Adaptive Application of Green s Functions:

Transcription

1 Adaptive Application of Green s Functions: Fast algorithms for integral transforms, with a bit of QM Fernando Pérez With: Gregory Beylkin (Colorado) and Martin Mohlenkamp (Ohio University) Department of Applied Mathematics University of Colorado, Boulder University of Michigan at Ann Arbor April 12, 2007

2 Outline 1 Context and motivation F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 2 / 49

3 Outline 1 Context and motivation 2 Multiresolution analysis in d = 1 F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 2 / 49

4 Outline 1 Context and motivation 2 Multiresolution analysis in d = 1 3 Multidimensional problems F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 2 / 49

5 Outline 1 Context and motivation 2 Multiresolution analysis in d = 1 3 Multidimensional problems 4 d 1: beyond separation of variables F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 2 / 49

6 Outline 1 Context and motivation 2 Multiresolution analysis in d = 1 3 Multidimensional problems 4 d 1: beyond separation of variables 5 Quantum Mechanics F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 2 / 49

7 Outline 1 Context and motivation 2 Multiresolution analysis in d = 1 3 Multidimensional problems 4 d 1: beyond separation of variables 5 Quantum Mechanics F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 3 / 49

8 Physical problems formulated as PDEs The Laplace/Poisson equations u = f The Schrödinger equation (for stationary states) ( 12 + V ) ψ = Eψ The modifed Stokes equation (time-stepping schemes for Navier-Stokes): αv µ v + p = f v = 0 A good fraction of the world s (scientific) computing time is devoted to the solution of this type of problem. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 4 / 49

9 Numerical approaches Broadly and loosely speaking, there are two main options; each has its own set of difficulties. Discretize differential operator and invert Sparse matrices (fast solvers) That represent unbounded operators... F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 5 / 49

10 Numerical approaches Broadly and loosely speaking, there are two main options; each has its own set of difficulties. Discretize differential operator and invert Sparse matrices (fast solvers) That represent unbounded operators... And hence are ill-conditioned and require pre-conditioning. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 5 / 49

11 Numerical approaches Broadly and loosely speaking, there are two main options; each has its own set of difficulties. Discretize differential operator and invert Sparse matrices (fast solvers) That represent unbounded operators... And hence are ill-conditioned and require pre-conditioning. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 5 / 49

12 Numerical approaches Broadly and loosely speaking, there are two main options; each has its own set of difficulties. Discretize differential operator and invert Sparse matrices (fast solvers) That represent unbounded operators... And hence are ill-conditioned and require pre-conditioning. Write integral formulation and apply integral operator (Green s functions) Well-conditioned objects... That lead to dense matrices F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 5 / 49

13 Numerical approaches Broadly and loosely speaking, there are two main options; each has its own set of difficulties. Discretize differential operator and invert Sparse matrices (fast solvers) That represent unbounded operators... And hence are ill-conditioned and require pre-conditioning. Write integral formulation and apply integral operator (Green s functions) Well-conditioned objects... That lead to dense matrices And don t easily generalize to multiple dimensions. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 5 / 49

14 What are we after? Immediate Goals Numerical algorithms with finite but controlled precision. Multiscale, fully adaptive algorithms. Approximations are a cousin of the Fast Multipole Method (FMM), but easier to generalize in dimension and kernel. Green s functions: G(r r ). Many fundamental physical processes are 2-body interactions (Nature cooperates). F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 6 / 49

15 What are we after? Immediate Goals Numerical algorithms with finite but controlled precision. Multiscale, fully adaptive algorithms. Approximations are a cousin of the Fast Multipole Method (FMM), but easier to generalize in dimension and kernel. Green s functions: G(r r ). Many fundamental physical processes are 2-body interactions (Nature cooperates). Overall program The curse of dimensionality : the exponential rise in complexity of many algorithms with the underlying physical dimension. Multiparticle Schrödinger equation, Navier-Stokes. A toolbox of reliable algorithms for the efficient application of integral transforms in multiple dimensions. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 6 / 49

