Some Thoughts on Guaranteed Function Approximation Satisfying Relative Error

Transcription

1 Some Thoughts on Guaranteed Function Approximation Satisfying Relative Error Fred J. Hickernell Department of Applied Mathematics, Illinois Institute of Technology mypages.iit.edu/~hickernell Joint work with Yuhan Ding (IIT) and Henryk Woźniakowski (Columbia U & U Warsaw) October 16, 2014 hickernell@iit.edu Function Approximation w/ Relative Error IIT, 10/16/ / 10

2 Function Approximation with Adaptive Linear Splines Given data px 0, y 0 q,..., px n, y n q with y fpx i q for f : r0, 1s Ñ R, Find A n pfq φpy 0,..., y n q : r0, 1s Ñ R such that f A n pfq is small. The linear spline is given by A n pfqpxq : We know that ˆ i f pi ` 1 nxq ` f n ˆi ` 1 n j pnx iq for i n ď x ď i ` 1 n. f A n pfq ď f 2 n 2 for f P W 2, (Clancy et al., 2014). hickernell@iit.edu Function Approximation w/ Relative Error IIT, 10/16/ / 10

3 Satisfying Error Tolerances for Balls f A n pfq ď f 2 n 2 for f P W 2,. Absolute error tolerances: the computational cost is bounded by f 2 : mintn : f A n pfq ď ε P B σ u c σ ε a, B σ : tf P W 2, such that f 2 ď σu. Hybrid error tolerances, the computational cost is the same as for the absolute error tolerance: c σ mintn : f A n pfq ď maxpε a, ε r f P B σ u. ε a Relative error tolerances, the computational cost is infinite: mintn : f A n pfq ď ε r f P B σ u hickernell@iit.edu Function Approximation w/ Relative Error IIT, 10/16/ / 10

4 Why a Relative Error Tolerance Doesn t Help for Balls Let Ãn be any algorithm using n data. Let ξ ď ζ be the consecutive data sites spaced furthest apart. Define f bump pxq : 1 4pζ ξq2 ` p4x 2ξ 2ζq 2 32 `p4x ξ 3ζq 4x ξ 3ζ p4x 3ξ ζq 4x 3ξ ζ s à n pf bump q 0, f 2 bump 1, f bump pζ ξq2 16 ě 1 16pn ` 1q 2! ) min n : f Ãnpfq ď maxpε a, ε r f P B σ! ) ě min n : σf bump Ãnpσf bump q ď maxpε a, ε r σf bump q " * c σ σ ě min n : 16pn ` 1q 2 ď ε a ε a hickernell@iit.edu Function Approximation w/ Relative Error IIT, 10/16/ / 10

5 Satisfying Error Tolerances for Cones Clancy et al. (2014) developed a way of choosing n based on function data to ensure that f A n pfq ď ε a without knowing f 2. Let C τ : tf P W 2, : f 2 ď τ f 1 fp1q ` fp0q u. By noting that for all f P W 2,, f 1 fp1q ` fp0q A n pfq 1 fp1q ` fp0q ď f 2 2n, it may be shown that f 2 ď τ A npfq 1 fp1q ` fp0q 1 τ{p2nq ÐÝ data-based hickernell@iit.edu Function Approximation w/ Relative Error IIT, 10/16/ / 10

6 Satisfying Error Tolerances for Cones Clancy et al. (2014) s algorithm chooses n to satisfy f A n pfq ď f 2 n 2 ď τ A npfq 1 fp1q ` fp0q ď ε a looooooooooooooooomooooooooooooooooon 4np2n τq data-based The computational cost for the absolute error tolerance is mintn : f A n pfq ď ε P C τ X B σ u Hybrid error tolerances, the computational cost is unknown: c σ ε a. mintn : f A n pfq ď maxpε a, ε r f τ P B σ u? What about relative error tolerances (ε a 0)? hickernell@iit.edu Function Approximation w/ Relative Error IIT, 10/16/ / 10

7 Bounding Weaker Norms in Terms of Stronger Ones Let g f A 1 pfq, and note that g 1 is continuous. Let ξ be chosen such that g 1 pξq g 1. Also, define ζ ξ ` 1{τ or ζ ξ 1{τ, whichever falls inside r0, 1s. It follows from integration by parts and the triangle inequality Thus 2 g ě gpξq gpζq g1 pxqpx ζq ξ ζ ż ξ ě g 1 pξq ξ ζ 1 2 g 1 pxq dx g 2 pxqpx ζq dx ζ ż ξ ζ g 2 ξ ζ 2 ě g 1 1 τ 1 2 ˆ τ g 1 ˆ 1 τ 2 g 1 1 2τ f ě 1 2 f A 1pfq ě 1 f 1 A 1 pfq 1 τ, hickernell@iit.edu Function Approximation w/ Relative Error IIT, 10/16/ / 10

8 Adaptive Hybrid Error Tolerances Now we choose n to satisfy f A n pfq ď f 2 n 2 ď τ A npfq 1 fp1q ` fp0q looooooooooooooooomooooooooooooooooon 4np2n τq data-based ˆ ď max ε a, ε r A n pfq 1 A 1 pfq 1 ˆ ď max ε a, ε r f 1 A 1 pfq 1 τ τ ď maxpε a, ε r f q and get mintn : f A n pfq ď maxpε a, ε r f P C τ X B σ u d ˆ σ min, 1 ε a ε r So either ε a or ε r positive gives bounded computational cost. hickernell@iit.edu Function Approximation w/ Relative Error IIT, 10/16/2014 / 10

9 Next We Consider W r`1, Use piecewise r th degree polynomials to approximate function. Maybe consider tensor product for d-variate functions. This idea does not work for integration problems. Why? hickernell@iit.edu Function Approximation w/ Relative Error IIT, 10/16/ / 10

10 References Clancy, N., Y. Ding, C. Hamilton, F. J. H., and Y. Zhang The cost of deterministic, adaptive, automatic algorithms: Cones, not balls, J. Complexity 30, Function Approximation w/ Relative Error IIT, 10/16/ / 10