multigrid techniques: past, present and future

The third Russian -Chinese Workshop on Numerical Mathematics and Scientific Computing

Moscow September 11-13, 2013

Multigrid Techniques: past, present and future

Galina MuratovaLev KrukierEvgenia AndreevaComputer CenterSouthern Federal [email protected]

2/37Moscow September 11-13, 2013

Southern Federal UniversityRostov-on-DonRussia

Computer Center of SFU

One of the 9 Federal Universities of Russia

The leading center of education, science and culture in the South of Russia

More than 50 thousands students

36 departments 16 research institutes


The main idea of the topic Multigrid 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


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


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.


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


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


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 'Ein iteratives Verfahren zur schnellen Auflosung

elliptischer Randwertprobleme', Report 76-12, (1976), Mathematisches Institut der Universitat zu 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).


Multigrid method researchers

The Russian Federation National Award 2003 was handed overfor 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. ShaidurovThe 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

http://en.wikipedia.org/wiki/File:RusStatePrize.jpg


Multigrid method

The basic components of multigrid technology are 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)


Two grid method

nhu

SmoothingCoarse-grid correction Compute the defect

Restrict the defect (fine-to-coarse transfer)Solve on coarse-gridInterpolate the correction

(coarse-to-fine transfer) Compute a new approximation

Postsmoothing

1nhhhh uLfd

hh2

hh2 dRd

h2h2h2 dvL

h2hh2h vPv

hnh

nh vuu

2 1nhu

hnhh fuL

hnhh fuL


Multigrid method

Cascade algorithm V – cycle W – cycle Full MGM

PIM 2013 July 1-5, 2013


The structure of the Multigrid method (V – cycle )


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.


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.


The fundamental basis of almost all CFD problems are the Navier–Stokes equations Navier-Stokes equations are also of great interest in

a mathematical sense mathematicians have not yet proved that, in three

dimensions, solutions always exist, or that if they do exist, then they do not contain any singularity

These are called the Navier–Stokes existence and smoothness problems.

The Clay Mathematics Institute has called this one of the seven Millennium Prize Problems in mathematics and has offered a US$1,000,000 prize for a solution or a counter-example.


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.


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


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


Navier-Stokes equation

(1)

(2)

(3)

,112

2

2

2

fyu

xu

RxP

yuv

xuu

tu

,122

2

2

2

fyv

xv

RyP

yvv

xvu

tv

0

yv

xu

,1,01,0

),,,(),,(

),,,(),,(

2

1

tyxgtyxv

tyxgtyxu

).0,,(),,(),0,,(),,(),0,,(),,(

3

2

1

yxgtyxpyxgtyxvyxgtyxu


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.


Modification of the equation

,121

12

2

2

22

fyu

xu

RxP

DtDu

u

,121

22

2

2

22

fyv

xv

RyP

DtDv

v

0

yv

xu

,)()(21

21 2222

yuv

xuu

tu

uDtDu

u

.)()(21

21 2222

yvv

xvu

tv

vDtDv

v


Space discretization

,12 12

2

2

2

Fyu

xu

RxPu

,12 22

2

2

2

Fyv

xv

RyPv

0

yv

xu

njijhyihxyx iiiih ,...,1,0,,,),,(

),(),( 11, jjiiji yyxx


MGM for obtained systemRestriction Prolongation

Smoother- Jacobi Method

h

h

hhR

2

2

121242121

161

h

h

hhP

2

2

121242121

41

h

h

hhR

22

1111

41

h

h

hhP

22 11

11

The operators of restriction and prolongation are realized for pressure components by other templates:

- iteration parameter Project in progress

)(1 fAuAuu nTnn


Model problem Convection-diffusion equation

f)u)(vuvu)(vu(v21

uΔPe1

y2y2x1x1

,0|u д [0,1][0,1]

V = (v1(x),v2(x))0

xvdivV

2

1k k

k

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 Galerkin discretisations 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.


MGM with TIM as the smoother

ululul KKRKKKKA

AAAAAAAAA

*1

1010 *)(210*)(

21

fAu

,1 fAyyy

B nnn

10 AA


Smoothers of MGM

TIM TIM1

TIM2

ul KEBorKEB 22

MKEBorKEB ul 22

luij

n

jiji

uili

KKAMmMnim

KEBorKEB

01

,0,

22


Numerical experiments

f)u)(vuvu)(vu(v21

uΔPe1

y2y2x1x1

xyy)e(sinx)(siny,xu

Pe = 10, 100, 1 000, 10 000, 100 000

32×32, 64 × 64, 128 × 128, 256 × 256, 512x512

,0|u д [0,1][0,1]

hhh fuA LU KKAAAAAAAAAA 1

*1

*010 ),(5.0,0)(5.0,


Velocity coefficients

Problem V1(x, y) V2 (x,y)

1 1 -1

2 1-2x 2y-1

3 X+Y X-Y

4 Sin (2Pi x) -2Pi cos (2Pi x)


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

Pe MGM(Seidel)

MGMTIM

MGMTIM1

MGMTIM2

K=Pe*h/2

10 13 35 30 30 0,1562

100 63 7 5 5 1,5625

1000 A 13 9 9 15,625

10000 A 75 58 58 156,25

100000 A 535 460 430 1562,5


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

Pe MGM(Seidel)

MGMTIM

MGMTIM1

MGMTIM2

K=Pe*h/2

10 22 72 53 50 0,1562100 18 24 19 14 1,56251000 A 16 12 6 15,62510000 A 59 51 32 156,25100000 A 384 522 165 1562,5


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

Pe MGM(Seidel)

MGMTIM

MGMTIM1

MGMTIM2

K=Pe*h/2

10 16 43 35 35 0,1562100 23 10 7 5 1,5625

1000 N 16 12 8 15,62510000 N 74 55 36 156,25

100000 N 570 441 258 1562,5


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

Pe MGM(Seidel)

MGMTIM

MGMTIM1

MGMTIM2

K=Pe*h/2

10 17 42 32 27 0,1562100 N 16 12 7 1,5625

1000 N 30 22 10 15,62510000 N 193 159 65 156,25

100000 N A A A 1562,5


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.Muratova Multigrid 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 relaxation method for solving elliptic difference equations

Russian J. Comput. Math. and Math. 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


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.


Спасибо Thanks 謝謝

Best wishes from Rostov on Don

multigrid techniques: past, present and future

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