 00:34:21.050 --> 00:34:32.300 it uses exactly-- this multiplies x, 00:34:32.300 --> 00:34:38.750 or I should say u, u_1 and u_2. 00:34:38.750 --> 00:34:45.010 Let me see if I'm-- if I'll follow the notes in-- the notes 00:34:45.010 --> 00:34:55.520 use v_1 and v_2 for the coarse mesh values to avoid h, 2h, 00:34:55.520 --> 00:34:56.850 extra subscripts. 00:34:56.850 --> 00:35:00.560 So this height is v_1, this height is v_2, 00:35:00.560 --> 00:35:07.910 and I'm going to take-- this'll be the new u_1, u_3, and u_5. 00:35:07.910 --> 00:35:10.850 And u_2 I'm going to save. 00:35:14.600 --> 00:35:20.260 I better put I and make this into a true equation, I*v. 00:35:20.260 --> 00:35:26.200 This is going to give me the values u_1, u_2, u_3, u_4, 00:35:26.200 --> 00:35:29.500 and u_5. 00:35:29.500 --> 00:35:34.720 So u_2 is just v_1. 00:35:34.720 --> 00:35:39.010 u_4 is just v_2. 00:35:39.010 --> 00:35:41.050 No problem. 00:35:41.050 --> 00:35:43.381 And what do I do with the other ones? 00:35:43.381 --> 00:35:43.880 Let's see. 00:35:43.880 --> 00:35:53.150 I guess u_1 is halfway between v_1 and 0. 00:35:53.150 --> 00:35:54.840 OK, so it's 1/2. 00:35:58.280 --> 00:36:04.490 u_1 is just 1/2 v_1, it doesn't involve v_2. 00:36:04.490 --> 00:36:09.010 u_3, which is sitting here is half of these, 00:36:09.010 --> 00:36:14.510 so it's half of this and half of that where this was nothing. 00:36:14.510 --> 00:36:19.630 Then u_4 was the same as v_2, just take v_2. 00:36:19.630 --> 00:36:20.940 And what's u_5? 00:36:20.940 --> 00:36:23.980 u_5 is half of v_2 and half of zero. 00:36:29.200 --> 00:36:36.110 So it's a simple matrix, the interpolation matrix. 00:36:36.110 --> 00:36:38.670 It's just reasonable to use the letter I 00:36:38.670 --> 00:36:42.670 and we all recognize, here it's a rectangular matrix, not 00:36:42.670 --> 00:36:44.280 the identity. 00:36:44.280 --> 00:36:45.970 I doesn't stand for identity here, 00:36:45.970 --> 00:36:48.500 it stands for interpolation. 00:36:48.500 --> 00:36:52.340 OK, so I hope you can focus on the matrix. 00:36:57.650 --> 00:37:02.510 And if it was much, much bigger, but still in 1D, 00:37:02.510 --> 00:37:04.210 it wouldn't look very different. 00:37:04.210 --> 00:37:07.420 Each column would have a 1/2, 1, 1/2. 00:37:07.420 --> 00:37:12.010 That's a typical part of the interpolation matrix. 00:37:12.010 --> 00:37:17.760 It uses the value v_1, saves it at the center point, 00:37:17.760 --> 00:37:20.830 uses half of it at the previous point, and half of it 00:37:20.830 --> 00:37:23.800 at the right-hand point. 00:37:23.800 --> 00:37:34.120 And similarly, you see it's sort of a typical rectangular matrix 00:37:34.120 --> 00:37:36.560 in signal processing as well. 00:37:39.720 --> 00:37:43.410 I have to say something about what happens in 2D 00:37:43.410 --> 00:37:47.500 because our problems are in 2D. 00:37:47.500 --> 00:37:52.520 Maybe I just draw a typical 2D mesh. 00:37:55.940 --> 00:38:00.160 These are the coarse values, this is the coarse mesh and now 00:38:00.160 --> 00:38:03.270 I'll-- and of course, we'll save those values. 00:38:03.270 --> 00:38:07.040 I will have a 1, so it'll save those values 00:38:07.040 --> 00:38:13.670 when it goes to the fine mesh, the fine mesh being half. 00:38:13.670 --> 00:38:16.530 So I just have to decide what to do there. 00:38:16.530 --> 00:38:18.160 So question mark. 00:38:18.160 --> 00:38:21.350 What value shall I take-- oh, I suppose 00:38:21.350 --> 00:38:23.520 I have to decide there, too. 00:38:23.520 --> 00:38:28.890 These are all new points and what shall I do. 00:38:28.890 --> 00:38:31.400 I'm just going to stay with linear interpolation, which 00:38:31.400 --> 00:38:34.250 is quite fast and satisfactory. 00:38:34.250 --> 00:38:37.650 This value will be the average of those two. 00:38:37.650 --> 00:38:39.754 This value will be the average of those 00:38:39.754 --> 00:38:41.420 and what will this one be in the center? 