Multigrid Method ZHONG-CI SHI. Institute of Computational Mathematics Chinese Academy of Sciences, Beijing, China. Joint work: Xuejun Xu

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1 Multigrid Method ZHONG-CI SHI Institute of Computational Mathematics Chinese Academy of Sciences, Beijing, China Joint work: Xuejun Xu

2 Outline Introduction Standard Cascadic Multigrid method(cmg) Economical Cascadic Multigrid method(ecmg) Summary Appendix

3 1. Introduction Multigrid techniques give algorithms that solve sparse linear systems Ax = b of N unknowns with O(N) computational complexity for large classes of problems. The systems arise from approximation of partial differential equations. The condition number is usually very large when the mesh size becomes small.

4 Background: Idea (Finite difference methods), 60.s: [Fedorenko, 1964], [Bachvalov, 1966] Applications, 70.s: [Brandt, 1973], etc. Theoretical Analysis, 80.s: [Bank and Dupont,1981], [Braess and Hackbusch,1983], [Bramble, Pasciak and Xu, 1987], etc. Features: For many iterative methods one can prove that the iterates {x i } satisfy: x i+1 x δ x i x for δ (0, 1) in certain norm. Jacobi, Gauss-Seidel: δ = 1 O(h 2 ). MG: δ < 1 is independent of the mesh size h.

5 Observation Two solution techniques Iterative method: Jacobi, Gauss-Seidel Easy to perform (matrix vector), but slow convergence with operations O(N 2 ) Reason: the higher frequencies of the residual can be supressed quickly, but lower frequencies are reduced very slowly. After few iterations, the residual becomes smooth, but not small. Direct method: Gauss elimination Good for small-size problem Rather difficult to perform for large-size problem, the accuracy is not high (round error accumulation)

6 MGM: Combination of the above two techniques Using different techniques on different levels of grid 1. Fine grid (smoothing) Iterative method to smooth the residual 2. Coarse grid (correction) Direct method to solve the residual equation 3. Add the correction to the initial residual a more precise residual 4. Back to 1. Total operations: O(N)

7 Two types of MGM: 1. W cycle: two corrections per cycle (1980 ) Difficult to carry out, easy to analyze 2. V cycle: one correction per cycle (1990 ) Easy to carry out, difficult to analyze New scheme Cascadic scheme No correction at all, only iterations on each level. Very easy to carry out, difficult to analyze

8 Model problem: The second order elliptic problem 2 i,j=1 The differential operator is uniformly elliptic: x i (a ij u x j ) = f in Ω, u = 0 on Ω, cξ t ξ d a ij ξ i ξ j Cξ t ξ (x, y) Ω, ξ R d. i,j=1 The variational form: find u H 1 0(Ω) such that a(u, v) = (f, v) v H 1 0(Ω), where the bilinear form is a(u, v) = d Ω i,j=1 a ij u x j v x i dx u, v H 1 (Ω). First Prev Next Last Go Back Full Screen Close Quit

9 Let Γ l (l 0) denote triangulation of Ω, which is generated uniformly from the initial triangulation Γ 0. The mesh size is h l = h 0 2 l. For instance, in 2D case, Γ l is obtained by linking the midpoints of three edges of triangle or the midpoints of counteredges of rectangle on Γ l 1. Let V l L 2 (Ω) denote the finite element space on Γ l. On each level, we define a l (u, v) = K Γ l d K i,j=1 a ij u x j v x i dx and v i,l = K Γ l v 2 i,k i = 0, 1, here v 0 = v 0,l.

10 It is easy to see that the following properties are satisfied for V l : (1) a l (u, v) C u 1,l v 1,l u, v V l. (2) a l (v, v) C v 2 1,l v V l. We assume that v 1,l is a norm over V l. The finite element approximation on V l : Find u l V l such that a l (u l, v) = (f, v) v V l. It is known that the above equation has a unique solution u l V l. Define the operator A l : V l V l as follows (A l u, v) = a l (u, v) u, v V l. Then we get a linear system A l u l = f l, here f l V l and (f l, v) = (f, v) v V l.