16 Multiresolution algorithms in multiple dimensions Key mathematical ideas: 1 Multiresolution analysis (wavelets): sparse matrix representations for a large class of kernels. 2 Separated representations: reduction of dimensionality cost. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 7 / 49

17 Multiresolution algorithms in multiple dimensions Key mathematical ideas: 1 Multiresolution analysis (wavelets): sparse matrix representations for a large class of kernels. 2 Separated representations: reduction of dimensionality cost. Group effort over many years (1988-today): 1 Gregory Beylkin, Lucas Monzón, Christopher Kurcz - CU Boulder 2 Martin Mohlenkamp - Ohio University 3 Robert Harrison, George Fann, Takeshi Yanai, Zhengting Gan - ORNL 4 Vani Cheruvu - (now at NCAR) 5 Robert Cramer - (now at Raytheon) F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 7 / 49

18 Functions Operators Operator application d = 1 recap Outline 1 Context and motivation 2 Multiresolution analysis in d = 1 3 Multidimensional problems 4 d 1: beyond separation of variables 5 Quantum Mechanics F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 8 / 49

19 Functions Operators Operator application d = 1 recap Multiresolution analysis, formally Let V j L 2 (R 3 ) denote subspaces of the multiresolution analysis corresponding to a multiwavelet basis of order p (p is the number of nodes per interval) We have V 0 V 1 V 2 V j or, for each V n, V n = V 0 W 0 W 1 W n 1 where V j W j = V j+1. The subspace V 0 is spanned by products of orthogonal polynomials up to degree p 1 in each variable, localized within cubes shifted by n Z 3. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 9 / 49

20 Functions Operators Operator application d = 1 recap Multiresolution analysis, intuitively Imagine a simple signal f(t) you want to study: F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 10 / 49

21 Functions Operators Operator application d = 1 recap Multiresolution analysis, intuitively Imagine a simple signal f(t) you want to study: At each scale n, divide the unit interval [0,1] into 2 n binary subintervals: F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 10 / 49

22 Multiresolution basics (2) And compute: Average (s) values of function at level n : space V n. Differences (d) between successive levels: space W n = V n+1 V n.

23 Multiresolution basics (2) And compute: Average (s) values of function at level n : space V n. Differences (d) between successive levels: space W n = V n+1 V n. f(t) can studied (compressed, denoised,...) from {V 0,W 0,W 1,...} : V2 V4 V9-full W2 W4 W The d coefficients are small and localized around changes. We ll use multiwavelets: p coefficients per subinterval ( high order Haar ).

24 Functions Operators Operator application d = 1 recap Adaptive subdivision of the unit interval in R d Simple recursive subdivision produces a d-binary tree on the unit interval, with p d coefficient blocks on the leaves: (0,1) (1,1) (0,0) (1,0) (2,1) (3,1) (2,0) (3,0) (4,1) (4,0) (5,0) (5,1) F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 12 / 49

25 Functions: adaptive, controlled accuracy decompositions Nnod = 12, ǫ = 1.0e 10, Nblocks = 21

26 Functions: adaptive, controlled accuracy decompositions Nnod = 12, ǫ = 1.0e 10, Nblocks = 21 Nnod = 10, ǫ = 5.0e 11, Nblocks = 634

27 Functions Operators Operator application d = 1 recap Applying an operator to a function: g = Tf Definitions Projections onto various subspaces For the subspaces V j, P j : L 2 ([0,1] d ) V j For the subspaces W j, the orthogonal projector Q j : L 2 ([0,1] d ) W j is defined as Q j = P j+1 P j. For an operator T : L 2 ([0,1] d ) L 2 ([0,1] d ), we define its projection T j : V j V j as T j = P j TP j. For a function f, f j = P j f. We can think of f j as a vector with 2 j p entries. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 14 / 49