00:38:41.420 --> 00:38:49.060 It's the average of these four, of course. 00:38:49.060 --> 00:38:50.490 And that's what we'll get. 00:38:50.490 --> 00:38:56.910 Actually, that Kronecker product business, a Kronecker product 00:38:56.910 --> 00:39:01.350 of a 1D matrix like that with itself 00:39:01.350 --> 00:39:05.590 would be a 2D interpolation that would do exactly this. 00:39:05.590 --> 00:39:08.260 We can write it out more fully, but it's not 00:39:08.260 --> 00:39:13.630 fun to write out really fully because if this is 5 by 2, 00:39:13.630 --> 00:39:17.970 then I2D would be 25 by 4. 00:39:20.970 --> 00:39:24.700 So we're right away, even on this tiny, tiny problem, 00:39:24.700 --> 00:39:28.510 we're right away up to 25 internal mesh 00:39:28.510 --> 00:39:29.680 points on the fine mesh. 00:39:32.850 --> 00:39:34.920 That's the matrix that I want you to see. 00:39:34.920 --> 00:39:43.760 And, the natural choice for R is the transpose. 00:39:46.450 --> 00:39:50.890 So one possible choice-- and it will, the answer will be yes, 00:39:50.890 --> 00:39:52.540 we do use neighbors. 00:39:52.540 --> 00:40:00.530 So R will be the transpose of I. Actually, 00:40:00.530 --> 00:40:03.620 I'll need a factor here. 00:40:03.620 --> 00:40:06.740 I think it's 1/2. 00:40:06.740 --> 00:40:09.390 Let me just see what it is. 00:40:09.390 --> 00:40:18.180 Let me see why we need a little multiple there to make things 00:40:18.180 --> 00:40:20.310 right. 00:40:20.310 --> 00:40:23.170 So there's my matrix I, so I'm just 00:40:23.170 --> 00:40:31.310 going to transpose it and let me factor 1/2 out of that 00:40:31.310 --> 00:40:34.760 so you see these matrices without fractions inside. 00:40:34.760 --> 00:40:37.280 That would be 1, 2, 1. 00:40:37.280 --> 00:40:38.640 So I'm going to transpose that. 00:40:38.640 --> 00:40:41.340 1, 2, 1, 0, 0. 00:40:41.340 --> 00:40:43.210 0, 0, 1, 2, 1. 00:40:45.800 --> 00:40:49.550 So I factored 1/2 out of I when I did that. 00:40:49.550 --> 00:40:52.060 And then I believe that I need another 1/2 here. 00:40:52.060 --> 00:40:54.410 I think I need 1/4. 00:40:54.410 --> 00:40:56.660 And what do I mean by needing 1/4? 00:41:02.010 --> 00:41:04.810 Of course, the main point is that this 00:41:04.810 --> 00:41:07.330 is the transpose of this. 00:41:07.330 --> 00:41:12.310 And you'll see that that preserves the symmetry. 00:41:12.310 --> 00:41:17.910 That's just sort of the natural thing to pick. 00:41:17.910 --> 00:41:20.580 Not the only possibility, but it's very good 00:41:20.580 --> 00:41:25.190 to connect the restriction with the interpolation. 00:41:25.190 --> 00:41:28.790 Of course, you can't take the restriction 00:41:28.790 --> 00:41:31.140 to be the inverse of the interpolation. 00:41:31.140 --> 00:41:32.970 Why not? 00:41:32.970 --> 00:41:39.990 I mean that sort of mental idea is that the restriction goes 00:41:39.990 --> 00:41:41.970 in one direction and the interpolation 00:41:41.970 --> 00:41:43.010 in the other direction. 00:41:43.010 --> 00:41:45.800 So why not just let one be the inverse of the other? 00:41:45.800 --> 00:41:50.100 Well of course, that matrix doesn't have an inverse. 00:41:54.680 --> 00:41:57.540 I mean its rank is only 2. 00:41:57.540 --> 00:42:01.450 When we do a restriction we're going to lose information, 00:42:01.450 --> 00:42:04.080 the interpolation can't put it all back 00:42:04.080 --> 00:42:10.430 because we threw it away when we restricted to the coarse mesh. 00:42:10.430 --> 00:42:12.030 Now why 1/4? 00:42:12.030 --> 00:42:17.970 I think it's just if we had all constants, like all ones, 00:42:17.970 --> 00:42:22.280 as our mesh values, we would want 00:42:22.280 --> 00:42:26.910 to get all ones for the restriction. 