Hickernell Department of Applied Mathematics Center for Interdisciplinary Scientific Computation Illinois Institute of Technology hickernell@iit.edu mypages.iit.edu/~hickernell with Sou-Cheng Choi, Yuhan Ding, Mac Hyman, Xin Tong, and the GAIL team partially supported by NSF-DMS-1522687 and NSF-DMS-1638521 (SAMSI) Thanks to Guohui Song for the invitation and hospitality Old Dominion University, March 5, 2020
References Problem Given black-box function routine f : X ⊆ Rd → R, e.g., output of a computer simulation Expensive cost of a function value, $(f) Want fixed tolerance algorithm ALG : C × (0, ∞) → L∞(X) such that f − ALG(f, ε) ∞ ε ∀f ∈ C candidate set cheap cost of an ALG(f, ε) value, e.g., spline 3/14
References Problem Given black-box function routine f : X ⊆ Rd → R, e.g., output of a computer simulation Expensive cost of a function value, $(f) Want fixed tolerance algorithm ALG : C × (0, ∞) → L∞(X) such that f − ALG(f, ε) ∞ ε ∀f ∈ C candidate set cheap cost of an ALG(f, ε) value design or node array X ∈ Xn ⊆ Rn×d, function data y = f(X) ∈ Rn xn+1 = argmax x∈X ACQ(x, X, y) acquisition function f − APP(X, y) ∞ ERR(X, y) data-driven error bound ∀n ∈ N, f ∈ C n∗ = min {n ∈ N: ERR(X, y) ε} stopping criterion ALG(f, ε) = APP(X, y) fixed budget approximation for this n∗ 3/14
References Problem Given black-box function routine f : X ⊆ Rd → R, e.g., output of a computer simulation Expensive cost of a function value, $(f) Want fixed tolerance algorithm ALG : C × (0, ∞) → L∞(X) such that f − ALG(f, ε) ∞ ε ∀f ∈ C candidate set cheap cost of an ALG(f, ε) value design or node array X ∈ Xn ⊆ Rn×d, function data y = f(X) ∈ Rn xn+1 = argmax x∈X ACQ(x, X, y) acquisition function f − APP(X, y) ∞ ERR(X, y) data-driven error bound ∀n ∈ N, f ∈ C n∗ = min {n ∈ N: ERR(X, y) ε} stopping criterion ALG(f, ε) = APP(X, y) fixed budget approximation for this n∗ Adaptive sample size, design, and fixed budget approximation Assumes that what you see is almost what you get 3/14
References Linear Splines X f : [a, b] → R a =: x0 < x1 < · · · < xn := b, X = xi n i=0 data sites function data y = f(X) linear spline APP(X, y) := x − xi xi−1 − xi yi−1 + x − xi−1 xi − xi−1 yi , xi−1 x xi , i ∈ 1:n f − APP(X, y) ∞,[xi−1,xi] (xi − xi−1 )2 f ∞,[xi−1,xi] 8 , i ∈ 1:n, f ∈ W2,∞ 4/14
References Linear Splines X f : [a, b] → R a =: x0 < x1 < · · · < xn := b, X = xi n i=0 data sites function data y = f(X) linear spline APP(X, y) := x − xi xi−1 − xi yi−1 + x − xi−1 xi − xi−1 yi , xi−1 x xi , i ∈ 1:n f − APP(X, y) ∞,[xi−1,xi] (xi − xi−1 )2 f ∞,[xi−1,xi] 8 , i ∈ 1:n, f ∈ W2,∞ Numerical analysis often stops here, leaving unanswered questions: How big should n be to make f − APP(X, y) ∞ ε? How big is f ∞,[xi−1,xi]? How best to choose X? 4/14
References Linear Splines Error f : [a, b] → R a =: x0 < x1 < · · · < xn := b, X = xi n i=0 data sites function data y = f(X) linear spline APP(X, y) := x − xi xi−1 − xi yi−1 + x − xi−1 xi − xi−1 yi , xi−1 x xi , i ∈ 1:n f − APP(X, y) ∞,[xi−1,xi] 1 8 (xi − xi−1 )2 f ∞,[xi−1,xi] , i ∈ 1:n, f ∈ W2,∞ f −∞,[xi−1,xi+1] yi+1−yi xi+1−xi − yi−yi−1 xi−xi−1 (xi+1 − xi−1 )/2 Di(X,y)=2|f[xi−1,xi,xi+1]| data based abs. 2nd deriv. of interp. poly. f ∞,[xi−1,xi+1] 5/14