28 Functions Operators Operator application d = 1 recap Applying an operator to a function: g = Tf Definitions Projections onto various subspaces Goal For the subspaces V j, P j : L 2 ([0,1] d ) V j For the subspaces W j, the orthogonal projector Q j : L 2 ([0,1] d ) W j is defined as Q j = P j+1 P j. For an operator T : L 2 ([0,1] d ) L 2 ([0,1] d ), we define its projection T j : V j V j as T j = P j TP j. For a function f, f j = P j f. We can think of f j as a vector with 2 j p entries. Build an efficient algorithm to compute g n = Tf n for a given input function f, with n chosen adaptively and controlled accuracy. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 14 / 49

29 Operators (d = 1): sparse representations Again, project the operator on each scale and use differences: T n = T 0 + (T 1 T 0 ) + (T 2 T 1 ) +... = T 0 + n j=1 D j. T j D j j = 3

30 Operators (d = 1): sparse representations Again, project the operator on each scale and use differences: T n = T 0 + (T 1 T 0 ) + (T 2 T 1 ) +... = T 0 + n j=1 D j. T j D j j = 3 j = 7

31 Operator application in d = 1 The non-standard form We wish to compute g = Tf using the telescopic expansion ĝ 0 = T 0 f 0 ĝ 1 = [T 1 (T 0 )]f 1 ĝ 2 = [T 2 (T 1 )]f ĝ j = [T j (T j 1 )]f j

32 Operator application in d = 1 The non-standard form We wish to compute g = Tf using the telescopic expansion ĝ 0 = T 0 f 0 ĝ 1 = [T 1 (T 0 )]f 1 ĝ 2 = [T 2 (T 1 )]f ĝ j = [T j (T j 1 )]f j For this, we must construct a redundant tree out of our adaptive one. Nnod = 8, ǫ = 1.0e 04, Nblocks = 17 Nnod = 8, ǫ = 1.0e 04, Nblocks = 33 =

33 Functions Operators Operator application d = 1 recap Natural-scale application The modified nsf Observations We do not have to apply the blocks of the lifted operator T j 1 on the finer scale j We first truncate the bands on scale j for the difference (T j T j 1 ). This defines a banded operator on scale j, T b j j with band b j. We then bring back T j 1 to its natural scale, j 1. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 17 / 49

34 Functions Operators Operator application d = 1 recap Natural-scale application The modified nsf Observations We do not have to apply the blocks of the lifted operator T j 1 on the finer scale j We first truncate the bands on scale j for the difference (T j T j 1 ). This defines a banded operator on scale j, T b j j with band b j. We then bring back T j 1 to its natural scale, j 1. Algorithmic implications All operations are done on indexing tables These tables are then used to drive the application of the operator F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 17 / 49

35 The Modified NSF A graphical illustration of what we gain for T j (j = 5 shown):

36 The Modified NSF A graphical illustration of what we gain for T j (j = 5 shown):

37 Natural-scale application Mathematically, this means that: ĝ 0 = [T 0 T b 1/2 0 ]f 0 ĝ 1 = [T 1 T b 2/2 1 ]f 1 ĝ 2 = [T 2 T b 3/2 2 ]f ĝ j = T b j j f j and the fully assembled result for any given scale n is given by: [( ) [( ) g n = T b n n f n+ T b n 1 n 1 T b n/2 n 1 f n 1 + T b n 2 n 2 T b n 1/2 n 2 f n [ [( ]] ]]... + T 0 T b 1/2 0 )f 0....

38 Natural-scale application Mathematically, this means that: ĝ 0 = [T 0 T b 1/2 0 ]f 0 ĝ 1 = [T 1 T b 2/2 1 ]f 1 ĝ 2 = [T 2 T b 3/2 2 ]f ĝ j = T b j j f j and the fully assembled result for any given scale n is given by: [( ) [( ) g n = T b n n f n+ T b n 1 n 1 T b n/2 n 1 f n 1 + T b n 2 n 2 T b n 1/2 n 2 f n [ [( ]] ]]... + T 0 T b 1/2 0 )f Observation Inter-scale interactions are dealt with by computationally inexpensive projections steps, not at operator application time.