00:42:26.910 --> 00:42:31.890 So if I multiply this matrix by the vector of all ones-- 00:42:31.890 --> 00:42:37.670 so assuming you remember where R comes in, 00:42:37.670 --> 00:42:43.540 if I just copy this equation, you 00:42:43.540 --> 00:42:51.170 remember that I go from the h mesh, it's fine to coarse. 00:42:55.770 --> 00:43:09.110 So R times ones on the fine mesh, all ones on the fine mesh 00:43:09.110 --> 00:43:13.410 gives ones on the coarse mesh. 00:43:13.410 --> 00:43:20.980 That's why I wanted that-- you see, if I multiply by all ones 00:43:20.980 --> 00:43:25.080 then that multiplication gives me 1 plus 2 plus 1 equal 4 00:43:25.080 --> 00:43:30.550 and I need to divide by 4 to get back to the 1. 00:43:30.550 --> 00:43:35.550 You quickly discover that that just arises 00:43:35.550 --> 00:43:37.760 because of a change of scale. 00:43:37.760 --> 00:43:40.200 It's just a factor of 1/2. 00:43:40.200 --> 00:43:45.730 In 2D that 1/2 there would change to 1/4. 00:43:45.730 --> 00:43:48.740 In 3D it'd be 1/8. 00:43:48.740 --> 00:43:54.550 It's just the scaling to preserve constants. 00:43:57.880 --> 00:44:01.650 And now the remaining question is, what's A_(2h). 00:44:01.650 --> 00:44:08.810 That's the other matrix here that we're not given. 00:44:08.810 --> 00:44:13.030 We're given the A_h, our problem. 00:44:13.030 --> 00:44:17.240 We choose an R and we choose an I and if we're smart, 00:44:17.240 --> 00:44:19.620 one's the transpose of the other. 00:44:19.620 --> 00:44:21.920 And then the question is, what's A_(2h)? 00:44:21.920 --> 00:44:27.680 And there's a beautiful answer, so let me just say what it is. 00:44:27.680 --> 00:44:46.240 A_(2h), the coarse mesh matrix, is the fine mesh matrix, 00:44:46.240 --> 00:44:57.330 but I need to do first-- if this is going to apply to the coarse 00:44:57.330 --> 00:45:03.660 mesh guys, the v's, I have to do an interpolation. 00:45:06.539 --> 00:45:07.330 You'll see it here. 00:45:07.330 --> 00:45:12.710 This is 5 by 5 in my problem. 00:45:12.710 --> 00:45:17.510 So I need to first do the interpolation 00:45:17.510 --> 00:45:22.670 to get from a vector of length 2 up to a vector of length 5. 00:45:22.670 --> 00:45:25.330 Then this and then the restriction. 00:45:25.330 --> 00:45:26.770 That's terrific. 00:45:26.770 --> 00:45:27.520 That's A_(2h). 00:45:32.900 --> 00:45:40.550 So this is A_(2h) and we'd better do an example. 00:45:40.550 --> 00:45:42.090 I guess it's going to be next time. 00:45:42.090 --> 00:45:46.160 So that'll be the first thing for next time, would be to see, 00:45:46.160 --> 00:45:50.690 okay, for the standard second difference A_h, 00:45:50.690 --> 00:45:55.630 for these piecewise linear interpolation and restriction, 00:45:55.630 --> 00:45:59.380 what comes out as A_(2h)? 00:45:59.380 --> 00:46:03.760 Do we get the standard second difference 00:46:03.760 --> 00:46:10.640 on the coarse mesh, which you really hope we do and we do. 00:46:10.640 --> 00:46:15.300 This will be the second difference, in our example, 00:46:15.300 --> 00:46:16.810 on the coarse mesh. 00:46:21.130 --> 00:46:26.390 So that this middle step of multigrid is exactly what 00:46:26.390 --> 00:46:29.630 we would have expected. 00:46:29.630 --> 00:46:32.330 What we would've had if we set up the problem originally 00:46:32.330 --> 00:46:33.940 on the coarse mesh. 00:46:33.940 --> 00:46:37.260 But now we're going to get an answer that we take back 00:46:37.260 --> 00:46:38.560 to the fine mesh. 00:46:38.560 --> 00:46:41.260 OK, so I'll do that next time. 00:46:41.260 --> 00:46:45.990 Some homeworks here that I held for an extra two days 00:46:45.990 --> 00:46:50.780 and now I'm returning and some thoughts to go up 00:46:50.780 --> 00:46:54.340 on to the website about possible projects, 00:46:54.340 --> 00:46:57.210 but actually, I expressed a lot of them 00:46:57.210 --> 00:46:59.280 at the beginning of the lecture. 00:46:59.280 --> 00:47:00.510 OK, thanks. 00:47:00.510 --> 00:47:01.760 Good.