Cubature Rules for High-dimensional Adaptive Integration January 24 th, 2001 Rudolf Schrer 1
The Problem Given a function f :C s, with C s denoting an s-dimensional hyper-rectangular region [r 1 ; t 1 ] [r s ; t s ] s. Calculate an approximation Qf for the multivariate integral If := f(x) dx. C s 2
Basic Principle of Numerical Integration: Q n f = n i=1 w i f(x i ) Q n f If 0 for n. Fast! All x i should be inside C s n i=1 w i should be as small as possible If some w i or x i depend on previously calculated f(x j ) with j < i, the algorithm is called adaptive. 3
Scope of this Talk Only about adaptive algorithms based on interpolatory cubature rules Especially on the impact of different cubature formulas on the standard algorithm 4
Adaptive Algorithm 1. Calculate estimation for result and error for the whole region. 2. Store region and estimations in region collection. 3. Take region with highest estimated error from collection 4. Split into subregions 5. Calculate estimation for result and error for these subregions 6. Store regions and estimations 7. If not finished, goto 2 Step 1 and 4 require Cubature Rules for estimation of result and error. 5
Parallelization Manager-Worker Approach Manager responsible for region management Workers check out regions do refinements (maybe multiple times) return results Does not scale: Bottleneck! 6
Decentralized Design No dedicated manager. equal. All nodes are Every node maintains its own region collection, containing one subset of a partition of the initial domain Topology: Multi-dimensional periodic mesh Nodes talk only to their neighbors to exchange bad regions Nodes perform local refinements, interrupted by redistribution procedure Performance inferior to sequential algorithm, but good scalability 7
Benchmark # Correct Digits 4.5 4 3.5 3 2.5 2 2 PNs 4 PNs 8 PNs 16 PNs 32 PNs LocalList - Corner Peak, dim=10 1.5 1 1 10 100 1000 10000 Time (msec) 35 30 25 2 PNs 4 PNs 8 PNs 16 PNs 32 PNs LocalList - Corner Peak, dim=10 Speed-up 20 15 10 5 0 1.5 2 2.5 3 3.5 4 4.5 # Correct Digits 8
Cubature Rules Take advantage of the smoothness of f! Construct Cubature Rules that are exact for all polynomials up to a certain degree ( Interpolatory Formulas) Smooth functions can be approximated by polynomials (Weierstrass et al.) Interpolatory formulas can be expected to give good results for smooth functions The integral of a polynomial can be calculated analytically 9
Error Estimation Use two rules Q n and Q m of different degree (deg Q n > deg Q m ) For most f, Q n f Q m f > Q n f If, which gives an upper bound on the integration error. Embedded Rules: Abscissas of Q m are a subset of abscissas of Q n, so no extra integrand evaluation are performed 10
Construction of multi-variate cubature rules There is no silver bullet! Momentum Equations: Invent basic structure One equation for each monomial Non-linear Solutions must not be complex Abscissas must be inside C s n i=1 w i should be small Rule should exist for arbitrary dimensions Building simple rules is guessing 11
Example Degree 5, C s = [ 1; 1] s Basic Structure: Abscissas w i #Abscissas (0,..., 0) W 0 1 (α, 0,..., 0) FS W 1 2s (α, α, 0,..., 0) FS W 2 2s(s 1) 2s 2 + 1 Momentum Equations: All monomials? 1, x 1,..., x s, x 2 1,..., x2 s, x 2 1 x3 2, x5 1,... Monomials: 1, x 2 1, x2 1 x2 2, x4 1 Equations, e. g. for monomial x 2 1 : W 0 0 + 2W 1 α 2 + 4(s 1)W 2 α 2 = 2s 3 12
Implemented Rules Name Deg n ni=1 w i Midpoint 1 1 1 Octahedron 3 2s 1 Product Gauss 3 2 s 1 Hammer & Stroud 5 O(s 2 ) O(s) Stroud 5 O(s 2 ) O(s) Phillips 7 O(s 3 ) O(s 2 ) Dobrodeev 7 O(s 3 ) O(s 2 ) Stenger 9 O(s 4 ) O(s 3 ) Genz & Malik 7 O(2 s ) O(s) High degree is not enough (Genz & Malik does better than Phillips) #abscissas in Genz & Malik increases exponentially, but for s 11 it uses less points than Phillips Genz & Malik is already an embedded Rule 13
Testing Accuracy Compare accuracy and take number of integrand evaluations into account Test setup: Calculate accuracy for given number of integrand evaluations Determine the number of possible rule evaluations Split C s. Distribute splitting evenly on all dimensions Apply rule to each sub-cube Add up results 14
Results # Correct Digits 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 Comp_1Midpoint Comp_3Gauss Comp_3Octahedron Comp_5Hammer Comp_5Stroud Comp_7-5Genz Comp_7Dobrodeev Comp_7Phillips Comp_9Stenger Corner Peak, dim=7-0.5 10 100 1000 10000 100000 1e+06 1e+07 # Integrand Evaluations # Correct Digits 12 10 8 6 4 Comp_1Midpoint Comp_3Gauss Comp_3Octahedron Comp_5Hammer Comp_5Stroud Comp_7-5Genz Comp_7Dobrodeev Comp_7Phillips Comp_9Stenger Product Peak, dim=15 2 0 10 100 1000 10000 100000 1e+06 1e+07 # Integrand Evaluations 15
Testing Error Estimation We do not need good approximations We do not need reliable upper bounds What we need is: implies E (B 1) f > E (B 2) f Q (B 1) f I (B 1 ) f > Q (B 2) f I (B 2 ) f 16
Testing Error Estimation Test procedure: Create k = 10000 random cubes C s Apply rule to each sub-cube and estimate integral and error Sort sub-cubes by real error (Position r i ) Sort sub-cubes by estimated error (Position e i ) Calculate quality parameter k (r i e i ) 2 i=1 17
Results # Integrand Evaluations 2500 2000 1500 1000 500 GenzMalik Phillips-Hammer Stenger-Phillips Hammer-Octahedron Phillips-Hammer-Stroud Corner Peak, dim=7 0 1e+08 1e+09 1e+10 1e+11 Errors # Integrand Evaluations 60000 50000 40000 30000 20000 10000 GenzMalik Phillips-Hammer Stenger-Phillips Hammer-Octahedron Phillips-Hammer-Stroud C0, dim=15 0 1e+06 1e+07 1e+08 Errors 18
Final Results 16 14 12 Discontinuous, dim=10, #PN=16 Adapt_7-5Genz_LL Adapt_9-7_LL Adapt_7-5-5_LL Adapt_5-3_LL Niederreiter_Net_Comp_NoLB # Correct Digits 10 8 6 4 2 # Correct Digits 0 11 10 9 8 7 6 10000 100000 1e+06 1e+07 # Integrand Evaluations Gaussian, dim=30, #PN=16 Adapt_9-7_LL Adapt_7-5-5_LL Adapt_5-3_LL Niederreiter_Net_Comp_NoLB 5 4 10000 100000 1e+06 1e+07 # Integrand Evaluations 19