A Deterministic Guaranteed Automatic Algorithm for Univariate 2014 June Function 9 1 Approximatio

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1 A Deterministic Guaranteed Automatic Algorithm for Univariate Function Approximation Yuhan Ding Advisor: Dr. Hickernell Department of Applied Mathematics Illinois Institute of Technology 2014 June 9 A for Univariate 2014 June Function 9 1 Approximatio

2 Outline Introduction Introduction Deterministic Guaranteed Automatic Algorithm A for Univariate 2014 June Function 9 2 Approximatio

3 My Questions Background My Questions for You If you are designing algorithms for others to use or performing numerical simulations Are the algorithms that you design or use automatic, i.e., requiring minimal user intervention to obtain the desired answer? Are your algorithms guaranteed to give the answer to the desired tolerance? A for Univariate 2014 June Function 9 3 Approximatio

4 Basic Assumption on [a, b] My Questions Background We consider the function f W 2, [a, b], where W 2, [a, b] = {u L [a, b] : D α u L [a, b], α 2}. We define a cone function set i.e. { C τ = f W 2, [a, b] : f τ [a,b] b a f f (b) f (a) b a }, where τ [a,b] a parameter depend on the length of interval [a, b], and τ [a,b] : [0, ) [0, ). A for Univariate 2014 June Function 9 4 Approximatio

5 Piecewise Linear Interpolation My Questions Background Goal: Find an algorithm A, such that Evenly spaced data: f A(f ) ε x i = (i 1) b a n 1 + a, f (x i) i = 1,..., n, Algorithm: x [x i, x i+1 ] A n (f )(x) = n 1 b a [f (x i)(x i+1 x) + f (x i+1 )(x x i )] A for Univariate 2014 June Function 9 5 Approximatio

6 Estimate Introduction f f (b) f (a) b a Error Bound Estimation Complexity of Multi-Step Automatic Algorithm Define F n (f ) := Bound f 0 f sup i=1,...,n 1 f (b) f (a) b a n 1 b a [f (x i+1) f (x i )] f (b) f (a) b a For all f C τ, if n > 1 + τ [a,b] /2 f f (b) f (a) b a F n (f ) f (b) f (a) b a b a f 2(n 1) F n (f ) 1 τ [a,b] /(2n 2).. A for Univariate 2014 June Function 9 6 Approximatio

7 Error Bound Estimation Complexity of Multi-Step Automatic Algorithm Bound f Define F n (f ) := For f C τ (n 1)2 (b a) 2 sup f (x i ) 2f (x i+1 ) + f (x i+2 ). i=1,...,n 2 F n (f ) f If n > 1 + τ [a,b] /2 τ min,n = τ [a,b] (b a) F n (f ) 1 τ [a,b] /(2n 2) F n (f ) F n (f )/(b a) + F n (f )/(2n i 2) τ [a,b] A for Univariate 2014 June Function 9 7 Approximatio

8 Error Bound Derivation Error Bound Estimation Complexity of Multi-Step Automatic Algorithm Error Bound Goal: f A n (f ) (b a)2 f 8(n 1) 2 f (b a)τ [a,b] 8(n 1) 2 f (b) f (a) b a τ [a,b] (b a) F n (f ) 4(n 1)(2n 2 τ [a,b] ) τ [a,b] (b a) F n (f ) 4(n 1)(2n 2 τ [a,b] ) ε f A n(f ) ε A for Univariate 2014 June Function 9 8 Approximatio

9 Multi-Step Automatic Algorithm Error Bound Estimation Complexity of Multi-Step Automatic Algorithm Initial Process Set i = 1. Choose n lo, n hi, where n hi n lo { ( ) 1 } nlo 1+b a n 1 = max n hi, 3. n hi Let τ = 2(n 1 1) 1. Given error tolerance ε and input function f. Stage 1. Estimate f f (b) f (a) b a Compute F ni (f ) and F ni (f ) as above. and bound f A for Univariate 2014 June Function 9 9 Approximatio