11 2. Cascadic Multigrid Method Bornemann and Deuflhard (1996) Shaidurov (1996) Shi and Xu (1998) Braess and Dahmen (1999) Stevensen (2002) Advantages: Simplicity No coarse grid corrections One way multigrid

12 Cascadic Multigrid Algorithm (1) Set u 0 0 = u 0 ˆ=u 0 and let u 0 l = I l u l 1. (2) For l = 1,..., L, do iterations: (3) Set u l ˆ=um l l. u m l l = C m l l u 0 l. Choose the iterative operator C l : V l V l on the level l and assume that there exists a linear operator T l : V l V l such that u l C m l l u 0 l = T m l l (u l u 0 l ). C l : Gauss-Seidel, Jacobi, CG.

13 Full Multigrid Algorithm (1) Set u 0 0 = u 0 ˆ=u 0 and let u 0 l = I l u l 1. (2) For l = 1,..., L, do iterations: u m l l = MG(l, u 0 l, r). (3) Set u l ˆ=um l l, where MG(l, u 0 l, r) denotes to do r-th multigrid iterations. Difference: No coarse grid corrections in CMG.

14 Optimal Cascadic Multigrid Method: If we obtain both the Accuracy u L u L L u u L L which means that the iterative error is comparable to the approximation error of the finite element method and the multigrid Complexity amount of work = O(n L ), n L = dimv L.

15 General Framework: (Shi and Xu, 1998) Three assumptions: Define the intergrid transfer operator I l : V l 1 V l which is assumed to satisfy the condition: ( H1) (1). v I l v t 1,l Ch l v t,l 1 v V l 1, (2). u l I l u l 1 t 1,l Ch 2 l f 0, where u l is the finite element solution on V l.

16 Choose the iterative operator C l : V l V l on the level l and assume that there exists a linear operator T l : V l V l such that u l C m l l u 0 l = T m l l (u l u 0 l ) ( H2) (1). T m l l v l C h 1 l m γ v t 1,l v V l, l (2). T m l l v l v l v V l, where m l is the number of iteration steps on the level l, and γ is a positive number depending on the given iteration.

17 Introduce a projection operator P l : V l 1 + V l V l defined by a l (P l u, v) = a l (u, v) v V l. From the definition, it is easily seen that P l v l v l 1 v V l 1. For the operator P l, we assume that (H3) v P l v t 1,l Ch l v t,l 1 v V l 1. Note that the project operator P l is used only in the convergence analysis, it is not needed in real computation.

18 Note that the mesh size on the level l is h l = h L 2 L l. Let m l (0 l L) be the smallest integer satisfying m l β L l m L (1) for some fixed β > 1, where m L is the number of iterations on the finest level L. Under the assumptions (H1), (H2) and (H3), if m l, the number of iterations on level l is given by (1), then the accuracy of the cascadic multigrid is u L u L L 1 C 1 ( 2 h L β ) m γ γ L f 0 for β > 2 1 γ, CL h L m γ f 0 for β = 2 1 γ. L

19 Complexity of CMG: The computational cost of the cascadic multigrid is proportional to L C 1 m l n l 1 β m L n L for β < 2 d, 2 l=0 d CLm L n L for β = 2 d, where d is the dimension of the domain Ω.

20 Main Results: Suppose (H1), (H2) and (H3) hold. (1) If γ = 1 2, d = 3, then the cascadic multigrid is optimal. (2) If γ = 1, d = 2, 3, then the cascadic multigrid is optimal. (3) If γ = 1 2, d = 2, and the number of iterations on the level L is then the error in the energy norm is m L = [m L 2 ], and the complexity is u L u L L C h L m 1 2 f 0, L m l n l cm n L (1 + logn L ) 3. l=0 It means that the cascadic multigrid is nearly optimal in this case.