39 Functions Operators Operator application d = 1 recap Adaptive natural-scale application A graphical representation F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 20 / 49

40 Functions Operators Operator application d = 1 recap Numerical example in 1D Consider the periodic analogue of the Hilbert transform. (Cf)(y) = p.v. applied to the periodic function Z 1 f(x) = e a(x+k 1/2)2 (Cf)(y) = i k Z 0 cot(π(y x))f(x)dx, π sign(n)e n2 π 2 /a e 2πiny a n Z F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 21 / 49

41 Functions Operators Operator application d = 1 recap Numerical example in 1D Consider the periodic analogue of the Hilbert transform. (Cf)(y) = p.v. applied to the periodic function Z 1 f(x) = e a(x+k 1/2)2 (Cf)(y) = i k Z 0 cot(π(y x))f(x)dx, π sign(n)e n2 π 2 /a e 2πiny a n Z F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 21 / 49

42 Functions Operators Operator application d = 1 recap d = 1, lessons learned The Good 1 Sparse representations of operators via multiwavelets lead to fast algorithms. 2 Accuracy is guaranteed by construction. 3 We can efficiently handle multi-scale interactions. 4 While the example was a convolution (fast via FFT?), we have an automatically adaptive algorithm. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 22 / 49

43 Functions Operators Operator application d = 1 recap d = 1, lessons learned The Good 1 Sparse representations of operators via multiwavelets lead to fast algorithms. 2 Accuracy is guaranteed by construction. 3 We can efficiently handle multi-scale interactions. 4 While the example was a convolution (fast via FFT?), we have an automatically adaptive algorithm. The Bad This approach does not directly extend to d > 1. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 22 / 49

44 Outline 1 Context and motivation 2 Multiresolution analysis in d = 1 3 Multidimensional problems 4 d 1: beyond separation of variables 5 Quantum Mechanics F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 23 / 49

45 The curse of dimensionality A simple observation: numerical algorithms (C = AB, y = Ax) in d physical dimensions scale exponentially with d in complexity. Not good. A typical example: Poisson s equation (electromagnetics, gravity,... ): 2 φ(r) = ρ(r) F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 24 / 49

46 The curse of dimensionality A simple observation: numerical algorithms (C = AB, y = Ax) in d physical dimensions scale exponentially with d in complexity. Not good. A typical example: Poisson s equation (electromagnetics, gravity,... ): 2 φ(r) = ρ(r) A Green s function solution (free space, d = 3, ignore constants) : Z φ(r) = Z G(r r )ρ(r )d 3 r = 1 r r ρ(r )d 3 r. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 24 / 49

47 The curse of dimensionality A simple observation: numerical algorithms (C = AB, y = Ax) in d physical dimensions scale exponentially with d in complexity. Not good. A typical example: Poisson s equation (electromagnetics, gravity,... ): 2 φ(r) = ρ(r) A Green s function solution (free space, d = 3, ignore constants) : Z φ(r) = Z G(r r )ρ(r )d 3 r = If we discretize using a global basis, this becomes: φ ijk = N i j k =1 G ii,jj,kk ρ i j k 1 r r ρ(r )d 3 r. Applying an integral kernel is a matrix-vector multiplication. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 24 / 49

48 Can we do this efficiently for d > 1? What if we could write: M G ii,jj,kk = w m F m ii F m jj F m kk. m=1 F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 25 / 49

49 Can we do this efficiently for d > 1? What if we could write: M G ii,jj,kk = w m F m ii F m jj F m kk. m=1 We could separate the different dimensions: φ ijk = M m=1 The problem partially factorizes. w m F m i ii j F m jj k F m kk ρ i j k F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 25 / 49