10 Multi-Step Automatic Algorithm Error Bound Estimation Complexity of Multi-Step Automatic Algorithm Stage 2. Check the necessary condition for f C τ τ min,ni = F ni (f ) F ni (f )/(b a) + F ni (f )/(2n i 2). If τ τ min,ni, go to stage 3. Otherwise, set τ = 2τ min,ni. If n i (τ + 1)/2, go to stage 3. Otherwise, choose τ + 1 n i+1 = 1 + (n i 1). 2n i 2 Go to Stage 1. A for Univariate 2014 June Function 9 10 Approximatio

11 Multi-Step Automatic Algorithm Stage 3. Check for convergence Check whether n i large enough such that Error Bound Estimation Complexity of Multi-Step Automatic Algorithm F ni (f ) 4ε(n i 1)(2n i 2 τ). τ(b a) If true, then return A ni (f ) and terminate the algorithm. If NOT true, choose n i+1 = 1 + (n i 1) max 2, 1 τ(b a) F ni (f ) (n i 1) 8ε Go to Stage 1. A for Univariate 2014 June Function 9 11 Approximatio

12 Error Bound Estimation Complexity of Multi-Step Automatic Algorithm Theorem ( τ ) + 1 (b a) max, 2 f ε max τ + 1 τ(b a) f, 2 8ε τ(b a) f 2ε f (b) f (a) b a cost(a, f ; ε) f (b) f (a) b a + 1 τ(b a) +τ+4 2 f +τ+4. 4ε A for Univariate 2014 June Function 9 12 Approximatio

13 What is GAIL? Introduction about GAIL Introduction about funappx g Syntax of funappx g Documentation and Test Numerical Examples GAIL is Guaranteed Automatic Integration Library A suite of algorithms In one, many, and infinite dimensions Answers guaranteed to be correct A for Univariate 2014 June Function 9 13 Approximatio

14 What is funappx g? Introduction about GAIL Introduction about funappx g Syntax of funappx g Documentation and Test Numerical Examples Built on 1-D guaranteed function recovery On closed interval [a,b] A for Univariate 2014 June Function 9 14 Approximatio

15 How to use funappx g Introduction about GAIL Introduction about funappx g Syntax of funappx g Documentation and Test Numerical Examples pp = funappx g(f ) pp = funappx g(f, a, b, abstol, nlo, nhi, nmax) pp = funappx g(f, a, a, b, b, abstol, abstol, nlo, nlo, nhi, nhi, nmax, nmax) [pp, out param] = funappxab g(f, in param) pp a piecewise polynomial structure out param a structure contained parameters of algorithm A for Univariate 2014 June Function 9 15 Approximatio

16 Introduction about GAIL Introduction about funappx g Syntax of funappx g Documentation and Test Numerical Examples Doc Test, Unit Test, Documentation Doc Test Unit Test Documentation A for Univariate 2014 June Function 9 16 Approximatio

17 f (x) = x 2 Introduction Introduction about GAIL Introduction about funappx g Syntax of funappx g Documentation and Test Numerical Examples Fix ε = 1e 7, nlo = 10, nhigh = 1000, nmax = x 2 ninit npoints errorbound exceedbudget flag [0, 1] e 10 No [ 1, 1] e 10 No [ 10, 10] e 08 Yes A for Univariate 2014 June Function 9 17 Approximatio

18 Fix Parameters Introduction about GAIL Introduction about funappx g Syntax of funappx g Documentation and Test Numerical Examples a = 1, b = 1, ε = 1e 7, nlo = 10, nhigh = 1000, nmax = Funct ninit npoints err bound flag observed err f = x e 10 No e 10 f = e 10000x e 08 Yes e 08 1 f = 1+(10x) e 07 Yes e 06 A for Univariate 2014 June Function 9 18 Approximatio

19 Develop locally adaptive algorithms Automatic algorithms with guaranteed relative error Develop high order algorithms A for Univariate 2014 June Function 9 19 Approximatio

20 Questions Introduction Thank you! A for Univariate 2014 June Function 9 20 Approximatio