Algorithms for Scientific Computing

Transcription

1 Algorithms for Scientific Computing Hierarchical Methods and Sparse Grids d-dimensional Hierarchical Basis Michael Bader Technical University of Munich Summer 208

2 Intermezzo/ Big Picture : Archimedes Quadrature Start with 2d example (compare tutorials): f := 6x (x )x 2 (x 2 ), Ω = [0, ] 2 f Ω = 0 Consider hierarchical surplus at grid points with n = 3, h 3 = Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer 208 2

3 Big Picture : Archimedes Quadrature (2) Ω f d x = 4/9 = 0. 4 = = Consider volume of subvolumes (pagodas) for quadrature Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer 208 3

4 Big Picture : Archimedes Quadrature (3) What, if we leave out (adaptively) all subvolumes with volume < ɛ =? 49 grid points (full grid) 7 grid points (sparse grid) 0 0 Approximation of volume: = = Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer 208 4

5 Part I Hierarchical Decomposition, d-dimensional Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer 208 5

6 Recall: Archimedes and Hierarchical Basis (in D) Archimedes Quadrature: Hierarchical Basis: t= t=2 ¼ 0 ½ ¼. x, Φ, Φ 2, Φ 2,3. x 2, x 2,3 0 ½ 0 ½ Φ 3, Φ 3,3 Φ 3,5 Φ 3,7. x 3, x 3,3 x 3,5 x 3,7 use nodal basis functions φ l,i with I l := {i : i < 2 l, i odd} hierarchical basis Ψ n := n l= {φ l,i : i I l } hierachical function spaces W l := span {φ l,i : i I l } and V l = V l W l unique hierachical representation u = n l= w l = n l= i I l v l,i φ l,i size of surpluses v l,i roughly decays with 4 n for smooth functions Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer 208 6

7 Hierarchical Decomposition Step by Step Now (and more formally), starting with d-dimensional hierarchical decompositions... Transfer from d = to d > Functions in multiple variables x = (x,..., x d ) Domain Ω := [0, ] d We consider only functions u which are 0 on Ω (on the edges of the square, faces of the cube,... ) Each hierarchical grid described by multi-index l = (l,..., l d ) N d Grids have different mesh-widths in different dimensions: h l := (h,..., h d ) := (2 l,..., 2 l d ) =: 2 l Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer 208 7

8 Hierarchical Decomposition, d > Introducing further notation (which we ll need later on): Grid points (for function evaluations): x l, i = (i h l,..., i d h ld ) Comparisons of multi-indices component-wise: Two norms for multi-indices l l i lk i k, k =,..., d index sum: l := l l d maximum index: l := max { l,..., l d } Note: taking the absolute values,, for l k N is not necessary, but is part of the usual definition of and Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer 208 8

9 Practicing Identifiers l, h l, x l, i l = l =2 l =3 l l 2 = l 2 =2 l 2 =3 l 2 Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer 208 9

10 Piecewise d-linear Functions Suitable generalization of piecewise linear functions Piecewise d-linear functions w.r.t. h l grid If you fix d coordinates, they are linear in remaining x j V l : space of all functions for given l Alternative point of view: Define suitable basis Φ l Regard V l as span of Φ l d-dimensional basis functions: products of one-dimensional hat functions: φ l, i ( x) = d φ lj,i j (x j ) = φ l,i (x ) φ l2,i 2 (x 2 )... φ ld,i d (x d ) j= Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer 208 0

11 d-dimensional Basis Functions Basis functions are pagoda functions (not pyramids!) Examples: φ (,),(,), and φ (2,3),(3,5) : x x x x Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer 208

12 Function Spaces V l and V n Basis for space of piecewise linear functions w.r.t. h l grid Φ l := {φ l, i, i < 2 l } Function space V l := span{φ l } with dim V l = (2 l )... (2 l d ) O(2 l ) Special case l =... = l d function space denoted as V n : V n := V (n,...,n) Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer 208 2

