Methods and Algorithms for Scientific Computing

On Local Fourier Analysis of Multigrid Methods for PDEs with Jumping and Random Coefficients

Abstract

In this paper, we propose a novel nonstandard local Fourier analysis (LFA) variant for accurately predicting the multigrid convergence of problems with random and jumping coefficients. This LFA method is based on a specific basis of the Fourier space rather than the commonly used Fourier modes. To show the utility of this analysis, we consider, as an example, a simple cell-centered multigrid method for solving a steady-state single phase flow problem in a random porous medium. We successfully demonstrate the predictive capability of the proposed LFA using a number of challenging benchmark problems. The information provided by this analysis could be used to estimate a priori the time needed for solving certain uncertainty quantification problems by means of a multigrid multilevel Monte Carlo method.

Keywords

  1. PDEs
  2. random coefficients
  3. multigrid
  4. local Fourier analysis
  5. multilevel Monte Carlo
  6. uncertainty quantification

MSC codes

  1. 65F10
  2. 65M22
  3. 65M55

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
M. Bolten and H. Rittich, Fourier analysis of periodic stencils in multigrid methods, SIAM J. Sci. Comput., 40 (2018), pp. A1642--A1668, doi:10.1137/16M1073959.
2.
J. P. Delhomme, Spatial variability and uncertainty in groundwater flow parameters: A geostatistical approach, Water Resources Research, 15 (1979), pp. 269--280, https://doi.org/10.1029/WR015i002p00269.
3.
R. A. Freeze, A stochastic-conceptual analysis of one-dimensional groundwater flow in nonuniform homogeneous media, Water Resources Research, 11 (1975), pp. 725--741, doi:10.1029/WR011i005p00725.
4.
R. J. Hoeksema and P. K. Kitanidis, Analysis of the spatial structure of properties of selected aquifers, Water Resources Research, 21 (1985), pp. 563--572, https://doi.org/10.1029/WR021i004p00563.
5.
I. Babuška, F. Nobile, and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal., 45 (2007), pp. 1005--1034, doi:10.1137/050645142.
6.
D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press, Princeton, NJ, 2010.
7.
B. Seynaeve, E. Rosseel, B. Nicola, and S. Vandewalle, Fourier mode analysis of multigrid methods for partial differential equations with random coefficients, J. Comput. Phys., 224 (2007), pp. 132--149, doi:10.1016/j.jcp.2006.12.011.
8.
K. Cliffe, M. B. Giles, R. Scheichl, and A. L. Teckentrup, Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients, Comput. Vis. Sci., 14 (2011), pp. 3--15, doi:10.1007/s00791-011-0160-x.
9.
M. B. Giles, Multilevel Monte Carlo path simulation, Oper. Res., 56 (2008), pp. 981--986, doi:10.1287/opre.1070.0496.
10.
M. B. Giles, Multilevel Monte Carlo methods, Acta Numer., 24 (2015), pp. 259--328, doi:10.1017/S096249291500001X.
11.
A. Brandt and O. E. Livne, Multigrid Techniques: $1984$ Guide with Applications to Fluid Dynamics, Classics in Appl. Math. 67, SIAM, Philadelphia, 2011.
12.
J. Molenaar, A simple cell-centered multigrid method for $3$D interface problems, Comput. Math. Appl., 31 (1996), pp. 25--33, doi:10.1016/0898-1221(96)00039-9.
13.
I. Yavneh, Coarse-grid correction for nonelliptic and singular perturbation problems, SIAM J. Sci. Comput., 19 (1998), pp. 1682--1699, https://doi.org/10.1137/S1064827596310998.
14.
W. Hackbusch, Multigrid Methods and Applications, Springer, New York, 1985.
15.
J. H. Bramble, R. E. Ewing, J. E. Pasciak, and J. Shen, The analysis of multigrid algorithms for cell centered finite difference methods, Adv. Comput. Math., 5 (1996), pp. 15--29, doi:10.1007/BF02124733.
16.