References Linear Splines Error f : [a, b] → R a =: x0 < x1 < · · · < xn := b, X = xi n i=0 data sites function data y = f(X) linear spline APP(X, y) := x − xi xi−1 − xi yi−1 + x − xi−1 xi − xi−1 yi , xi−1 x xi , i ∈ 1:n f − APP(X, y) ∞,[xi−1,xi] 1 8 (xi − xi−1 )2 f ∞,[xi−1,xi] , i ∈ 1:n, f ∈ W2,∞ f −∞,[xi−1,xi+1] yi+1−yi xi+1−xi − yi−yi−1 xi−xi−1 (xi+1 − xi−1 )/2 Di(X,y)=2|f[xi−1,xi,xi+1]| data based abs. 2nd deriv. of interp. poly. f ∞,[xi−1,xi+1] candidate set C := f ∈ W2,∞ : |f (x)| max C(h− ) |f (x − h− )| , C(h+ ) |f (x + h+ )| , 0 < h± < h, a < x < b inflation factor C(h) := C0 h h − h |f | does not change abruptly 5/14
References Linear Splines Error f : [a, b] → R a =: x0 < x1 < · · · < xn := b, X = xi n i=0 data sites function data y = f(X) f − APP(X, y) ∞,[xi−1,xi] 1 8 (xi − xi−1 )2 f ∞,[xi−1,xi] max ± ERRi,± (X, y), i ∈ 1:n, f ∈ C candidate set C := f ∈ W2,∞ : |f (x)| max C(h− ) |f (x − h− )| , C(h+ ) |f (x + h+ )| , 0 < h± < h, a < x < b inflation factor C(h) := C0 h h − h |f | does not change abruptly ERRi,− (X, y) = 1 8 (xi − xi−1 )2C(xi − xi−3 )Di−2 (X, y), ERRi,+ (X, y) = 1 8 (xi − xi−1 )2C(xi+2 − xi−1 )Di+1 (X, y) Di (X, y) = 2 |f[xi−1 , xi , xi+1 ]| data based, absolute 2nd derivative of interpoplating polynomial 5/14
References Adaptive Linear Spline Algorithm X Given ninit 4, C0 1: h = 3(b − a) ninit − 1 , C(h) = C0 h h − h n = ninit , xi = a + i(b − a)/n Choi, S.-C. T., Ding, Y., H., F. J. & Tong, X. Local Adaption for Approximation and Minimization of Univariate Functions. J. Complexity 40, 17–33 (2017). 6/14
References Adaptive Linear Spline Algorithm X Given ninit 4, C0 1: h = 3(b − a) ninit − 1 , C(h) = C0 h h − h n = ninit , xi = a + i(b − a)/n Step 1. Compute data based ERRi,± (X, y) for i = 1, . . . , n. Step 2. Construct I, the index set of subintervals that might be split: I = i ∈ 1:n : ERRi±j,∓ (X, y) > ε, j = 0, 1, 2} Choi, S.-C. T., Ding, Y., H., F. J. & Tong, X. Local Adaption for Approximation and Minimization of Univariate Functions. J. Complexity 40, 17–33 (2017). 6/14
References Adaptive Linear Spline Algorithm X Given ninit 4, C0 1: h = 3(b − a) ninit − 1 , C(h) = C0 h h − h n = ninit , xi = a + i(b − a)/n Step 1. Compute data based ERRi,± (X, y) for i = 1, . . . , n. Step 2. Construct I, the index set of subintervals that might be split: I = i ∈ 1:n : ERRi±j,∓ (X, y) > ε, j = 0, 1, 2} Step 3. If I = ∅, return ALG(f, ε) = APP(X, y) as the approximation satisfying the error tolerance. Otherwise split those intervals in I with largest width and go to Step 1 (acquisition function). Choi, S.-C. T., Ding, Y., H., F. J. & Tong, X. Local Adaption for Approximation and Minimization of Univariate Functions. J. Complexity 40, 17–33 (2017). 6/14
References Adaptive Linear Spline Algorithm X Given ninit 4, C0 1: h = 3(b − a) ninit − 1 , C(h) = C0 h h − h n = ninit , xi = a + i(b − a)/n Step 1. Compute data based ERRi,± (X, y) for i = 1, . . . , n. Step 2. Construct I, the index set of subintervals that might be split: I = i ∈ 1:n : ERRi±j,∓ (X, y) > ε, j = 0, 1, 2} Step 3. If I = ∅, return ALG(f, ε) = APP(X, y) as the approximation satisfying the error tolerance. Otherwise split those intervals in I with largest width and go to Step 1 (acquisition function). Choi, S.-C. T., Ding, Y., H., F. J. & Tong, X. Local Adaption for Approximation and Minimization of Univariate Functions. J. Complexity 40, 17–33 (2017). 6/14
References Highlights of Adaptive Linear Spline Algorithm X Defined for cone candidate set, C, whose definition does not depend on the algorithm Choi, S.-C. T., Ding, Y., H., F. J. & Tong, X. Local Adaption for Approximation and Minimization of Univariate Functions. J. Complexity 40, 17–33 (2017). 7/14