13 Hierarchical Increments W l Analogous to d: Omit grid points with even index (exist on coarser grid) Now in all directions I l := { i : i < 2 l, all i j odd} Hierarchical increment spaces l 2 = l 2 =2 l = l =2 l =3 l W l := span{φ l, i } i I l contain all functions of V l that vanish at all grid points of all coarser grids l 2 =3 l 2 Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer 208 3

14 Hierarchical Increments W l vs. Nodal Basis Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer 208 4

15 Hierarchical Subspace Decomposition For l N d we obtain a unique representation of each u V l as u = l l w l with w l W l Representation in the hierarchical basis u = w l = v l, i φ l, i l l l l i I l with d-dimensional hierarchical surpluses v l, i Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer 208 5

16 Determining the Hierarchical Surpluses in 2D We now compute the hierarchical surpluses v l, i for some u V n = V (n,n) : u( x) = φ l, i Φ (n,n) u(x l, i ) φ l, i ( x) = 2 n i = 2 n i 2 = First step Hierarchization in x -direction (fix x 2 and employ d hierarchization in x -direction): u( x) = n 2 n l = i I l i 2 = u(x n, i ) φ n,i (x )φ n,i2 (x 2 ) v l,i (x n,i2 ) φ l,i (x ) φ n,i2 (x 2 ) with d surplus (still depending on x 2, evaluated at all x 2 = x n,i2 ) v l,i (x 2 ) = u(x l,i, x 2 ) u(x l,i, x 2 ) + u(x l,i +, x 2 ) 2 Note: the indices i ± of the grid points x l,i and x l,i + are even, such that the corresponding hierarchical basis functions belong to a parent/ancestor level. Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer 208 6

17 Determining the Hierarchical Surpluses in 2D (2) A bit more intuitive: We mark the grid points of the corresponding ansatz functions we use (before and after) l = l =2 l =3 l l = l =2 l =3 l l 2 = l 2 = l 2 =2 l 2 =2 l 2 =3 l 2 =3 l 2 l 2 Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer 208 7

18 Determining the Hierarchical Surpluses in 2D (3) Second step Hierarchize every v l,i (x 2 ) (separately) in x 2 dimension:: n n u( x) = v (l,l 2 ),(i,i 2 ) φ l,i (x ) φ l2,i 2 (x 2 ) l = l 2 = i 2 I l2 i I l l = l =2 l =3 l l = l =2 l =3 l l 2 = l 2 = l 2 =2 l 2 =2 l 2 =3 l 2 =3 l 2 l 2 Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer 208 8

19 Determining the Hierarchical Surpluses (the general d-dimensional case) Now: compute the d-dim. hierarchical surpluses v l, i for some u( x) V n : u( x) = u(x l, i ) φ l, i ( x) = u(x l, i ) φ l,i (x )... φ ld,i d (x d ) φ l, i Φ (n,...,n) φ l, i Φ (n,...,n) First step Hierarchization in x d -direction (fix x,..., x d and employ d hierarchization): n u = v ld,i d (x n,(i,...,i d )) φ ld,i d (x d ) φ l,i (x )... φ ld,i d (x d ) φ l, i Φ (n,...,n) l d = i d I ld with d surplus evaluted at (x,..., x d ) = x n,(i,...,i d ): v ld,i d (x,..., x d ) = u(x,..., x d, x ld,i d ) u(x,..., x d, x ld,i d ) + u(x,..., x d, x ld,i d +) 2 Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer 208 9

20 Determining the Hierarchical Surpluses (the general d-dimensional case, second step) Second step Hierarchize every v ld,i d : R d R (separately) in its first argument: n n ( u( x) = l d = i d I ld l d = i d I ld φ l, Φ i (n,...,n) ) v ld,i d (x n,(i,...,i d 2 )) φ ld,i d (x d ) φ ld,i d (x d ) φ l,i (x )... φ ld 2,i d 2 (x d 2 ) Steps 3 to d All steps correspondingly for each remaining dimension Afterwards we have computed surpluses v l, i (functions in zero parameters / scalar values) u( x) = v l, i φ l,i (x )... φ ld,i d (x d ) = v l, i φ l, i ( x) = l i I l l i I l where l i I l is short for n l d = i d I ld n l = i I l w l ( x) Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer l