50 Can we do this efficiently for d > 1? What if we could write: M G ii,jj,kk = w m F m ii F m jj F m kk. m=1 We could separate the different dimensions: φ ijk = M m=1 The problem partially factorizes. w m F m i And if we can construct sparse F m ii multidimensional algorithm. ii j F m jj k F m kk ρ i j k representations, we may have a fast, F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 25 / 49

51 Operators(d > 1): Gaussians to the rescue We can approximate a wide class of kernels as sums of Gaussians: M 1 r r w m e τ m r r 2, m=1 with controlled accuracy ε over a large dynamic range [M O( logε)]: This gives us the factorization we wanted for our kernel: M G ii,jj,kk = w m F m ii F m jj F m kk. m=1

52 Sparsity in d > 1 Problem How do we construct sparse multidimensional operators? F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 27 / 49

53 Sparsity in d > 1 Problem How do we construct sparse multidimensional operators? Solution Use the separated structure of the modified ns-form T l j (T l j 1) = = M m=1 w m F j;ml 1 F j;ml 2 F j;ml 3 ( M w m F j 1;ml 1 F j 1;ml 2 F j 1;ml 3 m=1 M w m F j;ml 1 F j;ml 2 F j;ml 3 m=1 M m=1 w m (F j 1;ml 1 ) (F j 1;ml 2 ) (F j 1;ml 3 ). ) F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 27 / 49

54 Norm considerations for sparsity in d > 1 Solution... The norm of T l j (Tl j 1 ) decays rapidly with l, l = (l 1,l 2,l 3 ). We can define With these, we can estimate T l j (T l j 1) 1 6 N j;m;l = F dif j;m;l (F j 1;m;l ), N j;m;l F = F j;m;l, N j;m;l F = (F j 1;m;l ), M m=1 [ w m sym N j;m;l 1 F N j;m;l 2 dif N j;m;l 1 dif N j;m;l 2 F N j;m;l 3 F + N j;m;l 3 F + N j;m;l 1 F N j;m;l 2 F N j;m;l 3 dif ].

55 Norm considerations for sparsity in d > 1 Solution... The norm of T l j (Tl j 1 ) decays rapidly with l, l = (l 1,l 2,l 3 ). We can define With these, we can estimate Note T l j (T l j 1) 1 6 N j;m;l = F dif j;m;l (F j 1;m;l ), N j;m;l F = F j;m;l, N j;m;l F = (F j 1;m;l ), M m=1 [ w m sym N j;m;l 1 F N j;m;l 2 dif N j;m;l 1 dif This is strictly based on one-dimensional quantities. N j;m;l 2 F N j;m;l 3 F + N j;m;l 3 F + N j;m;l 1 F N j;m;l 2 F N j;m;l 3 dif ].

56 Sparsity also in the m direction? The Gaussian expansion gave us separation of directions... F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 29 / 49

57 Sparsity also in the m direction? The Gaussian expansion gave us separation of directions... at the cost of a new internal degree of freedom, the separation index m. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 29 / 49

58 Sparsity also in the m direction? The Gaussian expansion gave us separation of directions... at the cost of a new internal degree of freedom, the separation index m. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 29 / 49

59 Sparsity also in the m direction? The Gaussian expansion gave us separation of directions... at the cost of a new internal degree of freedom, the separation index m. Do we really need all these terms? F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 29 / 49

60 The 2-scale differences cancel most terms F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 30 / 49

61 The multidimensional algorithm Use the Gaussian approximation to express the kernel in separable form. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 31 / 49

62 The multidimensional algorithm Use the Gaussian approximation to express the kernel in separable form. Construct all the one-dimensional sparse structures as for d = 1. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 31 / 49

63 The multidimensional algorithm Use the Gaussian approximation to express the kernel in separable form. Construct all the one-dimensional sparse structures as for d = 1. Compute the norm estimates for the separation series effective separation range. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 31 / 49

64 The multidimensional algorithm Use the Gaussian approximation to express the kernel in separable form. Construct all the one-dimensional sparse structures as for d = 1. Compute the norm estimates for the separation series effective separation range. Observe that different values of l have different effective ranges this defines various regions (based on how much the singularity of the kernel contributes). F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 31 / 49

