Multigrid Techniques: past, present and future

1. Multigrid Techniques: past, present and future Galina Muratova Lev Krukier Evgenia Andreeva Computer Center Southern Federal University Rostov-on-Don muratova@sfedu.ru

2. Southern Federal University • One of the 9 Federal Universities of Russia • The leading center of education, science and culture in the South of Russia Rostov-on-Don Russia • More than 50 thousands students • 36 departments • 16 research institutes Computer Center of SFU

4. The main idea of the topicMultigrid Techniques: past, present and future About some results of the developing modern numerical mathematics Prof. Yuri Laevsky Novosibirsk, “Computer Technologies”, volume 7 №2, 2002, p. 74-83

5. Outline • Introduction • Multigrid method • Multigrid basics and development stages • Geometric and algebraic multigrid • MGM for computational fluid dynamic problems (CFD) • MGM for Navier-Stokes equations • MGM for convection- diffusion problem • Special Smoothers of MGM for Strongly Nonsymmetric Linear Systems • Conclusions

6. Multigrid - history and main development stages • Iterative methods like the Gauss-Seidel or the Jacobi iteration have been used from the beginning of the numerical treatment of partial differential equations. • An important step was Young's successive over-relaxation method (1950) which is much faster than the closely related Gauss-Seidel iteration. This method shares with direct elimination methods the disadvantage that the amount of work does not remain proportional to the number of unknowns; the computer time needed to solve a problem grows more rapidly than the size of the problem. • Multigrid methods were the first ones to overcome this complexity barrier. The starting point of the multigrid method is the following “golden rule”: • The amount of computational work should be proportional to the amount of real physical changes in the computed system.

7. Multigrid - history and main development stages • The general ideas of MGM was suggested by R.P. Fedorenko Fedorenko R.P., A relaxation method for solving elliptic difference equations, Russian J. Comput. Math. Math. Phys. 1 (1961) p.p.1092 - 1096. • Prof. Fedorenko – alumnus of Rostov State University 11.03.1930 -13.09.2009

8. Multigrid - history and main development stages • N.S. Bachvalov (1966)considered the theoreticallymuch more complex case of variable coefficients. N.S. Bachvalov. 'On the convergence of a relaxation method with natural constraints on the elliptic operator' (in Russian), USSR Comput. Math, and Math. Phys. (1966),v.6 №5, p.p. 101-135. 29.05.1934 — 29.08.2005

9. Multigrid - history and main development stages • Although the basic idea of combining discretization on different grids in an iterative scheme appears to be very natural, the potential of this idea was not recognized before the middle of the 1970s. • The report of Hackbusch (1976) and the paper of Brandt (1977) were the historical breakthrough. • The first big multigrid conference in 1981 in Koln was a culmination point of the development; the conference proceedings edited by Hackbusch and Trottenberg (1982) are still a basic reference. • With Hackbusch's 1985 monograph, the first stage in MGM theory came to an end • W. Hackbusch 'EiniterativesVerfahrenzurschnellenAuflosungelliptischerRandwertprobleme', Report 76-12, (1976), MathematischesInstitutderUniversitatzu Koln. • A. Brandt (1977), 'Multi-level adaptive solutions to boundary-value problems',Math. Comput. 31, 333-390. • W. Hackbusch and U. Trottenberg, eds (1982), Multigrid Methods, Proceedings,Koln 1981, Lecture Notes in Mathematics 960, Springer (Berlin, Heidelberg,NY). • W. Hackbusch (1985), Multigrid Methods and Applications, Springer (Berlin, Heidelberg, New York).

10. Multigrid method researchers The Russian Federation National Award 2003 was handed over for a cycle of fundamental works on creation and the subsequent heading highly effective multigrid method for the numerical solution of a wide class of mathematical physics problems G. Astahancev, N. Bahvalov, R. Fedorenko, V. Shaidurov The significant contribution to MGM development: • A. Brandt, P.Wesseling, U. Trottenberg, CW. Oosterlee, A Schüller, W. Briggs, W. Hackbusch P. Vassilevsky, Y. Kuznecov, Z. Cao, M. Olshanskiy and others

11. Multigrid method The basic components of multigrid technologyare • Sequence of hierarchically nested grids • Smoothing procedure • Operators of transition from a fine grid on coarse grid and back (restriction, prolongation) • the procedure of the decision connecting these components (coarse-grid correction)