21 Comparison 2D with Wavelet Transform Apply Level-Wise Hierarchisation First level: First step: split into nodal basis and hierarchical surpluses (st argument) Second step: split into nodal basis and hierarchical surpluses (2nd argument) l = l =2 l =3 l l = l =2 l =3 l l 2 = l 2 = l 2 =2 l 2 =2 l 2 =3 l 2 =3 l 2 l 2 Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer 208 2

22 Comparison 2D with Wavelet Transform (2) Apply Level-Wise Hierarchisation Second level: First step: split into nodal basis and hierarchical surpluses (st argument) Second step: split into nodal basis and hierarchical surpluses (2nd argument) l = l =2 l =3 l l = l =2 l =3 l l 2 = l 2 = l 2 =2 l 2 =2 l 2 =3 l 2 =3 l 2 l 2 Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer

23 Part II Hierarchical Decomposition Outlook on Cost and Accuracy Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer

24 Analysis of Hierarchical Decomposition Contribution of summands in hierarchical decomposition in D: n n u = w l = v l,i φ l,i l= l= i I l in dd: u = l w l = l start analysis in univariate setting i I l v l, i φ l, i ( x) and port to mulitvariate setting Cost/benefit analysis quantifies reduction of effort Need several norms to measure w l Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer

25 Norms of Functions As always, we assume sufficiently smooth functions u : [0, ] R, then: Maximum norm L 2 norm u := max x [0,] u(x) for the L 2 scalar product u 2 := 0 u(x) 2 dx, (u, v) 2 := 0 u(x)v(x) dx Energy norm u E := u 2 Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer

26 Norms of Basis Functions For the basis functions φ l,i, we obtain φ l,i = φ l,i 2 = φ l,i E = 2hl 3 2 h l h -2 Φ Φ 2 (Φ')2... x l,i- x l,i x l,i+ x l,i- x l,i x l,i+ x l,i- x l,i x l,i+ Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer

27 Estimation of Surpluses Consider surplus v l,i of basis function φ l,i : u two times differentiable v l,i := u(x l,i ) 2 (u(x l,i ) + u(x l,i+ )) We can then write v l,i as (see separate proof) v l,i = 0 ψ l,i (x)u (x) dx with ψ l,i := h l 2 φ l,i v l,i depends on u, thus we define for future use µ 2 (u) := u 2 and µ (u) := u. note: µ 2 (u) and µ (u) are properties of the function u Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer

28 Estimation of Surplusses (2) With integral representation v l,i = 0 h l 2 φ l,i(x)u (x) dx, we can bound ( ) v l,i h l 2 φ l,i dx µ (u) = h2 l 2 µ (u) O(hl 2 ) 0 and, via Cauchy-Schwartz inequality (u, v) u v, v l,i h l 2 φ hl 3 l,i 2 µ 2 (u Ti ) = 6 µ 2(u Ti ), where u Ti restricts u to the support T i = [x l,i, x l,i+ ] of φ l,i Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer

29 Estimation of w l Estimate contribution of entire level l in hierarchical decomposition of u, i.e. w l = i I l v l,i φ l,i Use that supports of φ l,i are pairwise disjoint Maximum norm w l h2 l 2 µ (u) O(h 2 l ) L 2 norm w l 2 2 = v l,i 2 φ l,i 2 2 h3 l 6 2h l 3 µ 2 (u Ti ) 2 = h4 l 9 µ 2(u) 2 i I l i I l w l 2 O(h 2 l ) Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer

30 Estimation of w l (2) Energy norm w l 2 E = i I l v l,i 2 φ l,i 2 E = i I l v l,i 2 2 h l 2 h4 l h l 4 µ (u) 2 = h2 l 2h l 4 µ (u) 2 (2 l = /(2h l ) summands) w l E O(h l ) Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer

31 Estimation of w l (3) We can write u (twice differentiable) as infinite series u = l= w l Convergent in all three norms Approximation error given as u u n := u in maximum and L 2 norm: O(h 2 n) in energy norm: O(h n ) n w l = l= l=n+ w l Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer 208 3