65 The multidimensional algorithm Use the Gaussian approximation to express the kernel in separable form. Construct all the one-dimensional sparse structures as for d = 1. Compute the norm estimates for the separation series effective separation range. Observe that different values of l have different effective ranges this defines various regions (based on how much the singularity of the kernel contributes). Build all the necessary indexing tables (within blocks, in separation index, per region). F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 31 / 49

66 The multidimensional algorithm Use the Gaussian approximation to express the kernel in separable form. Construct all the one-dimensional sparse structures as for d = 1. Compute the norm estimates for the separation series effective separation range. Observe that different values of l have different effective ranges this defines various regions (based on how much the singularity of the kernel contributes). Build all the necessary indexing tables (within blocks, in separation index, per region). Apply the lot, much like in the d = 1 case. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 31 / 49

67 Poisson s equation: an example Let s solve Poisson s equation for a simple case with a known solution, a sum of Gaussians: ρ(r;α) = 3 i=1 (6α 4α 2 r 2 i )e αr 2 i = φ(r;α) = 3 i=1 e αr 2 i, α = 300. We compute φ(r) = 1 Z ρ(r ) 4π R 3 r r dr. Timings: done on a Pentium 4, 2.8 GHz machine Tolerance Resulting ε L 2 Basis order App. time (s) MFLOPS F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 32 / 49

68 Poisson s equation: an example Let s solve Poisson s equation for a simple case with a known solution, a sum of Gaussians: ρ(r;α) = 3 i=1 (6α 4α 2 r 2 i )e αr 2 i = φ(r;α) = 3 i=1 e αr 2 i, α = 300. We compute φ(r) = 1 Z ρ(r ) 4π R 3 r r dr. Timings: done on a Pentium 4, 2.8 GHz machine Tolerance Resulting ε L 2 Basis order App. time (s) MFLOPS F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 32 / 49

69 Outline 1 Context and motivation 2 Multiresolution analysis in d = 1 3 Multidimensional problems 4 d 1: beyond separation of variables 5 Quantum Mechanics F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 33 / 49

70 The Separated Representation The standard separation of variables: f(x 1,x 2,...,x d ) = φ 1 (x 1 ) φ 2 (x 2 ) φ d (x d ) only leads to exact solutions in a limited number of cases.

71 The Separated Representation The standard separation of variables: f(x 1,x 2,...,x d ) = φ 1 (x 1 ) φ 2 (x 2 ) φ d (x d ) only leads to exact solutions in a limited number of cases. Definition For a given ε, we represent a matrix A = A(j 1,j 1 ;...;j d,j d ) in dimension d as r l=1 s l A l 1(j 1,j 1)A l 2(j 2,j 2) A l d(j d,j d), where s l is a scalar, s 1 s r > 0, and A l i are matrices with entries A l i (j i,j i ) and norm one. We require: A r l=1 s l A l 1 A l 2 A l d ε. We call the scalars s l separation values and the rank r the separation rank.

72 Remarks on the Separated Representation Fundamentally developed by G. Beylkin and M. Mohlenkamp. For d = 2, it reduces to the well-known SVD. Separated representations are not unique: if d 3 then the analogy with SVD breaks down: by changing ε we change all terms rather than add/delete terms. They can be ill-conditioned: we have developed algorithms for reducing the separation rank while controlling the condition number. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 35 / 49

73 Remarks on the Separated Representation Fundamentally developed by G. Beylkin and M. Mohlenkamp. For d = 2, it reduces to the well-known SVD. Separated representations are not unique: if d 3 then the analogy with SVD breaks down: by changing ε we change all terms rather than add/delete terms. They can be ill-conditioned: we have developed algorithms for reducing the separation rank while controlling the condition number. Efficient computation in this framework requires a combined consideration of both operators and functions. An operator can be also written as T = r l=1 s l B l 1... B l d. A major computational challenge lies in reducing the naïve rank rr of g = T f if f is represented in separated form. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 35 / 49