12. Two grid method • Smoothing • Coarse-grid correction • Compute the defect • Restrict the defect (fine-to-coarse transfer) • Solve on coarse-grid • Interpolate the correction (coarse-to-fine transfer) • Compute a new approximation • Postsmoothing

13. Multigrid method • Cascade algorithm • V – cycle • W – cycle • Full MGM

14. The structure of the Multigrid method (V – cycle )

15. Geometric MGM (GMG) and algebraic multilevel method (AMG) • The main difference between AMG and GMG is related to the manner of constructing the coarser grids: the AMG method requires no knowledge of the problem geometry. • The application of the AMG method includes problems in which the use of the GMG method is difficult or even impracticable, such as: unstructured grids, large matrix equations which are not at all derived from continuous problems, extreme anisotropic equations and so on. A remarkable use of the AMG method takes place when there is none information about the problem geometry. • A. Brandt, Algebraic multigrid theory: the symmetric case, Appl. Math. Comput. 19 (1986) 23–56. • V.E. Henson, P.S. Vassilevski, Element-free AMGe: general algorithms for computing interpolation weights in AMG, SIAM J. Sci. Comput. 23 (2001) 629–650 • R.D. Falgout, An introduction to algebraic multigrid, Comput. Sci. Eng. 8 (2006) 24–33. • Y. Xiao, S. Shu, P. Zhang, M. Tan, An algebraic multigrid method for isotropic linear elasticity problems on anisotropic meshes, Int. J. Numer. Biomed.Eng. 26 (2010) 534–553.

16. Fourier analysis as a tool for analyzing multigrid method • The convergence behavior of a multigrid algorithm depends strongly on the smoother, which must have the smoothing property • A convenient tool for the study of smoothing efficiency is Fourier analysis (MPA and LFA) • The efficiency of smoothing methods is problem-dependent • U. Trottenberg, C.W. Oosterlee, A. Schuller, Multigrid, Academic Press, New York, 2001. • V.E. Henson, P.S. Vassilevski, Element-free AMG: general algorithms for computing interpolation weights in AMG, SIAM J. Sci. Comput. 23 (2001) 629–650 • R.D. Falgout, An introduction to algebraic multigrid, Comput. Sci. Eng. 8 (2006) 24–33.

17. The fundamental basis of almost all CFD problems are the Navier–Stokes equations • Navier-Stokes equations arealso of greatinterestina mathematical sense • mathematicianshavenotyetprovedthat, inthreedimensions, solutionsalwaysexist, orthatiftheydoexist, thentheydonotcontainanysingularity • Thesearecalledthe Navier–Stokesexistenceandsmoothnessproblems. • The ClayMathematicsInstitutehascalledthisone of thesevenMillenniumPrizeProblemsinmathematicsandhasoffereda US$1,000,000 prizeforasolutionoracounter-example.

18. MGM for solving incompressible unsteady Navier–Stokes equations • Classical multigrid method have been proved to be extremely efficient on solving pressure Poisson equation, enabling solution to the level of discretization errors in just a few minimal work units, so that the total work invested in the solution grows linearly with the number of variable flow, such as pre-optimization techniques which accelerate the multigrid process before the coarse grid procedure. • A.Brandt, Multigrid techniques:1984 guide with applications to fluid dynamics, Weizmann Institute of Science, 1995. • A.Brandt, I.Yavneh, On multigrid solution of high Reynolds incompressible entering flows, J.Comp.Phys.101 (1992) pp. 151–164.

19. A hybrid multigrid method for incompressible unsteady Navier–Stokes equations • This approach is presented for the high Reynolds incompressible flow, based on multigrid method and sequential regularization method. • The velocity–pressure increment and sequential regular equations are derived from the Navier–Stokes equation. The convergence speed is accelerated by using the pressure increment method and the optimum relaxation sweep methods. • Zhang Shesheng , Department of Applied Mathematics, The Weizmann Institute of Science, Israel, A hybrid multigrid method for the unsteady incompressible Navier–Stokes equations. Applied Mathematics and Computation 138 (2003). Pp. 341–353

20. Algebraic multigrid for effective GPGPU-based solution of nonstationary hydrodynamics problems • The modification is easy to implement and allows us to reduce number of times when the multigrid setup is performed, thus saving up to 50% of computation time with respect to unmodified algorithm. D.E. Demidov, D.V. Shevchenko Russian Academy of Sciences, Kazan Branch of Joint Supercomputer Center Russia. Modification of algebraic multigrid for effective GPGPU-based solution of nonstationary hydrodynamics problems, Journal of Computational Science 3 (2012) 460–462

21. Navier-Stokes equation (1) (2) (3)

22. Approach for solving Navier-Stokes equations • To approximate the time derivative and inertial first space derivatives a method of characteristics is used • Space discretization is carried out by finite element method. It's used a mixed formulation in the finite element method, when a combination of simple finite elements (bilinear for velocities and constant elements for pressure) is applied. • This combination provides stability of pressure calculation with additional application of a numerical filtration. • MGM is used for solving obtained linear equation system • Pironneau, O. On the Transport-Diffusion Algorithm and Its Applications to the Navier-Stokes Equations. Numerische Mathematics 38 (1982): p.p. 309-332.

