A Comparison of a Second-Order versus a Fourth- Order Laplacian Operator in the Multigrid Algorithm

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1 A Comparison of a Second-Order versus a Fourth- Order Lapacian Operator in the Mutigrid Agorithm Kaushik Datta (kdatta@cs.berkeey.edu Math Project May 9, 003 Abstract In this paper, the mutigrid agorithm was used to sove Poisson s equation for various right-hand sides. However, in order to cacuate the discretized apacian vaue, both a second-order and a fourth-order apacian operator was used. The fourth-order apacian operator achieves higher accuracy at the cost of more cacuations and a onger execution time. In order to compensate for this, the grid spacing for the fourth-order operator was doubed, and the convergence resuts with the second-order operator were compared for simiar program running times.. Preiminaries. The Mutigrid Agorithm Mutigrid is an iterative agorithm that is generay used to sove eiptic partia differentia equations. In our case, it wi be used to sove Poisson s equation. Specificay, we seek the soution ϕ in the equation: ϕ ϕ ϕ ϕ = + + = ρ x y z Convergence to a fixed error in mutigrid takes O(og N steps, where N is the number of variabes in grid ϕ. In addition, the agorithm operates in O(N time since it can communicate data across many grid ces in one step. The three main operators in the mutigrid agorithm are restrict (coarsen, interpoate, and reax. The restrict operation coarsens the grid to one with haf the dimensions of the origina grid. In the three-dimensiona case, each vaue in the smaer grid is a weighted average of the eight vaues in the origina grid that correspond to that point. Conversey, interpoation takes a coarse grid and distributes a weighted average of each vaue onto the eight corresponding points in the fine grid. Thus, restriction and interpoation are compementary operations. Finay, the reax operation is some iterative method that is

2 effective in smoothing the current soution, usuay by subtracting off a fraction of the residua. Note that a three operators ony do nearest-neighbor computations by averaging neighboring grid vaues. However, since adjacent points in the coarsened grids represent points that are far away in the fine grids, information sti moves quicky across the grid. Three-dimensiona mutigrid, in its simpest impementation, usuay requires an equa, power-of-two number of ces in each dimension of the grid. The base two ogarithm of the number of ces in one dimension can be considered the eve of that grid. For instance, a 64 x 64 x 64 grid is a eve 6 grid. The coarsen operation owers the grid eve by one, whie interpoate increases it by one. The eve of a grid is usefu in understanding which phase of mutigrid is being executed. The mutigrid agorithm begins by cacuating the residua associated with the current soution. This residua is then iterativey refined using any reaxation operator, incuding Jacobi, Gauss-Seide, and Gauss-Seide red-back. In our case, three iterations of a modified weighted Jacobi method were used. To be precise, for the starting grid eve, each grid point in ϕ executed the foowing: ϕ + λ ϕ ρ ( The cacuation of the apacian operator wi be discussed ater. The quantity ( ϕ ρ( i, is the negative of the residua, so a fraction of the residua is being subtracted from the soution during each iteration. The vaue of λ was chosen by using the rue-of-thumb formua λ = h /, where h is the grid spacing. After reaxing the soution, a new residua grid is computed for this eve, and this grid is then coarsened through the restrict operator. Mathematicay, for each point in the coarsened residua grid R at eve, the foowing is performed: R = ( 8 R, j+,k k j+,k +, k j+,k, k+ k +, j+,k + The coarsened residua grid is recursivey reaxed and coarsened is this manner unti a eve 0 grid is achieved. At that point, the grid is again reaxed three times. Then, the correction from this eve 0 grid is interpoated onto the eve correction grid, which is initiay a zeros. In genera, for each point in the coarsened correction grid at eve, interpoate does the foowing: ( i, k k ϕ ( i +, j+,k, j+,k + ( i +, k, k ( i, j+,k j+,k ( i, k+ k + ( i +, k +, k + ( i, j+,k + j+,k + ( i +, j+,k +, j+,k +