74 An illustration: a new trigonometric identity Consider the expression of d variables (x 1,x 2,...,x d ): ( d j=1xj ) f(x) = sin. If expanded using textbook trig identities, this produces 2 d 1 terms. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 36 / 49

75 An illustration: a new trigonometric identity Consider the expression of d variables (x 1,x 2,...,x d ): ( d j=1xj ) f(x) = sin. If expanded using textbook trig identities, this produces 2 d 1 terms. However, the following is true: f(x) = d j=1 sin(x j ) d k=1,k j sin(x k + α k α j ) sin(α k α j ) for all choices of {α j } such that α j α k for all j k. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 36 / 49

76 An illustration: a new trigonometric identity Consider the expression of d variables (x 1,x 2,...,x d ): ( d j=1xj ) f(x) = sin. If expanded using textbook trig identities, this produces 2 d 1 terms. However, the following is true: f(x) = d j=1 sin(x j ) d k=1,k j sin(x k + α k α j ) sin(α k α j ) for all choices of {α j } such that α j α k for all j k. This is an identity, not an approximate result. It was discovered via numerical experimentation. The separated representation is not unique... But many choices are ill-conditioned. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 36 / 49

77 Outline 1 Context and motivation 2 Multiresolution analysis in d = 1 3 Multidimensional problems 4 d 1: beyond separation of variables 5 Quantum Mechanics F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 37 / 49

78 Quantum mechanics (H atom) Schrödinger s equation in an integral (Lippman-Schwinger) formulation: We write 1 2 ψ 1 r ψ = Eψ, φ = 2G µ Vφ, where G µ = ( + µ 2 I ) 1 is the Green s function for some µ and V = 1/r is the nuclear potential. This can be solved iteratively. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 38 / 49

79 Quantum mechanics (H atom) Schrödinger s equation in an integral (Lippman-Schwinger) formulation: We write 1 2 ψ 1 r ψ = Eψ, φ = 2G µ Vφ, where G µ = ( + µ 2 I ) 1 is the Green s function for some µ and V = 1/r is the nuclear potential. This can be solved iteratively. 1 Initialize with some value µ 0 and function φ. 2 Apply the Green s function G µ to the product Vφ to compute: 3 Compute the energy for φ new, E new = φ new = 2G µ Vφ. 1 φ 2 new, φ new + Vφ new,φ new. φ new,φ new 4 Set µ = 2E new, φ = φ new / φ new and return to Step 2. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 38 / 49

80 Error for the Hydrogen atom ground state eigenvalue F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 39 / 49

81 The Multiparticle Schrödinger equation Consider the Hamiltonian of the system with K nuclei and N e electrons, H = N e j=1 ( j K k=1 Z k x j R k ) + i>j 1 x i x j. The ultimate goal is to solve the eigenvalue problem H ψ = Eψ without resorting to a reduced model. The curse of dimensionality, the additional requirement of antisymmetry, size extensivity are just several of many issues that need to be resolved. We need need numerical methods to transition from one-particle theories (e.g. Kohn-Sham, HF) to a two particle approach. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 40 / 49

82 Antisymmetry Electrons are fermions: the wave function must be antisymmetric, e.g., ψ(γ 2,γ 1,γ 3,...) = ψ(γ 1,γ 2,γ 3,...), where γ = ((x,y,z),σ) and σ represents the spin degrees of freedom. For a function of N variables, the antisymmetrizer A is defined as A = 1 N! p S N ( 1) p P, where S N is the permutation group on N elements. If A is applied to a separable function, then the result can be expressed as a Slater determinant, φ 1 (γ 1 ) φ 1 (γ 2 ) φ 1 (γ N ) A N j (γ j ) = j=1φ 1 φ 2 (γ 1 ) φ 2 (γ 2 ) φ 2 (γ N ) N! φ N (γ 1 ) φ N (γ 2 ) φ N (γ N ) F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 41 / 49

