Motivating Example Compute expectation E ( g ( X ( t 0 ))) for - PowerPoint PPT Presentation

G. W. Wasilkowski ∞ -Variate Linear Problems ∞ -Variate Linear Problems G. W. Wasilkowski Department of Computer Science University of Kentucky RICAM 2018 1

G. W. Wasilkowski ∞ -Variate Linear Problems In this presentation: Information-Based Complexity approach, i.e., study of the worst case complexity with respect to a whole normed space F of functions RICAM 2018 2

G. W. Wasilkowski ∞ -Variate Linear Problems In this presentation: Information-Based Complexity approach, i.e., study of the worst case complexity with respect to a whole normed space F of functions There is the Curse of Dimensionality for isotropic spaces . To brake this curse, Non-isotropic spaces are needed. RICAM 2018 3

G. W. Wasilkowski ∞ -Variate Linear Problems In this presentation: Information-Based Complexity approach, i.e., study of the worst case complexity with respect to a whole normed space F of functions There is the Curse of Dimensionality for isotropic spaces . To brake this curse, Non-isotropic spaces are needed. Weighted Spaces are good candidates. Introduced by [Sloan and Wo´ zniakowski 1998], assign different importance to different variables. RICAM 2018 4

G. W. Wasilkowski ∞ -Variate Linear Problems Motivating Example Compute expectation E ( g ( X ( t 0 ))) for stochastic process X ( t ) = � ∞ j =1 x j · ξ j ( t ) , where x j i.i.d. random variables w.r.t. µ and ξ j ( t ) → 0 fast. RICAM 2018 5

G. W. Wasilkowski ∞ -Variate Linear Problems Motivating Example Compute expectation E ( g ( X ( t 0 ))) for stochastic process X ( t ) = � ∞ j =1 x j · ξ j ( t ) , where x j i.i.d. random variables w.r.t. µ and ξ j ( t ) → 0 fast. Equivalent to computing an ∞ -variate integral   ∞ � � R N f ( x ) d µ N ( x ) for f ( x ) = g x j · ξ j ( t 0 )   j =1 RICAM 2018 6

G. W. Wasilkowski ∞ -Variate Linear Problems   ∞ � In g x j ξ j ( t 0 )   j =1 “importance” of x j is quantized by the size of | ξ j ( t 0 ) | . The larger | ξ j ( t 0 ) | the more important x j . This leads to WEIGHTED SPACES of ∞ -VARIATE FUNCTIONS RICAM 2018 7

G. W. Wasilkowski ∞ -Variate Linear Problems Initial complexity attempts: [W. and Wo´ zniakowski 96 and 00], [Plaskota, W. and Wo´ zniakowski 00], and [Hickernell and Wang 01] [Hickernell, M¨ uller-Gronbach, Niu, Ritter 10] Since 2010, many papers have been written by, e.g., J. Baldeaux, A. Cohen, J. Creutzig, W. Dahmen, S. Dereich, R. DeVore, J. Dick, D. D˜ ung, A. Gilbert, M. Gnewuch, M. Griebel, M. G. Gunzburger, M. Hefter, S. Heinrich, F. J. Hickernell, A. Hinrichs, Z. Kacewicz, P. Kritzer, F. Y. Kuo, S. Mayer, T. M¨ uller-Gronbach, P. Morkisz, B. Niu, J. Nichols, D. Nuyens, F. Pillichshammer, L. Plaskota, P. Przybylowicz, K. Ritter, Ch. Schwab, I. H. Sloan, H. Wo´ zniakowski RICAM 2018 8

G. W. Wasilkowski ∞ -Variate Linear Problems However, majority of the papers dealing with: Hilbert spaces , integration problem , and too simplistic cost model In this presentation: Banach or just normed spaces , linear tensor-product problems , and more realistic cost model RICAM 2018 9