3 So, the correction from the grid at each eve is interpoated onto the correction grid of the next higher eve (initiay a zeros. The higher eve grid is then reaxed by again performing three Jacobi iterations. This is done for consecutivey higher grid eves unti the correction is interpoated onto the origina soution grid. After three more Jacobi iterations, one mutigrid V-cyce is competed. The key to mutigrid is having the reax operation effectivey suppresses the high frequency (osciatory Fourier modes of the residua for that eve. By coarsening, the number of ces in each dimension is haved, so ow frequency error becomes higher frequency error. Performing reax operations shoud again suppress the higher Fourier modes in the error. If this is executed over a grid eves, then mutigrid shoud reduce error over a frequencies.. The Lapacian Operators In the mutigrid agorithm, the apacian vaue needs to be computed both in cacuating the residua and in performing reaxes. There are two different apacian operators that were used in doing so. Let us adopt the convention that for the 3 x 3 x 3 cube surrounding a given point in ϕ, then ϕ m, ϕ f, ϕ e, ϕ c represent a midde, face-centered, edge-centered, and corner point respectivey. A simiar convention hods for grid points in ρ. Then, the apacian for the first formua is cacuated by: ϕ = ( 6ϕ m + Σϕ f h This is a second-order formua. The second formua is of fourth-order, and cacuates the apacian as foows: ϕ = ( Σϕ e + Σϕ f 4ϕ m Σρ f 3h 6 The fourth-order formua shoud aow mutigrid to converge more rapidy than the second-order formua. 3 However, it wi aso take onger to execute a V-cyce since more cacuations need to be performed. As a resut, we wi compare whether using twice the grid spacing for the fourth-order formua wi resut in better convergence resuts than the second-order formua for a given program running time..3 The Programming Language and Patform The mutigrid code was written in the Titanium programming anguage. Titanium is an object-oriented high performance Java diaect written at UC Berkeey. To improve performance, Titanium code is first compied to C, and then to native binary. Whie Titanium can be used for parae processing, a the resuts in this paper are seria. 4

4 This code was run on the distributed-memory IBM SP machine seaborg.nersc.gov. Each node of seaborg contains 6 IBM Power3 processors, each of which has a peak fop rate of.5 GFops. In addition, each processor has two eves of cache. The L cache has a data cache size of 64 KB, with a cache ine size of 8 B and a atency of cyce. The L cache has a size of 8 MB with a cache ine size of 8 B and a atency of 9 cyces. 5. The Test Functions There were four test functions used for the right-hand side that were soved using mutigrid to compare the apacian operators. In order to simpify matters, periodic boundary conditions were impemented in a three dimensions, so the chosen functions needed to conform to this constraint. In addition, having functions with different Fourier modes woud be usefu in order to observe how we mutigrid suppresses ow, medium, and high frequency error. However, we need to make sure that the high frequency modes do not resut in aiasing. Aiasing is a phenomenon where the waveength of a Fourier mode is ess than h. The resuting wave appears to have a waveength greater than h on the grid since there are too few grid points to represent the wave accuratey. With these considerations in mind, the foowing four functions were chosen: Freq. ϕ (exact ρ (Test Functions Domain Low cos( π xcos(πycos(πz π cos(πx cos(πycos(πz (-.5,-.5,-.5 to (.5,.5,.5 Med. cos( 8π x cos(8πy cos(8πz 9π cos(8πx cos(8πy cos(8πz (-.5,-.5,-.5 to (.5,.5,.5 High cos( 6π xcos(6πycos(6πz 08π cos(6πxcos(6πycos(6πz (-.5,-.5,-.5 to (.5,.5,.5 A (cos(πxcos(πycos(πz + 3π (cos(πxcos(πycos(πz + (-.5,-.5,-.5 4 to cos(8πx cos(8πy cos(8πz + (.5,.5,.5 (cos(8πx cos(8πy cos(8πz + cos(6πxcos(6πycos(6πz 64 (cos(6πx cos(6πy cos(6πz 676 Having the exact soution ϕ aows us to cacuate the error directy rather than use the residua.