23. Modification of the equation

24. Space discretization

25. MGM for obtained system Restriction Prolongation The operators of restriction and prolongation are realized for pressure components by other templates: Smoother- Jacobi Method - iteration parameter Project in progress

26. Model problem Convection-diffusion equation V=(v1(x),v2(x)) W. Hackbusch, T. Probst, Downwind Gauss–Seidel smoothing for convection dominated problems, Numer.Linear Algebra Appl. 4 (1997)85–102. • G. Kanschat, Robust smoothers for high-order discontinuous Galerkindiscretisations of advection–diffusion problems, J. Comput. Appl. Math. 218 (2008) 53–60. L.A. Krukier, L.G. Chikina, T.V. Belokon, Triangular skew-symmetric iterative solvers for strongly nonsymmetric positive real systems of equations, Appl. Numer. Math. 41 (2002) 89 -105. G.V. Muratova, E.M. Andreeva, Multigrid method for solving convection–diffusion problems with dominant convection, J.Comput.Appl.Math. 226(2009) 77–83.

27. MGM with TIM as the smoother

28. Smoothersof MGM • TIM • TIM1 • TIM2

29. Numerical experiments Pe = 10, 100, 1 000, 10 000, 100 000 32×32, 64 × 64, 128 × 128, 256 × 256, 512x512

30. Velocity coefficients

31. MGM iteration number Problem1: v1= 1, v2=-1

32. MGM iteration number and CPU-time Problem2 : V1= 1-2x V2= 2y-1

33. MGM iteration number and CPU-time Problem 3: v1= x+y, v2 = x-y

34. MGM iteration number and CPU-time Problem4: v1 = sin (2Pi x), v2 =-2Pi ycos(2Pix)

35. The proof of the convergence of MGM-modification • G.V. Muratova, E.M. Andreeva, Multigrid method for solving convection–diffusion problems with dominant convection, J.Comput.Appl.Math. 226(2009) 77–83. • G.MuratovaMultigrid method for convection-diffusion problems with a small parameter. Math. Modelling, 2001, V.13, N3, P. 69-76 • L.Krukier, G.Muratova The solution of convection – diffusion stationary problem with dominant convection by Multigrid method with special smoothers –Math. Modelling, 2006,V.18, N5, P 63-72. We have used the results • R.P.Fedorenko. A relaxationmethodforsolvingellipticdifferenceequationsRussian J. Comput. Math. andMath. Phys., 1961, V.1, N5,P.1092-1096 • W. Hackbusch. Multigrid method and application - Springer - Verlag, Berlin, 1985, p.293 - 299. • Cao, Z. Convergence of Multigrid Methods for nonsymmetricindefinite problems. Appl.Math.Comp. N28 P.269-288, 1988 • Mandel, J. Multigrid Convergence for nonsymmetric indefinite variational problems and one smoothing step. -Appl. Math. Comput., N19, P.201-216, 1986 

36. Conclusions • Originally introduced as a way to numerically solve elliptic boundary-value problems, multigrid methods, and their various multiscale descendants, have since been developed and applied to various problems in many disciplines. • Two different approaches can be accomplished employing the multigrid method according to the kind of data and information employed and also how the operators deal with them: the geometric multigrid (GMG) and the algebraic multigrid (AMG). • MGM is very actively used in computational fluid dynamics. There are many different hybrid multigrid methods (using preconditioners, domain decompositions, parallel MGM and other ) • For the Navier-Stokes equations it has been shown that by mixing the method of characteristics and the finite element method we are able to obtain first and second order accurate conservative schemes of the upwinding type. • Fourier analysis is a powerful tool to analyze multigrid method quantitatively. Fourier smoothing analysis provides an easy way to optimize values of damping parameters and to predict smoothing efficiency of suggested smoothing methods.

37. Спасибо Thanks 謝謝 Best wishes from Rostov on Don