G. W. Wasilkowski ∞ -Variate Linear Problems In our model, f ( x ) can be evaluated for x with only finitely many x j � = 0   � for finite w ⊂ N f ( x w ) = g x j ξ j ( t 0 )  j ∈ w Cost of obtaining the sample f ( x w ) should depend on the size of w , i.e. cost ( f ( x w )) = $( | w | ) Positive and sharp results for Ω( k ) = $( k ) = e O ( k ) RICAM 2018 10

G. W. Wasilkowski ∞ -Variate Linear Problems Notation: D N set of sequences x = ( x 1 , x 2 , . . . ) Domain: with x j ∈ D ; e.g., D = [0 , 1] , R + or R finite subsets of N + w listing the “variables in action”, e.g., x w = ( x j : j ∈ w ) given x = ( x 1 , x 2 , . . . ) ,  x j if j ∈ w ,  [ x w ; 0 ] = ( y 1 , y 2 , . . . ) with y j = 0 if j / ∈ w  For p ∈ [1 , ∞ ] we use p ∗ = p p − 1 RICAM 2018 11

G. W. Wasilkowski ∞ -Variate Linear Problems Weighted Spaces F γ f = � Any f ∈ F γ has decomposition : w f w such that f w depends only on x w , f w ∈ F w , where F w is a Banach (or just normed) space and F w ∩ F v = { 0 } if w � = v It is Anchored if f w ( x ) = 0 x j = 0 for j ∈ w when RICAM 2018 12

G. W. Wasilkowski ∞ -Variate Linear Problems The space F γ is endowed with � 1 /p �� γ − p w � f w � p � f � F = F w w p ∈ [1 , ∞ ] γ w ≥ 0 are weights Here and zniakowski 1998] : Product Weights [Sloan and Wo´ � j − β γ w = c j ∈ w Product Order Depedent Weights [Kuo, Schwab, and Sloan 14] : γ w = ( | w | !) α � j − β β > max( α, 1 /p ∗ ) j ∈ w RICAM 2018 13

G. W. Wasilkowski ∞ -Variate Linear Problems Problem to Approximate S ( f ) , S : F γ → G where Approximate is a linear operator into a normed space G Example: � D N f ρ N S ( f ) = d � � = lim D d f ( x 1 , . . . , x d , 0 , . . . ) ρ ( x j ) d( x 1 , . . . , x d ) d →∞ j =1 where ρ is a probability density on D RICAM 2018 14

G. W. Wasilkowski ∞ -Variate Linear Problems For every w , ASSUMPTION 1: �S | F w � F w ≤ C | w | 1 Continuity of S ASSUMPTION 2: � 1 /p ∗ � 1 /p ∗ �� �� w C | w | p ∗ γ p ∗ w �S | F w � p ∗ γ p ∗ �S� F γ = ≤ < ∞ F w 1 w w For tensor product problems: �S | F w � F w = C | w | with C 1 = �S | F 1 � F 1 1 1 + ( C 1 j − β ) p ∗ �� 1 /p ∗ �� ∞ � and for product weights: �S� F γ = j =1 RICAM 2018 15

G. W. Wasilkowski ∞ -Variate Linear Problems Standard Integration Problem: F γ endowed with the norm � 1 /p �� p ∂ � � f ( w ) = γ − p � � f ( w ) ([ · w ; 0 ]) � f � F = , f � � w ∂x j � L p ( D | w | ) j ∈ w w D = [0 , 1] and ρ ≡ 1 (most often Hilbert case p = 2 ) � S = I = and D N Then 1 � γ w ≃ | ξ j ( t 0 ) | δ j ∈ w and if p ∗ = ∞ ) C 1 = �I | F 1 � F 1 = (1 + p ∗ ) − 1 /p ∗ (= 1 RICAM 2018 16

G. W. Wasilkowski ∞ -Variate Linear Problems How to Cope with So Many Variables? Truncate the Dimension , i.e., replace f ( x ) f ( x 1 , . . . , x k , 0 , 0 , . . . ) by for a “relatively” small k = k ( error ) or (even better) use Multivariate Decomposition Method i.e., approximate a small number of integrals each with a small number of variables RICAM 2018 17