5 3. Convergence Resuts The foowing resuts pot the base 0 og of the maximum error against the number of mutigrid V-cyces executed (execution time is not yet incuded. Examining the maximum error gives us a good sense of the worst-case convergence rate of mutigrid. The four runs that are shown are for the second and fourth-order apacian stenci being used for mutigrid that begins on a eve 7 grid and eve 8 grid. Note that starting at eve 7 resuts in twice the grid spacing as that of eve Low Frequency Data Here is the graph for ow frequency data: Low Frequency Data nd order, eve 8 start 4th order, eve 8 start 4th order, eve 7 start nd order, eve 7 start -.00 Number of V-cyces Whie a the runs seem to converge at a constant rate, the 4 th order method that starts on a eve 7 grid definitivey outperforms the other runs. 3. Medium Frequency Data Here is the graph for medium frequency data:

6 Medium Frequency Data nd order, eve 8 start 4th order, eve 8 start 4th order, eve 7 start nd order, eve 7 start Number of V-cyces Whie the rates of convergence are faster than for the ow frequency data, the fourthorder apacian operator that starts on a eve 7 grid again outstrips the other runs. 3.3 High Frequency Data Here is the same graph for high frequency data: High Frequency Data nd order, eve 8 start 4th order, eve 8 start 4th order, eve 7 start nd order, eve 7 start Number of V-cyces The suppression of the osciatory Fourier modes, whie converging at a reativey rapid constant rate for a number of V-cyces, seems to stop converging at a certain point. This seems to indicate that whie high-frequency error is quicky suppressed, it is difficut to

7 competey eiminate. Ony the second-order apacian operator that starts at eve 8 continues to converge at a constant rate for at east ten V-cyces, but it is aso the sowest converging run. 3.4 A Frequency Data Here is the graph for a frequency data: A Frequency Data nd order, eve 8 start 4th order, eve 8 start 4th order, eve 7 start nd order, eve 7 start -.60 Number of V-cyces Like the ow and medium frequency data graphs, the fourth-order apacian operator that starts on the eve 7 grid seems to do better than the other runs. 4. Anaysis It is unnecessary to factor time into the anaysis since the resuts seem to indicate that, for the same number of V-cyces, starting on a eve 7 grid produces faster convergence than a eve 8 grid. This is a somewhat surprising resut, and there are severa possibe expanations. First, the appropriate norm was not chosen for the probem at hand. Whie this anaysis uses the maximum error over the soution grid, it may have been better to choose the L norm of the soution grid instead. The L norm is a better indication of the error over the entire grid, and is more typicay used in this type of anaysis. If there were a sma area of the grid that produced unusuay arge error, the maximum error norm woud be arge, whie the L norm woud not increase significanty. This type of error coud possiby account for the resuts above. In addition, a good vaue for λ may not have been used for the eve 8 grids. Whie λ = h / is typicay a good vaue, a Fourier error anaysis woud have to be performed

8 to make sure that the high-frequency error is being effectivey damped at each eve of mutigrid. In addition, other test functions shoud be used. Whie each function isted isoates a specific Fourier mode, it may be worthwhie to use non-trigonometric functions as the right-hand side, as they may exhibit different types of convergence. Finay, the condition number of the matrix needs to be accounted for. In this case, starting at a eve 7 grid produces faster convergence for both the second and fourth order apacian operators than starting at a eve 8 grid, so that sti does not expain our resuts. However, it sti needs to be factored into the anaysis. 5. Further Anaysis As stated above, the anaysis shoud be attempted first with the L norm, as that may expain why eve 7 grids converge more rapidy than eve 8 grids. If that produces simiar resuts, then the other options isted above shoud be tried. Using a arger grid spacing shoud not resut in faster convergence, as this woud impy that, for the same number of V-cyces, using fewer mesh points woud increase the rate of convergence. References Yeick, Kathy. Lecture 3. Briggs, Wiiam L., Van Emden Henson, Steve F. McCormick. A Mutigrid Tutoria, nd edition. SIAM, Phiadephia, PA, Coatz, Lothar. The Numerica Treatment of Differentia Equations, 3 rd Edition. Springer-Verag, Berin, Germany, 960, pp Titanium programming anguage. 5 IBM SP.