J. H. Bramble, J. E. Pasciak, and J. Xu, The analysis of multigrid algorithms with non-nested spaces or non-inherited quadratic forms, Math. Comp., 56 (1991), pp. 1--34, doi:10.2307/2008527.
17.
M. Brezina, R. Falgout, S. MacLachlan, T. Manteuffel, S. McCormick, and J. Ruge, Adaptive smoothed aggregation ($\alpha$SA) multigrid, SIAM Rev., 47 (2005), pp. 317--346, doi:10.1137/S1064827502418598.
18.
K. Stüben, Appendix A: An introduction to algebraic multigrid, in Multigrid, U. Trottenberg, C. W. Oosterlee, and A. Schüller, eds., Academic Press, San Diego, CA, 2001, pp. 413--532.
19.
P. Vaněk, M. Brezina, and J. Mandel, Convergence of algebraic multigrid based on smoothed aggregation, Numer. Math., 88 (2001), pp. 559--579, doi:10.1007/s211-001-8015-y.
20.
D. Braess, Towards algebraic multigrid for elliptic problems of second order, Computing, 55 (1995), pp. 379--393, doi:10.1007/BF02238488.
21.
P. Vaněk, J. Mandel, and M. Brezina, Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems, Computing, 56 (1996), pp. 179--196, https://doi.org/10.1007/BF02238511.
22.
R. E. Alcouffe, A. Brandt, J. E. Dendy, Jr., and J. W. Painter, The multigrid methods for the diffusion equation with strongly discontinuous coefficients, SIAM J. Sci. Statist. Comput., 2 (1981), pp. 430--454, doi:10.1137/0902035.
23.
J. Dendy, Black box multigrid, J. Comput. Phys., 48 (1982), pp. 366--386, doi:10.1016/0021-9991(82)90057-2.
24.
J. Dendy, Black box multigrid for non-symmetric problems, Appl. Math. Comput., 13 (1983), pp. 261--283, doi:10.1016/0096-3003(83)90016-4.
25.
M. Khalil and P. Wesseling, Vertex-centered and cell-centered multigrid for interface problems, J. Comput. Phys., 98 (1992), pp. 1--10, doi:10.1016/0021-9991(92)90168-X.
26.
P. Wesseling, Cell-centered multigrid for interface problems, J. Comput. Phys., 79 (1988), pp. 85--91, doi:10.1016/0021-9991(88)90005-8.
27.
J. E. Dendy and J. D. Moulton, Black box multigrid with coarsening by a factor of three, Numer. Linear Algebra Appl., 17, pp. 577--598, doi:10.1002/nla.705.
28.
U. Trottenberg, C. W. Oosterlee, and A. Schuller, Multigrid, Academic Press, San Diego, CA, 2000.
29.
S. Knapek, Matrix-dependent multigrid homogenization for diffusion problems, SIAM J. Sci. Comput., 20 (1998), pp. 515--533, doi:10.1137/S1064827596304848.
30.
J. Moulton, J. E. Dendy, and J. M. Hyman, The black box multigrid numerical homogenization algorithm, J. Comput. Phys., 142 (1998), pp. 80--108, doi:10.1006/jcph.1998.5911.
31.
S. P. MacLachlan and J. D. Moulton, Multilevel upscaling through variational coarsening, Water Resources Research, 42 (2006), doi:10.1029/2005WR003940.
32.
M. S. Handcock and J. R. Wallis, An approach to statistical spatial-temporal modeling of meteorological fields (with discussion), J. Amer. Statist. Assoc., 89 (1994), pp. 368--390, doi:10.2307/2290832.
33.
R. J. Adler, The Geometry of Random Fields, Classics in Math. Appl. 62, SIAM, Philadelphia, 2010, doi:10.1137/1.9780898718980.
34.
F. Nobile and F. Tesei, A multilevel Monte Carlo method with control variate for elliptic PDEs with log-normal coefficients, Stoch. Partial Differ. Equ. Anal. Comput., 3 (2015), pp. 398--444, doi:10.1007/s40072-015-0055-9.
35.
J. Charrier, R. Scheichl, and A. L. Teckentrup, Finite element error analysis of elliptic PDEs with random coefficients and its application to multilevel Monte Carlo methods, SIAM J. Numer. Anal., 51 (2013), pp. 322--352, doi:10.1137/110853054.
36.
J. Galvis and M. Sarkis, Approximating infinity-dimensional stochastic Darcy's equations without uniform ellipticity, SIAM J. Numer. Anal., 47 (2009), pp. 3624--3651, doi:10.1137/080717924.
37.
C. R. Dietrich and G. N. Newsam, Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix, SIAM J. Sci. Comput., 18 (1997), pp. 1088--1107, doi:10.1137/S1064827592240555.
38.