G. W. Wasilkowski ∞ -Variate Linear Problems Low Truncation Dimension Anchored Decomposition [Kritzer, Pillichshammer, W.2016] ⊂ [Hinrichs, Kritzer, Pillichshammer, W.2018] Let f k ( x 1 , . . . , x k ) = f ( x 1 , . . . , x k , 0 , 0 , . . . ) dim trnc ( ε ) ε -truncation dimension is the smallest k such that �S ( f ) − S ( f k ) � G ≤ ε � f � F γ for all f ∈ F γ Our concept of Truncation Dimension is different than the one in Statistics. RICAM 2018 18

G. W. Wasilkowski ∞ -Variate Linear Problems If �S ( f ) − S ( f k ) � G ≤ ε � f � F �S ( f k ) − A k ( f k ) � G ≤ ε � f � F then and �S ( f ) − A k ( f k ) � G ≤ 2 ε � f � F the smaller dim trnc ( ε ) the better Hence j ∈ w j − β For product weights γ w = � THEOREM 1 � ε − 1 / ( β − 1+1 /p ) � dim trnc ( ε ) = O for p > 1 , and � ε − 1 /β � dim trnc ( ε ) = O for p = 1 . RICAM 2018 19

G. W. Wasilkowski ∞ -Variate Linear Problems Values of dim trnc ( ε ) for Standard Integration Problem j ∈ w j − β with p = 1 and γ w = � 10 − 1 10 − 2 10 − 3 10 − 4 10 − 5 ε dim trnc ( ε ) 2 9 31 99 316 β = 2 dim trnc ( ε ) 2 4 9 21 46 β = 3 dim trnc ( ε ) 1 3 5 9 17 β = 4 For instance, for the error demand ε = 10 − 3 with β = 4 , only five variables instead of ∞ -many! Worst Case Error of QMC or Sparse Grids Methods is: � ln 4 n � ≤ O n RICAM 2018 20

G. W. Wasilkowski ∞ -Variate Linear Problems Arbitrary Decomposition Theorem 1 holds true for “Quasi”-Truncation Dimension defined by  � �    � �    � q − dim trnc ( ε ) := min � �  k : S ( f ) − S f w ≤ ε � f � F γ �  � � �  w ⊆ 1: k � � G of interest in Statistics ( ANOVA Decomposition ) However for non-Anchored Decomposition � f ( x 1 , . . . , x k , 0 , 0 , . . . ) � = f w ( x ) w ⊆ 1: k and small q − dim trnc ( ε ) need not lead to efficient algorithms. Unless..... we’ll see it later. RICAM 2018 21

G. W. Wasilkowski ∞ -Variate Linear Problems Multivariate Decomposition Method Introduced by [Kuo, Sloan, W., and Wo´ znakowski 2010] MDM replaces one ∞ -variate integral by only few integrals each with only few variables RICAM 2018 22

G. W. Wasilkowski ∞ -Variate Linear Problems More precisely: Given the error demand ε > 0 , construct an “active set” Act ( ε ) of subsets w such that � �   � � ≤ ε � � � S f w 2 � f � F γ for all f ∈ F . � �   � � w / ∈ Act ( ε ) � � G S ( f w ) with w / ∈ Act ( ε ) , Do nothing for and concentrate on S ( f w ) with w ∈ Act ( ε ) . RICAM 2018 23

G. W. Wasilkowski ∞ -Variate Linear Problems For w ∈ Act ( ε ) , choose n w and algorithms A w ,n w to approximate S ( f w ) such that �S ( f w ) − A w ,n w ( f w ) � G ≤ ε � 2 � f � F for all f ∈ F . w ∈ Act ( ε ) The algorithms A w ,n w could be | w | -variate Sparse Grids Algorithms RICAM 2018 24