18.336J/6.335J: Fast methods for partial differential and integral equations. Lecture 13. Instructor: Carlos Pérez Arancibia

Transcription

1 18.336J/6.335J: Fast methods for partial differential and integral equations Lecture 13 Instructor: Carlos Pérez Arancibia MIT Mathematics Department Cambridge, October 24, 2016

2 Fast Multipole Method (FMM) We consider the problem of e ciently compute integrals of the form Z u(x) = G(x, y)q(y)ds y, or its discrete counterpart u i = X j G(x i, y j )q j, i =1,...,N. We start by considering the simplified problem of evaluating u i = X G(x i, y j )q j, x i 2 A y j 2B where A and B are well-separated boxes. B contains N B charges or sources q j at y j 2 B A contains N A evaluation points at x i 2 A

3 Examples of projection/interpolation 1. Polynomial interpolation 2. Collocation 3. Multipole expansions

4 Multipole Expansions We consider x, y 2 R 2 and G(x, y) = log x y, with x 2 A and y 2 B, wherea and B are well-separated boxes. We let r>0 be the radius of the circumscribed circle of B and d>2r be the distance from the center of B to the box A. Theorem. Consider A and B as described above. For all p>0 there exist functions f k (x) and g k (y) for 1 apple k apple 2p + 1, and a constant C p such that 2p+1 max log x y X (x,y)2a B k=1 f k (x)g k (y) apple C p r d p.

5 Why is it called multipole expansion? Note that 1/z x decomposes into a sum of two functions of x =(x 1,x 2 ): cos x x = x 1 x x2 1 log x and sin x x = x 2 x x2 2 log x. These are the expressions of two dipoles oriented horizontally and vertically, respectively. At higher order, there are again two multipoles cos(k x ) x k and sin(k x ) x k corresponding to the non-mixed higher partials derivatives of log.

6 Multipole expansion: three-dimensional case The fundamental solution of the Laplace equation admits the expansion: 1 G(x, y) = 4 x y = 1 1X x n 4 y n+1 = 1X n=0 n=0 1 2n +1 nx m= n x n y n+1 which is valid for x < y. (n m)! (n + m)! nx m= n e im x P m n (cos x ) e im y P m n (cos y ) Y m n ( x, x)y m n ( y, y)

7 Multipole expansion: three-dimensional case From the inequality 4 2n +1 nx m= n Y m n ( x, x)y m n ( y, y) apple 1 we obtain 1 4 x y NX n=0 1 2n +1 x n y n+1 nx m= n Y m n ( x, x)y m n ( y, y) apple 1 4 (1 x / y ) y N+1 x y where x < y.

8 Multipole Expansions: Moments The multipole expansion can be used to compute interactions from charges in box B to potentials in box A in an e cient way. Let u(x i )=Reu(z xi )= X log(z xi z yj )q j y j 2B where z xi = x i1 + ix i2 and z yj = y j1 + iy j2.then u(z xi ) = log z xi X y j 2B q j 1 X k=1 (k 1)! z k x i X y j 2B z k y j k! q j We then define the moments associated with the charges q j at y j 2 B as µ 0 = X y j 2B q j and µ k = X y j 2B z k y j k! q j, k 0,

9 Fast algorithm based on the multipole expansion Here is a simple algorithm to evaluate u i = X G(x i, y j )q j, y j 2B x i 2 A A and B are well-separated boxes. 1. Compute r>0 moments µ k requiring O(rN B ) operations. 2. Assign the moments to multipole components log z xi and (k 1)!/zx k i evaluated at z xi requiring O(rN A ) operations. The total cost amounts to O(r(N A + N B )) operations, instead of O(N A N B ).

10 Multipole expansion as a projection rule In order to see that the multipole expansion provides a projection rule, we interpret (k 1)!/z k x as the following partial derivative of log(z x z y )with respect to z y : (k 1)! z k y k log(z x z y ) zy=0, k 0, which is approximated by a finite di erence formula: log(z y where y B m =(Rez y B m, Im z y B m ) 2 B. z y ) zy=0 X T m,k log(z x z y B m ), x 2 A, k 1, m

11 Multipole expansion as a projection rule Replacing the finite di erence formula in the multipole expansion we recognize the projection rule u(z xi ) log(z xi ) X j Q 0,j q j + X m log(z xi z y B m ) X j Q B m(z yj )q j with projection functions Q 0,j =1, Q B m(z yj )= X k T mk k! z k y j.

12 Multipole expansion as an interpolation rule Let x A be the center of the evaluation box. We then perform a Taylor expansion of log(z x z y ) around z x = z x A to obtain log(z xi z y ) = log(z x A z y )+ X k 1 (z xi z x A) x k log(z x z y ) zx=zxa. Approximating the derivatives by the finite di x k log(z x z y ) zx=zxa X n T nk log(z x A z y ) and replacing it in the multipole expansion we recognize the interpolation rule log(z xi z y ) P 0,i log(z x A z y )+ X n P A n (z xi ) log(z x A n,z y ) with interpolation functions P 0,i =1, P A n (z xi )= X k T nk k! (z x i z x A) k.