References Highlights of Adaptive Linear Spline Algorithm X Defined for cone candidate set, C, whose definition does not depend on the algorithm Guaranteed to succeed for all f ∈ C Choi, S.-C. T., Ding, Y., H., F. J. & Tong, X. Local Adaption for Approximation and Minimization of Univariate Functions. J. Complexity 40, 17–33 (2017). 7/14
References Highlights of Adaptive Linear Spline Algorithm X Defined for cone candidate set, C, whose definition does not depend on the algorithm Guaranteed to succeed for all f ∈ C Candidate set C excludes spikes, i.e., two nearby inflection points Choi, S.-C. T., Ding, Y., H., F. J. & Tong, X. Local Adaption for Approximation and Minimization of Univariate Functions. J. Complexity 40, 17–33 (2017). 7/14
References Highlights of Adaptive Linear Spline Algorithm X Defined for cone candidate set, C, whose definition does not depend on the algorithm Guaranteed to succeed for all f ∈ C Candidate set C excludes spikes, i.e., two nearby inflection points C formalizes what you see is almost what you get Choi, S.-C. T., Ding, Y., H., F. J. & Tong, X. Local Adaption for Approximation and Minimization of Univariate Functions. J. Complexity 40, 17–33 (2017). 7/14
References Highlights of Adaptive Linear Spline Algorithm X Defined for cone candidate set, C, whose definition does not depend on the algorithm Guaranteed to succeed for all f ∈ C Candidate set C excludes spikes, i.e., two nearby inflection points C formalizes what you see is almost what you get Impossible to have an algorithm for all f ∈ W2,∞ since W2,∞ contains arbitrarily large functions that look like 0 Choi, S.-C. T., Ding, Y., H., F. J. & Tong, X. Local Adaption for Approximation and Minimization of Univariate Functions. J. Complexity 40, 17–33 (2017). 7/14
References Highlights of Adaptive Linear Spline Algorithm X Defined for cone candidate set, C, whose definition does not depend on the algorithm Guaranteed to succeed for all f ∈ C Candidate set C excludes spikes, i.e., two nearby inflection points C formalizes what you see is almost what you get Impossible to have an algorithm for all f ∈ W2,∞ since W2,∞ contains arbitrarily large functions that look like 0 Adaptive algorithms do not help for ball candidate sets C = {f : f ∞ R} Choi, S.-C. T., Ding, Y., H., F. J. & Tong, X. Local Adaption for Approximation and Minimization of Univariate Functions. J. Complexity 40, 17–33 (2017). 7/14
References Highlights of Adaptive Linear Spline Algorithm X Defined for cone candidate set, C, whose definition does not depend on the algorithm Guaranteed to succeed for all f ∈ C Candidate set C excludes spikes, i.e., two nearby inflection points C formalizes what you see is almost what you get Impossible to have an algorithm for all f ∈ W2,∞ since W2,∞ contains arbitrarily large functions that look like 0 Adaptive algorithms do not help for ball candidate sets C = {f : f ∞ R} cost(ALG, f, ε, C) C0 f 1 2 ε comp(f, ε, C) optimal Choi, S.-C. T., Ding, Y., H., F. J. & Tong, X. Local Adaption for Approximation and Minimization of Univariate Functions. J. Complexity 40, 17–33 (2017). 7/14