A. T. A. Wood and G. Chan, Simulation of stationary Gaussian processes in $[0, 1]^d$, J. Comput. Graph. Statist., 3 (1994), pp. 409--432, doi:10.2307/1390903.
39.
R. G. Ghanem and P. D. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer, New York, 1991, doi:10.1007/978-1-4612-3094-6.
40.
Ch. Schwab and R. A. Todor, Karhunen-Loeve approximation of random fields by generalized fast multipole methods, J. Comput. Phys., 217 (2006), pp. 100--122, doi:10.1016/j.jcp.2006.01.048.
41.
R. Wienands and C. W. Oosterlee, On three-grid Fourier analysis for multigrid, SIAM J. Sci. Comput., 23 (2001), pp. 651--671, doi:10.1137/S106482750037367X.
42.
S. Mishra and Ch. Schwab, Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random initial data, Math. Comp., 81 (2012), pp. 1979--2018, doi:10.1090/S0025-5718-2012-02574-9.
43.
S. Mishra, Ch. Schwab, and J. Šukys, Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions, J. Comput. Phys., 231 (2012), pp. 3365--3388, doi:10.1016/j.jcp.2012.01.011.
44.
S. Mishra, Ch. Schwab, and J. Šukys, Multi-level Monte Carlo finite volume methods for uncertainty quantification of acoustic wave propagation in random heterogeneous layered medium, J. Comput. Phys., 312 (2016), pp. 192--217, doi:10.1016/j.jcp.2016.02.014.
45.
D. Drzisga, B. Gmeiner, U. Rüde, R. Scheichl, and B. Wohlmuth, Scheduling massively parallel multigrid for multilevel Monte Carlo methods, SIAM J. Sci. Comput., 39 (2017), pp. S873--S897, doi:10.1137/16M1083591.
46.
J. Šukys, S. Mishra, and Ch. Schwab, Static load balancing for multilevel Monte Carlo finite volume solvers, in Parallel Processing and Applied Mathematics, Springer, Berlin, 2012, pp. 245--254, doi:10.1007/978-3-642-31464-3_25.
47.
P. Luo, C. Rodrigo, F. J. Gaspar, and C. W. Oosterlee, Uzawa smoother in multigrid for the coupled porous medium and Stokes flow system, SIAM J. Sci. Comput., 39 (2017), pp. S633--S661, doi:10.1137/16M1076514.
48.
P. Kumar, P. Luo, F. J. Gaspar, and C. W. Oosterlee, A multigrid multilevel Monte Carlo method for transport in the Darcy-Stokes system, J. Comput. Phys., 371 (2018), pp. 382--408, doi:10.1016/j.jcp.2018.05.046.
49.
C. Rodrigo, P. Salinas, F. J. Gaspar, and F. J. Lisbona, Local Fourier analysis for cell-centered multigrid methods on triangular grids, J. Comput. Appl. Math., 259 (2014), pp. 35--47, doi:10.1016/j.cam.2013.03.040.
50.
C. Rodrigo, F. J. Gaspar, X. Hu, and L. T. Zikatanov, Stability and monotonicity for some discretizations of the Biot's consolidation model, Comput. Methods Appl. Mech. Engrg., 298 (2016), pp. 183--204, doi:10.1016/j.cma.2015.09.019.
51.
M. L. Ravalec, B. Noetinger, and L. Y. Hu, The FFT Moving Average (FFT-MA) generator: An efficient numerical method for generating and conditioning Gaussian simulations, Math. Geol., 32 (2000), pp. 701--723, doi:10.1023/A:1007542406333.
52.
I. G. Graham, F. Y. Kuo, D. Nuyens, R. Scheichl, and I. H. Sloan, Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications, J. Comput. Phys., 230 (2011), pp. 3668--3694, doi:10.1016/j.jcp.2011.01.023.

Information & Authors

Information

Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: A1385 - A1413
ISSN (online): 1095-7197

History

Submitted: 5 March 2018
Accepted: 19 March 2019
Published online: 2 May 2019

Keywords

  1. PDEs
  2. random coefficients
  3. multigrid
  4. local Fourier analysis
  5. multilevel Monte Carlo
  6. uncertainty quantification

MSC codes

  1. 65F10
  2. 65M22
  3. 65M55

Authors

Affiliations

Funding Information

Diputacion General de Aragon : E24_17R
Horizon 2020 Framework Programme https://doi.org/10.13039/100010661 : 705402
Federación Española de Enfermedades Raras https://doi.org/10.13039/501100002924 : MTM2016-75139-R

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.