13 The Fast Multipole Method We present a fast multipole method (FMM) based on interpolation. The FMM presented here applies to potentials given in terms of kernels G that exhibit smoothness properties similar to those of 1/ x y or log x y. We continue assuming that G is symmetric and we consider the two dimensional case only.

14 FMM setup We consider the problem of e ciently compute integrals of the form Z u(x) = G(x, y)q(y)ds y, x 2 [0, 1] 2, [0, 1] 2. We consider a dyadic tree partitioning of the domain [0, 1] 2 into target boxes A and source boxes B of side length 2 ` with ` =0,...,L. The collection of boxes has associated a quadtree structure: call A p the parent of a box A, and call B c the four children of a box B. The highest level of the tree (` = 0) is called root, while the lowest-level boxes are called leafs. The tree may not be perfect, in the sense that not all leaf boxes occur at the same level.

15 FMM setup

16 FMM setup Definition (well-separated boxes). Two boxes at the same scale ` are said to be well-separated if they are not adjacent to one another, i.e., if B is neither A nor any of its eight nearest neighbors. In that case we say that the boxes are in the far-field of one another: B 2 far(a), or equivalently A 2 far(b). Definition (interaction list). The interaction list of a box A is defined as the set of boxes B, at the same scale as A, in the far-field of A, and such that B p is not in the far field of A p. We denote the interactions list of box A as IL(A). Definition (neighbor list). The neighbor list of a box A is defined as the box A itself and its 8 nearest neighbors at the same scale. The neighbor list of A is denoted as NL(A).

17 FMM setup: far(a) B A

18 A FMM setup: interaction list

19 A p FMM setup: interaction list

20 B FMM setup: interaction list

21 B p FMM setup: interaction list

22 A FMM setup: neighbor list

23 FMM setup In what follows we will make use of the following notation: For a source box B we let the partial potentials from B be denoted by Z u B (x) = G(x, y)q(y)ds y, x 2 far(b). B\ For each evaluation box A, we let the partial potential to A be denoted by Z u far(a) (x) = G(x, y)q(y)ds y, x 2 A. far(a)\ Finally, we use a projection rule G(x, y) = X m G(x, y B m)p B m(y), x 2 far(b), y 2 B, and an interpolation rule G(x, y) = X n P A n (x)g(x A n, y), x 2 A, y 2 far(a).

24 FMM: basic architecture The three main steps of the fast multipole method (FMM) are: 1. The projection rule is successively applied from finer to coarser scales to create canonical charges in sources boxes at all the levels of the tree. 2. The kernel G is used to compute potentials, at selected locations, from these canonical charges. 3. The interpolation rule is successively applied from coarser to finer scales to compute canonical potentials in evaluation boxes at all levels of the tree.

25 FMM basic architecture: multipole-to-multipole Consider canonical charges q B m at nodes y B m, expected to obey u B (x) = X m G(x, y B m)q B m, x 2 far(b), where q B m = Z B\ P B m(y)q(y)ds y. Then observe that the canonical charge qm B at y B m can be expressed as the projection of the canonical charges q B c m of the child boxes B 0 c : q B m = X c X m 0 P B m(y Bc m 0 )q B c m 0. This relation is called a multipole-to multipole (M2M) operation, or M2M translation, or moment-to-moment (M2M) translation. The first step of the FMM is the cascade of M2M translations in an upward step of the quad tree.

26 FMM basic architecture: multipole-to-local At every scale, for every box A, and every box B in the interaction list of A, the canonical charges are converted into potentials via the so-called multipoleto-local (or moment-to-local)(m2l) conversion rule: u A,M2L n = X B2IL(A) X m G(x A n, y B m)q B m. A B

27 FMM basic architecture: local-to-local Consider canonical potentials u A n at the nodes x A n which are expected to obey Z u A n = G(x A n, y)q(y)ds y, far(a)\ The interpolation rule gives a relation between the potentials in A and the canonical potentials u A n : u far(a) (x) = X n P A n (x)u A n, x 2 A. Apply this rule in the case when the canonical potentials are those of the parent box A p : u A,L2L n = X P A p n (x A 0 n )u A p n. 0 n 0 This relation is called a local-to-local (L2L) operation, or L2L translation.

28 FMM basic architecture: upward step Since far(a) = far(a p ) [ IL(A), it remains to add the M2L and L2L potentials to obtain the canonical potentials for box A: u A n = u A,M2L n + u A,L2L n. The last step of the FMM is the cascade of L2L translations, added to the M2L conversions, in a downward pass of the quad tree.