References Highlights of Adaptive Linear Spline Algorithm X Defined for cone candidate set, C, whose definition does not depend on the algorithm Guaranteed to succeed for all f ∈ C Candidate set C excludes spikes, i.e., two nearby inflection points C formalizes what you see is almost what you get Impossible to have an algorithm for all f ∈ W2,∞ since W2,∞ contains arbitrarily large functions that look like 0 Adaptive algorithms do not help for ball candidate sets C = {f : f ∞ R} cost(ALG, f, ε, C) C0 f 1 2 ε comp(f, ε, C) optimal Does not allow for smoothness to be inferred Choi, S.-C. T., Ding, Y., H., F. J. & Tong, X. Local Adaption for Approximation and Minimization of Univariate Functions. J. Complexity 40, 17–33 (2017). 7/14
References Highlights of Adaptive Linear Spline Algorithm X Defined for cone candidate set, C, whose definition does not depend on the algorithm Guaranteed to succeed for all f ∈ C Candidate set C excludes spikes, i.e., two nearby inflection points C formalizes what you see is almost what you get Impossible to have an algorithm for all f ∈ W2,∞ since W2,∞ contains arbitrarily large functions that look like 0 Adaptive algorithms do not help for ball candidate sets C = {f : f ∞ R} cost(ALG, f, ε, C) C0 f 1 2 ε comp(f, ε, C) optimal Does not allow for smoothness to be inferred Not multivariate Choi, S.-C. T., Ding, Y., H., F. J. & Tong, X. Local Adaption for Approximation and Minimization of Univariate Functions. J. Complexity 40, 17–33 (2017). 7/14
References Approximation via Reproducing Kernel Hilbert Spaces (RKHSs) X F is a Hilbert space with reproducing kernel K : X × X → R K(X, X) positive definite ∀X K(·, x) ∈ F, f(x) = K(·, x), f F ∀x ∈ X, e.g., K(t, x) = (1 + t − x 2 ) exp − t − x 2 Matérn Fasshauer, G. E. Meshfree Approximation Methods with M . (World Scientific Publishing Co., Singapore, 2007), Fasshauer, G. E. & McCourt, M. Kernel-based Approximation Methods using MATLAB. (World Scientific Publishing Co., Singapore, 2015). 8/14
References Approximation via Reproducing Kernel Hilbert Spaces (RKHSs) X F is a Hilbert space with reproducing kernel K : X × X → R K(X, X) positive definite ∀X K(·, x) ∈ F, f(x) = K(·, x), f F ∀x ∈ X, e.g., K(t, x) = (1 + t − x 2 ) exp − t − x 2 Matérn Optimal (minimum norm) interpolant is APP(X, y) = K(·, X) K(X, X) −1 y, y = f(X) f − APP(X, y) 2 ∞ K(·, ·) − K(·, X) K(X, X) −1 K(X, ·) ∞ f − APP(X, y) 2 F known Fasshauer, G. E. Meshfree Approximation Methods with M . (World Scientific Publishing Co., Singapore, 2007), Fasshauer, G. E. & McCourt, M. Kernel-based Approximation Methods using MATLAB. (World Scientific Publishing Co., Singapore, 2015). 8/14
References Approximation via Reproducing Kernel Hilbert Spaces (RKHSs) X F is a Hilbert space with reproducing kernel K : X × X → R K(X, X) positive definite ∀X K(·, x) ∈ F, f(x) = K(·, x), f F ∀x ∈ X, e.g., K(t, x) = (1 + t − x 2 ) exp − t − x 2 Matérn Optimal (minimum norm) interpolant is APP(X, y) = K(·, X) K(X, X) −1 y, y = f(X) f − APP(X, y) 2 ∞ K(·, ·) − K(·, X) K(X, X) −1 K(X, ·) ∞ f − APP(X, y) 2 F known candidate set C = f ∈ F : f − APP(X, y) F C(X) f F Fasshauer, G. E. Meshfree Approximation Methods with M . (World Scientific Publishing Co., Singapore, 2007), Fasshauer, G. E. & McCourt, M. Kernel-based Approximation Methods using MATLAB. (World Scientific Publishing Co., Singapore, 2015). 8/14