21 Applications: 1. Lagrange conforming elements In this case, V l 1 V l. Therefore, both the transfer operator I l and the projection operator P l are the identity I, so that (H1)-1 and (H3) are trivial. By finite element error estimates, so (H1)-2 also holds. u l I l u l 1 0 = u l u l 1 0 u l u 0 + u u l 1 0 Ch 2 l f 0, (H2) has been proved by Bank and Dupont (1981)

22 2. Nonconforming elements Let V l be the P 1 nonconforming finite element space. Define the intergrid transfer operator I l : V l 1 V l as follows: let m be the midpoint of a side of triangles in Γ l. (1) If m lies in the interior of a triangle in Γ l 1, then (I l v)(m) := v(m). (2) If m lies on the common edge of two adjacent triangles T 1 and T 2 in Γ l 1, then (I l v)(m) := 1 2 (v T 1 (m) + v T2 (m)). (H1) is valid for the P1 nonconforming element. Using a duality argument, we can show that (H3) is valid for the P1 nonconforming element. (H2) is obvious. Other nonconforming elements: the Wilson element, the Carey element.

23 3. The plate bending problem Let Ω be a convex polygon in R 2, the variational form of the plate bending problem is to find u H 2 0(Ω) such that a(u, v) = Ω a(u, v) = (f, v) v H 2 0(Ω), { u v + (1 σ)(2 2 u 2 v 2 u 2 v x 1 x 2 x 1 x 2 x 2 1 x 2 2 (f, v) = fvdx, 2 u x 2 2 where f L 2 (Ω), and 0 < σ < 1 2 is the Possion ratio. In this case V = H2 0(Ω). Ω 2 v )}dx, x 2 1

24 Morley element The Morley finite element space is nonconforming and nonnested. We define the intergrid transfer operator as follows: (1) I l v = 0 at the vertex of Γ l along Ω and I lv n = 0 at the midpoint of each edge of Γ l along Ω. (2) Let p be a vertex of Γ l inside Ω. If p is also a vertex of Γ l 1, then (I l v)(p) = v(p). If p is the midpoint of a common edge of two triangles K 1 and K 2 in Γ l 1, then (I l v)(p) = 1 2 (v K 1 (p) + v K2 (p)).

25 (3) Let m be a midpoint of the edge e of Γ l inside Ω and n denote the unit normal of e. If m is inside of a triangle in Γ l 1, then I l v v (m) = n n (m). If m is on the common edge of two triangles K 1 and K 2 in Γ l 1, then We can prove (Shi and Xu 1998) I l v n (m) = 1 2 [ v K 1 n (m) + v K 2 n (m)]. (H1)-(H3) hold for the Morley element.

26 3. Economical Cascadic Multigrid Method Advantages: Less operations on the each level. Computational costs can be greatly reduced.

27 Example: CMG: m l = [m L β L l ] l = 1,, L, ECMG: a new criterion for choosing smoothing steps on each level. When d = 2, that is: (i) If l > L 0, then (ii) If l L 0, then m l = [m L β L l ]. m l = [m 1 2 (L (2 ε 0 )l)h 2 l ], here L 0 is a positive integer which will be determined later, and 0 < ε 0 1 is a fixed positive number, m L = m 0 (L L 0 ) 2. Note that in the standard cascadic algorithm m L = m 0 L 2.

28 Consider the second order elliptic problem and use Gauss-Seidel iteration as a smoother. For simplicity, we only consider 2D case. Then L 0 = L/2 = 4, ε 0 = 1 2, h 0 = 1 4, and m 0 = 2, m = 1. The following table shows the iteration steps on each level for the two methods. Level CMG ECMG We will prove that the new cascadic multigrid algorithm is still optimal in both accuracy and complexity as the standard CMG.

29 Basic smoothers: For the basic iterative smoothers, we have where T m l l v l ɛ(m l, h l ) v 0 v V l, C h 1 l, for m ɛ(m l, h l ) = m 1 l < κ l, 2 l C2 m l 1 κ l, for m l κ l, where κ l denotes the condition number of the matrix A l.

30 New criteria for choosing iteration parameters Determine the largest positive integer L 0, which satisfies the following inequality β L L 0 m L κ L0. For simplicity, we assume that κ l = h 2 l, then by a simple manipulation, L 0 Llogβ + logm L + 2logh 0. logβ + 2log2 Define the level parameter L 0 as the largest positive integer, which satisfies the following inequality: L 0 min{ Llogβ + logm L + 2logh 0 dl, }. (2) logβ + 2log2 2 + d

31 New criteria: 1. When d = 2, (i) If l > L 0, then (ii) If l L 0, then m l = [m L β L l ]. m l = [m 1 2 (L (2 ε 0 )l)κ l ]. In practical implementation, because κ l h 2 l, the above terms can be replaced by: m l = [m 1 2 (L (2 ε 0 )l)h 2 l ].

