Bootstrap AMG. Kailai Xu. July 12, Stanford University

Transcription

1 Bootstrap AMG Kailai Xu Stanford University July 12, 2017

2 AMG Components A general AMG algorithm consists of the following components. A hierarchy of levels. A smoother. A prolongation. A restriction. Coarse grid operators. Fine grid P T P Coarse grid

3 AMG Components Explained Consider two layer AMG to solve Ax = b, which can be easily extended to more layers. The general structure of AMG for one iteration is x x + S(b Ax) x x + PB c P T (b Ax) relaxation correction If we perform relaxation and correction once, the iteration matrix will be (I SA)(I PB c P T A) Smoother. The smoother S is actually pre-conditioner for the relaxation step, for example, modified Jacobi ωd 1. We may perform relaxation step for several times.

4 AMG Components Explained, cont. Prolongation and restriction. Prolongation is also called interpolation in some literatures. We usually define restriction as the conjugate of prolongation, i.e. if the prolongation is P, the restriction will be P T. One way to view AMG is that given the smoother S, or even the sparsity pattern of P, we want to find the best P so that the convergence rate will be optimal. P should work together with S and the grid structure to yield an optimal result. Coarse grid operators. Coarse grid operators is constructed from the prolongation operator. A c = P T AP. On the coarse level, we aim at solving the projected linear system A c x c = P T b which has an approximate inverse(or pre-conditioner) B c A 1 c. Then we lift x c back to the fine space PB c x c.

5 Efficiency of Prolongation According to [1], we have the following remarks on the efficiency of prolongation operator. The notion of strength-of-connection used in the coarsening variables and forming interpolation can be defined from the entries of the system matrix. The eigenvectors with small absolute value eigenvalues are locally smooth in directions of such strong connections. These lowest eigenvectors of the system matrix provide a sufficiently accurate local representation of the other low eigenvectors not effectively treated by the MG relaxation scheme.

6 Bootstrap AMG Bootstrap: not necessarily smart, but try very hard. Bootstrap: (from wiki) a technique of loading a program into a computer by means of a few initial instructions that enable the introduction of the rest of the program from an input device. Bootstrap AMG: We will introduce two bootstrap method here.

7 Computing Prolongation Operator: Least Squares Given the sparsity pattern of P, for example, we may use the weighted sum of the coarse neighbors of every fine grids. v (κ) j fine grid pij coarse grid i We follow the strategy of least squares: for a given set of test vectors {v (1), v (2),..., v (k) } such that the residual is small, or more specifically, algebraically smooth, i.e. Av (i) 0, we want to minimize L(p i ) = for every row p i of P. k κ=1 ω k v (κ) i p ij v (k), j C i j

8 Hierarchy Structure of Multi-grid Method For the multi-grid method, it has a set of prolongation operators P0 1, P2 1,..., PL 1 L 2. We define P l = P0 1 P Pl 1 l, l = 1, 2,..., L 1. Thus the coarse grid operators are naturally defined by A l = Pl T AP l, and we have (x l, x l ) Al = (P l x l, P l x l ) A. Define T l = P T l P l. If A l ω l = λ l T l ω l, we would have Raleigh Quotient of (P l ω l ) = (P lω l, P l ω l ) A (P l ω l, P l ω l ) 2 = λ l

9 Computing Generalized Eigenvalue It is convenient to use ω l s as the test vectors instead of the eigenvalues of A l. It is actually generalized eigenvalue problem. Assume we already have ω l, λ l (there are k e eigenvectors and eigenvalues). Note that A l ω l = λ l T l ω l (P l l 1 )T A l 1 A l l 1 ω l = λ l (P l l 1 )T P l l 1 ω l, and this indicates P l l 1 ω l is an approximation of generalized eigenvectors on level l 1. Thus we can do a relax on the following system (A l 1 λ l 1 T l 1 )ω l 1 = 0, λ l 1 = λ l, for several loops and then update λ l 1 = (A l 1ω l 1, ω l 1 ) (T l 1 ω l 1, ω l 1 ) 2.

10 Computing Generalized Eigenvalue, cont. The algorithm will look like following steps. Compute the eigenvalues of A L 1 ω (κ) L directly at the coarsest level. = λ (κ) L 1 = λ(κ) L 1 T L 1ω (κ) L 1, κ = 1, 2,..., k e At level l = L 2, L 3,..., 0, relax the following linear system (A l 1 λ l 1 T l 1 )ω l 1 = 0, λ l 1 = λ l, using the scheme ω l 1 ω l 1 S l (A l 1 λ l T l 1 )ω l 1 where S l is a pre-conditioner for A l 1 λ l T l 1.

11 Algorithm Overview The above graph illustrate the ideas of the algorithm. At first, we do not have any information about A l so we use a random set of test vectors to compute P0 1. Then we restrict the test vectors onto coarser grid and can be used to form a new set of test vectors on the coarse grid with relaxation. For example, v (κ) i v (κ) i 1 a ii r (κ) i

12 Yet another Bootstrap algorithm Consider 0 a ii e i + j N i a ij e j (1) Therefore, we intend to compute prolongation weights w ij such that e i = j P i ω ij e j, where P i is the set of C nodes around i. In consideration of (1), we have e i = j P i ω ij e j = 1 a ii Thus we let a ij e j j N i j N i a ij a ii j P i a ij e j j P i a ij = j P i a ij j N i a ω ij = ij, i F, j C a ij a ii j P i j N i a ij j P i a ij a ij a ii e j

13 Reference Achi Brandt, J Brannick, Karsten Kahl, and Irene Livshits. Bootstrap amg. SIAM Journal on Scientific Computing, 33(2): , 2011.