32 Towards d Dimensions: Norms of φ l, i Estimating the w l will enable us to select those subspaces that contribute most to overall solution (best cost-benefit ratios) Same procedure as for d =, but slightly more complicated functions Start with norms Maximum norm: L 2 norm: φ l, i 2 := [0,] d φ l, i ( x) 2 d x = φ l, i := max φ l, i ( x) = x [0,] d d ) d d (2 ) d φ lj,i j 2 = ( 2 h lj = l j= j= Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer

33 Norms of φ l, i (2) Energy norm (defined as L 2 norm of the Euclidean norm of the gradient φ l, i ): φ l, i E := φ l, i ( x) φ l, i ( x) d x =... = [0,] d ( ) d d = 2 h... h d 2 3 h 2 (here always: h j := h lj ) j= j ( ) d = 2 d l 2 2l j For the two-dimensional settings (d = 2), we obtain ( 4 h φ l, i E = + h ) 2 3 h 2 h j= Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer

34 Estimation of Surpluses Hierarchical surpluses now depend on mixed 2nd derivatives If we define ψ l, i := 2d 2d u u := x 2... x d 2 d d ψ lj,i j = j= j= h j 2 φ l, i = ( ) d 2 l d φ l, i we can derive an integral representation similar to D: v l, i = ψ l, i 2d u d x [0,] d (Proof: Fubini s theorem and d integral representation) Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer

35 Estimation of Surpluses (2) We define (correspondingly to d) We can thus bound v l, i as ( d v l, i j= µ 2 (u) := 2d u 2 and µ (u) := 2d u ) ( ) ( d h j h 2 ) j φ l, 2 i d x µ (u) = µ (u) = 2 2 l d µ (u) 2 [0,] d j= and v l, i ( d j= ) h j h 3 φ l, 2 i 2 µ 2 (u T i ) =... h3 d 6 d µ 2 (u T i ) = ( 6) d/2 2 3 l /2 µ 2 (u T i ) Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer

36 Estimation of w l Obtain estimates for w l in subspace W l analogously as in d: Make use of the fact that supports of basis functions for a grid are disjoint (apart from the boundaries) Maximum norm L 2 norm d w l j= d w l 2 Energy norm w l E ( ) d d 4 2 j= j= h 2 j 2 h 2 j 3 h 4... h4 d h 2 j µ (u) = 2 2 l d µ (u), µ 2 (u) = 3 d 2 2 l µ 2 (u), µ (u) = ( ) d d l 2 2l j µ (u) j= Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer

37 Analysis of Cost-Benefit Ratio Consider not individual basis functions, but whole hierarchical increments From the tableau of subspaces, select those subspaces that minimize the cost, or maximize the benefit respectively, for u : [0, ] d R (u sufficiently often differentiable) Cost Measure cost in number of grid points ( coefficients ) c( l) = I l = 2 l d Benefit How to measure benefit? interpolation error Let L N d be the set of indices of all selected grids, then u L := l L w l and u u L = l L w l Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer

38 Analysis of Cost-Benefit Ratio (2) For each component w l, we have derived bounds of the type w l s( l) µ(u) with s( l) = 2 d 2 2 l or s( l) = 3 d 2 2 l and appropriate indices for norm and µ We obtain u u L ( ) w l s( l) µ(u) l L l L = ( l N d ) ( ) s( l) s( l) µ(u) l L st factor depends only on selected subspaces, 2nd factor only on u Justifies to interpret s( l) as benefit/contribution of subspace W l Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer

39 Quality of Approximation of Full Grid V n Examine cost c( l) and benefit s( l) for full grid Regular grid with mesh-width h = 2 n in each direction (full grid) for function space V n total cost dim V n O(2 dn ), or dim V n O(h d ) Considered subset of hierarchical increments: L n := { l : l n}. Bounds in L 2 and maximum norm involve factor s( l) = C 2 2 l In the following estimation, leave out l-independent factor C can be appended to the estimate in the end Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer

40 Quality of Approximation of Full Grid V n (2) We can estimate s( l) = 2 2 l = l Ln l Ln = ( 4 n 4 4 n l = ( n n ) d 2 2(l + +l n) = 2 2k l d = k= ) d ( = d ( 3) ) 2 2n d ( ) d ( ) 3 d 2 2n using ( ɛ) d dɛ for 0 ɛ and d N For n we obtain ( ) d s( l) = and thus s( 3 l) = l N d l Ln ( ) d d 2 2n 3 Leads to bounds for the approximation error in L 2 - and maximum norm u u Ln C s( l) C d 3 d 2 2n O(hn) 2 l Ln with constant C (independent of n) Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer

41 Sparse Grids Final steps to high-dimensional numerics Consider sum of benefits/contributions (for L 2 and maximum norm) l Ln 2 2 l Equal benefit of hierarchical increments W l for constant l Same for cost c( l) = 2 l d (number of grid points of W l ) Constant cost-benefit ratio c( l)/s( l) for constant l Full grids? Quadratic extract of subspaces is not economical: We take large subgrids with low contribution We could have taken others with much higher contribution Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer 208 4

42 Part III Sparse Grids Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer

43 Sparse Grids! cost-benefit analysis: equal contribution of hierarchical increments W l for constant l Best choice: Cut diagonally in tableau of subspaces: L n := { l : l n + d } Resulting sparse grid space Vn := l n+d W l l 2 = l 2 =2 l 2 =3 l 2 l = l =2 l =3 l Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer

44 Sparse Grids! Diagonall cut in tableau of subspaces: L n := { l : l n + d } Resulting sparse grid space Vn := l n+d W l Sparse grid for d = 2 and overall level n = 5 Grid points x l, i of same cost/benefit ratio in same color Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer

45 Sparse Grids Cost Number of grid points? For d = 2: dim V n = For d = 3: l n+ dim V n = dim W l = n k= l n+ k(k + ) 2 2 l 2 = n k 2 k = 2 n (n ) +, k= ( n 2 k = 2 n 2 2 n ) 2 +, Both in O(2 n n d ) Holds for general d as well (proof with some combinatorics) Expressed in terms of N = 2 n (max. points per dimension): O ( N(log N) d ) Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer

46 Sparse Grids Cost (2) In numbers... Compare cost for full grid V n and sparse grid V n : d = 2: n dim V n = (2 n ) ,046,529 dim V n = 2 n (n ) ,27 Even more distinct for d = 3: n dim V n = (2 n ) ,375...,070,590,67 ( ) dim Vn = 2 n n ,03 Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer

47 Sparse Grids Cost (3)... and for overall level n = 5 in different dimensions d V 5 V , , ,629,5, ,503,68 2, ,52,64, 4, ,89,037,44 6, ,439,622,60,67 9, ,628,286,980,80 3,44 The higher the dimension, the higher the benefit of sparse grids! Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer

48 Sparse Grids Examples Sparse Grids of overall level n = 6 in d = 2 and d = 3 Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer

49 Sparse Grids Accuracy Much fewer grid points much lower accuracy? Would force us to choose larger n to obtain similar accuracy (and spoil everything) Error in L 2 and maximum norm: Compute sum ( l = k + ): l L n s( l) = k=n+ And for d = 3 (with l = k + 2): l L n s( l) = k=n+3 k(k + ) 2 k 2 2(k+) = 2 2(k+2) = ( n 2 + ) 2 2n 9 ( n n ) 2 2n 27 Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer

50 Sparse Grids Accuracy (2) In general, it can be shown Error of interpolation in L 2 and maximum norm is O(2 2n n d ) or, expressed in mesh size h := 2 n : O ( h ( ) 2 log d ) h Only polynomial (in n) factor worse than full grid with O(2 2n ) or, expressed in mesh size h := 2 n : O(h 2 ) Outlook on Energy norm: ( Algorithms for Scientific Computing II) Analysis is more complicated (lines through subspaces with similar s( l), and thus c( l)/s( l), are more complicated) Overall result even better: obtain accuracy of O(2 n ) with only O(2 n ) grid points no polynomial terms (of type n d ) left! Michael Bader Algorithms for Scientific Computing d-dim. Hierarchical Basis Summer