32 2. When d = 3, (i) If l > L 0, then m l = [m L β L l ]. (ii) If l L 0, there are two cases: (a) If (2 ε 0 )L 0 L, then m l = [m 1 2 (L (2 ε 0 )l)κ l ]. (b) If(2 ε 0 )L 0 > L, then there exists a largest positive integer L < L 0 such that (2 ε 0 )L L. For all l L, we choose m l as follows: m l = [m 1 2 (L (2 ε 0 )l)κ l ], for all L < l L 0, we choose m l as follows: m l = [m 1 2 κ l ].

33 Economical Cascadic Multigrid (1) Set u 0 0 = u 0 ˆ=u 0 and let u 0 l = I l u l 1. (2) For l = 1,..., L, do u m l l (3) Set u l ˆ=um l l, where the m l is determined by the new criteria. = C m l l u 0 l.

34 Main Results: Suppose (H1) and (H3) hold. (1) If d = 3, then the cascadic multigrid is optimal, i.e.,the error in the energy norm is u L u L L Ch L f 0 ( h 0 m 1 2 L + h 0 m 1 2 ) and the complexity is L l=0 m l n l C(h 2 0 m β 2 d m L )n L.

35 (2) If d = 2, and the number of iterations on the level L is m L = [m 0 (L L 0 ) 2 ], then the error in the energy norm is and the complexity of computation is L l=0 u L u L L Ch L f 0 ( h 0 m h 0 m ) m l n l C(m h m 0 (L L 0 ) 3 )n L. It means that the cascadic multigrid is nearly optimal in this case.

36 CG method as a smoother: Main results: Suppose (H1) and (H2) hold. For d = 2, 3, the cascadic multigrid with CG smoother is optimal, i.e., the error in the energy norm is and the complexity of computation is u L u L L Ch L f 0 ( h 0 m L + h 0 m ) L l=0 m l n l C(h β 2 d )n L. It means that the cascadic multigrid is optimal in this case.

37 Applications: Conforming and Nonconforming elements for the second order problem. Nonconforming plate elements, Morley element, Adini element.

38 Numerical experiments: We use ECMG and standard CMG algorithms to solve the following problem: { u = f, in Ω = ( 1, 1) ( 1, 1), u = 0, on Ω, where f is chosen such that the exact solution of the problem is u(x, y) = (1 x 2 )(1 y 2 ).

39 ECMG and CMG with GS smoother (P1 conforming element) CMG ECMG Mesh Energy error CPU Energy error CPU e-03 38(s) e-03 20(s) e (s) e (s) e (s) e (s)

40 ECMG and CMG with CG smoother (P1 conforming element) CMG ECMG Mesh Energy error CPU Energy error CPU e-03 14(s) e-03 10(s) e-03 66(s) e-03 42(s) e (s) e (s)

41 ECMG and CMG with GS smoother (P1 nonconforming element) CMG ECMG Mesh Energy error CPU Energy error CPU e-02 40(s) e-02 19(s) e (s) e-03 98(s) e (s) e (s)

42 ECMG and CMG with CG smoother (P1 nonconforming element) CMG ECMG Mesh Energy error CPU Energy error CPU e-02 18(s) e-03 14(s) e-03 72(s) e-03 48(s) e (s) e (s)

43 4. Summary Simplicity No coarse grid corrections, one way mutigrid ECMG: less operations on the each level. ECMG: computational costs can be greatly reduced.

44 5. Appendix Efficiency Computer vs Algorithm Computer from 10 3 (Kilo) (T era), increase 10 9 (Giga) Algorithm (solution of linear system) 1950 (Gauss elimination) 2000 (multigrid method) Convergence rate: from O(N 3 ) O(N), increase O(N 2 ) 3D problem: N = (10 2 ) 3 (Mega) Operations: , reduce (T era)!!

45 THANK YOU!