83 The Wavefunction: an unconstrained sum of Slater determinants We consider approximations to the wavefunction of the form φ l 1 ψ (r) = A r N l l=1s i=1φ l i(γ i ) = 1 r (γ 1) φ l 1 (γ 2) φ l 1 (γ N) φ l 2 N! s l (γ 1) φ l 2 (γ 2) φ l 2 (γ N) l=1... φ l N (γ 1) φ l N (γ 2) φ l N (γ N). F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 42 / 49

84 The Wavefunction: an unconstrained sum of Slater determinants We consider approximations to the wavefunction of the form φ l 1 ψ (r) = A r N l l=1s i=1φ l i(γ i ) = 1 r (γ 1) φ l 1 (γ 2) φ l 1 (γ N) φ l 2 N! s l (γ 1) φ l 2 (γ 2) φ l 2 (γ N) l=1... φ l N (γ 1) φ l N (γ 2) φ l N (γ N). We impose no constraints (such as orthogonality). Without constraints, we can have ψ = A N i=1 φ i (γ i ) +Ac N i=1 (φ i (γ i ) + φ i+n (γ i )) where {φ j } 2N j=1 form an orthonormal set. Conventional methods that constrain the factors to come from an orthogonal set are forced to multiply out the second term and thus obtain 2 N terms. F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 42 / 49

85 Weak Formulation The number of terms in A N j=1 φ j (γ j ) grows exponentially fast and, although this number can algebraically be reduced somewhat. However, if we care only about computing inner products with A N j=1 φ j (γ j ), then the so-called Löwdin rules provide a solution, φ 1, φ 1 φ 1, φ 2 φ 1, φ N A N φ j (γ j ),A N j=1 φ j (γ j ) = 1 φ 2, φ 1 φ 2, φ 2 φ 2, φ N j=1 N! φ N, φ 1 φ N, φ 2 φ N, φ N Computing determinant costs at most O(N 3 ). For large N the matrix is banded and the cost is O(N). F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 43 / 49

86 Computed slices of a He wavefunction with r = 4 F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 44 / 49

87 Computed φ 1 1 (,,0; 1/2), s 3 = F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 45 / 49

88 Computed φ 2 1 (,,0; 1/2), s 3 = F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 46 / 49

89 Computed φ 3 1 (,,0; 1/2), s 3 = F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 47 / 49

90 Summary A new fast, adaptive algorithm for applying integral kernels via multiwavelets with guaranteed precision. Implementation exists for d = 1,2,3. Core algorithm is kernel-independent, already implemented for several physically important kernels. Our current big goal: multiparticle Schrödinger equation (in progress). F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 48 / 49

91 Summary Outlook A new fast, adaptive algorithm for applying integral kernels via multiwavelets with guaranteed precision. Implementation exists for d = 1,2,3. Core algorithm is kernel-independent, already implemented for several physically important kernels. Our current big goal: multiparticle Schrödinger equation (in progress). The structure of the wavefunction and of the multiparticle Green s function: there s still much to learn. Navier-Stokes: a project of its own. Other boundary conditions, more operators: we want a high-level toolbox based on these ideas for end-users. Complex geometries, scattering, time-dependent problems, etc. Undeveloped... F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 48 / 49

92 References Appendix References Beylkin, Coifman, Rokhlin Comm. Pure App. Math. 44, (1991) Alpert SIAM J. Math. Anal. 24, (1993) Alpert, Beylkin, Gines, Vozovoi J. Comp. Phys. 182, (2002) Beylkin and Mohlenkamp Proc. Nat. Acad. Sci. 99, (2002) Beylkin and Monzón App. Comp. Harmonic Analysis 12, (2002) Beylkin, Cheruvu and Pérez J. Comp. Phys. (submitted). Beylkin, Mohlenkamp and Pérez J. Comp. Phys. (in preparation). F. P. (App. Math - CU Boulder) Adaptive Green s Functions UMich 49 / 49