References Approximation via Reproducing Kernel Hilbert Spaces (RKHSs) X F is a Hilbert space with reproducing kernel K : X × X → R K(X, X) positive definite ∀X K(·, x) ∈ F, f(x) = K(·, x), f F ∀x ∈ X, e.g., K(t, x) = (1 + t − x 2 ) exp − t − x 2 Matérn Optimal (minimum norm) interpolant is APP(X, y) = K(·, X) K(X, X) −1 y, y = f(X) f − APP(X, y) 2 ∞ K(·, ·) − K(·, X) K(X, X) −1 K(X, ·) ∞ f − APP(X, y) 2 F known K(·, ·) − K(·, X) K(X, X) −1 K(X, ·) ∞ C2(X) 1 − C2(X) APP(X, y) 2 F =: ERR2(X, y) candidate set C = f ∈ F : f − APP(X, y) F C(X) f F Fasshauer, G. E. Meshfree Approximation Methods with M . (World Scientific Publishing Co., Singapore, 2007), Fasshauer, G. E. & McCourt, M. Kernel-based Approximation Methods using MATLAB. (World Scientific Publishing Co., Singapore, 2015). 8/14
References Error and Acquisition for Optimal RKHS Approximation X F is a Hilbert space with reproducing kernel K : X × X → R e.g., K(t, x) = (1 + t − x 2 ) exp − t − x 2 Matérn APP(X, y) = K(·, X) K(X, X) −1 y, y = f(X) candidate set C = f ∈ F : f − APP(X, y) F C(X) f F f − APP(X, y) 2 ∞ K(·, ·) − K(·, X) K(X, X) −1 K(X, ·) ∞ f − APP(X, y) 2 F known K(·, ·) − K(·, X) K(X, X) −1 K(X, ·) ∞ C2(X) 1 − C2(X) APP(X, y) 2 F =: ERR2(X, y) ACQ(x, X, y) := K(x, x) − K(x, X) K(X, X) −1 K(X, x) C2(X) 1 − C2(X) APP(X, y) 2 F yT(K(X,X))−1y xn+1 = argmax x∈X ACQ(x, X, y) acquisition function ALG(f, ε) = APP(X, y) for n∗ = min {n ∈ N: ERR(X, y) ε} stopping criterion 9/14
References Error and Acquisition for Optimal RKHS Approximation X X X F is a Hilbert space with reproducing kernel K : X × X → R e.g., K(t, x) = (1 + t − x 2 ) exp − t − x 2 Matérn APP(X, y) = K(·, X) K(X, X) −1 y, y = f(X) candidate set C = f ∈ F : f − APP(X, y) F C(X) f F f − APP(X, y) 2 ∞ K(·, ·) − K(·, X) K(X, X) −1 K(X, ·) ∞ f − APP(X, y) 2 F known K(·, ·) − K(·, X) K(X, X) −1 K(X, ·) ∞ C2(X) 1 − C2(X) APP(X, y) 2 F =: ERR2(X, y) ACQ(x, X, y) := K(x, x) − K(x, X) K(X, X) −1 K(X, x) C2(X) 1 − C2(X) APP(X, y) 2 F yT(K(X,X))−1y xn+1 = argmax x∈X ACQ(x, X, y) acquisition function ALG(f, ε) = APP(X, y) for n∗ = min {n ∈ N: ERR(X, y) ε} stopping criterion 9/14
References Must Infer Kernel from y = f(X) X X X Fθ is a Hilbert space with reproducing kernel Kθ C = f ∈ ∪ θ Fθ : f − APP(X, y) Fθ∗ C(X) f Fθ∗ ∀X, y = f(X), θ∗(X, y) given below e.g., Kθ(t, x) = (1 + θ (t − x) 2 ) exp − θ (t − x) 2 Choose the θ (inspired by empirical Bayes) by minimizing the ellipsoid in Rn of function data yielding interpolants with no greater norm than that observed: θ∗ = argmin θ 1 n log det(Kθ) + log yT K−1 θ y f − APP(X, y) 2 ∞ K(·, ·) − K(·, X) K(X, X) −1 K(X, ·) ∞ C2(X) 1 − C2(X) yT(K(X, X))−1y =: ERR2(X, y) ACQ(x, X, y) := K(x, x) − K(x, X) K(X, X) −1 K(X, x) C2(X) 1 − C2(X) yT(K(X, X))−1y xn+1 = argmax x∈X ACQ(x, X, y) acquisition function 10/14
References Must Infer Kernel from y = f(X) X X X Fθ is a Hilbert space with reproducing kernel Kθ C = f ∈ ∪ θ Fθ : f − APP(X, y) Fθ∗ C(X) f Fθ∗ ∀X, y = f(X), θ∗(X, y) given below e.g., Kθ(t, x) = exp(bT(t + x)) × (1 + a (t − x) 2 ) exp − a (t − x) 2 , θ = (a, b) Choose the θ (inspired by empirical Bayes) by minimizing the ellipsoid in Rn of function data yielding interpolants with no greater norm than that observed: θ∗ = argmin θ 1 n log det(Kθ) + log yT K−1 θ y f − APP(X, y) 2 ∞ K(·, ·) − K(·, X) K(X, X) −1 K(X, ·) ∞ C2(X) 1 − C2(X) yT(K(X, X))−1y =: ERR2(X, y) ACQ(x, X, y) := K(x, x) − K(x, X) K(X, X) −1 K(X, x) C2(X) 1 − C2(X) yT(K(X, X))−1y xn+1 = argmax x∈X ACQ(x, X, y) acquisition function 10/14
References Must Infer Kernel from y = f(X) X X X Fθ is a Hilbert space with reproducing kernel Kθ C = f ∈ ∪ θ Fθ : f − APP(X, y) Fθ∗ C(X) f Fθ∗ ∀X, y = f(X), θ∗(X, y) given below e.g., Kθ(t, x) = exp(bT(t + x)) × (1 + a (t − x) 2 ) exp − a (t − x) 2 , θ = (a, b) Choose the θ (inspired by empirical Bayes) by minimizing the ellipsoid in Rn of function data yielding interpolants with no greater norm than that observed: θ∗ = argmin θ 1 n log det(Kθ) + log yT K−1 θ y f − APP(X, y) 2 ∞ K(·, ·) − K(·, X) K(X, X) −1 K(X, ·) ∞ C2(X) 1 − C2(X) yT(K(X, X))−1y =: ERR2(X, y) ACQ(x, X, y) := K(x, x) − K(x, X) K(X, X) −1 K(X, x) C2(X) 1 − C2(X) yT(K(X, X))−1y xn+1 = argmax x∈X ACQ(x, X, y) acquisition function 10/14
References Must Infer Kernel from y = f(X) X X X Fθ is a Hilbert space with reproducing kernel Kθ C = f ∈ ∪ θ Fθ : f − APP(X, y) Fθ∗ C(X) f Fθ∗ ∀X, y = f(X), θ∗(X, y) given below e.g., Kθ(t, x) = exp(bT(t + x)) × (1 + a (t − x) 2 ) exp − a (t − x) 2 , θ = (a, b) Choose the θ (inspired by empirical Bayes) by minimizing the ellipsoid in Rn of function data yielding interpolants with no greater norm than that observed: θ∗ = argmin θ 1 n log det(Kθ) + log yT K−1 θ y f − APP(X, y) 2 ∞ K(·, ·) − K(·, X) K(X, X) −1 K(X, ·) ∞ C2(X) 1 − C2(X) yT(K(X, X))−1y =: ERR2(X, y) ACQ(x, X, y) := K(x, x) − K(x, X) K(X, X) −1 K(X, x) C2(X) 1 − C2(X) yT(K(X, X))−1y xn+1 = argmax x∈X ACQ(x, X, y) acquisition function 10/14
References What Are the Right Ingredients for Adaptive Function Approximation? A fixed budget homogeneous approximation, APP : Xn × Rn → L∞(X), with an error bound, e.g., linear splines, RKHS approximation An unbounded, non-convex candidate set, C, for which the error bound can be bounded in data-driven way; what you see is almost what you get Necessary conditions for f to lie in C; will not have sufficient conditions A rich enough candidate set from which the right approximation can be inferred; attention to underfitting and overfitting 12/14
References What Are the Right Ingredients for Adaptive Function Approximation? A fixed budget homogeneous approximation, APP : Xn × Rn → L∞(X), with an error bound, e.g., linear splines, RKHS approximation An unbounded, non-convex candidate set, C, for which the error bound can be bounded in data-driven way; what you see is almost what you get Necessary conditions for f to lie in C; will not have sufficient conditions A rich enough candidate set from which the right approximation can be inferred; attention to underfitting and overfitting More work is needed on What makes a good initial sample Balancing the richness of the candidate set with overfitting Numerical instability and computational effort challenges for larger numbers of data sites. 12/14
References References Choi, S.-C. T., Ding, Y., H., F. J. & Tong, X. Local Adaption for Approximation and Minimization of Univariate Functions. J. Complexity 40, 17–33 (2017). Fasshauer, G. E. Meshfree Approximation Methods with M . (World Scientific Publishing Co., Singapore, 2007). Fasshauer, G. E. & McCourt, M. Kernel-based Approximation Methods using MATLAB. (World Scientific Publishing Co., Singapore, 2015). Bingham, D. & Surjano, S. Virtual Library of Simulation Experiments. 2013. https://www.sfu.ca/~ssurjano/. Cheng, H. & Sandu, A. Collocation least-squares polynomial chaos method. in Proceedings of the 2010 Spring Simulation Multiconference, Society for Computer Simulation International. (2010). 14/14
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