Radial basis functions topics in slides

Transcription

1 Radial basis functions topics in slides Stefano De Marchi Department of Mathematics Tullio Levi-Civita University of Padova Napoli, 22nd March 2018

2 Outline 1 From splines to RBF 2 Error estimates, conditioning and stability Strategies for controlling errors 3 Stable basis approaches WSVD Basis 4 Rescaled, Rational RBF and some applications 2 of 41

3 From cubic splines to RBF 3 of 41

4 Cubic splines Let S 3 (X) = {s C 2 [a, b] : s [xi,x i+1 ] P 3, 1 i N 1, s [a,x1 ], s [xn,b] P 1 } be the space of natural cubic splines. S 3 (X) has the basis of truncated powers ( x j ) 3 +, 1 j N plus an arbitrary basis for P 3 (R) s(x) = N a j (x x j ) j=1 3 b j x j, x [a, b]. (1) Using the identity x 3 + = ( x 3 +x 3 ) 2 and the fact that s [a,x1 ], s [xn,b] P 1 j=0 4 of 41

5 Cubic splines Let S 3 (X) = {s C 2 [a, b] : s [xi,x i+1 ] P 3, 1 i N 1, s [a,x1 ], s [xn,b] P 1 } be the space of natural cubic splines. S 3 (X) has the basis of truncated powers ( x j ) 3 +, 1 j N plus an arbitrary basis for P 3 (R) s(x) = N a j (x x j ) j=1 3 b j x j, x [a, b]. (1) Using the identity x+ 3 = ( x 3 +x 3 ) and the fact that s 2 [a,x1 ], s [xn,b] P 1 Every natural cubic spline s has the representation N s(x) = a j φ( x x j ) + p(x), x R j=1 j=0 where φ(r) = r 3, r 0 and p P 1 (R) s.t. N a j = j=1 N a j x j = 0. (2) j=1 4 of 41

6 Extension to R d Going to R d is straightforward: radiality becomes more evident s(x) = N a j φ( x x j 2 ) + p(x), x R d, (3) i=1 where φ : [0, ) R is a univariate fixed function and p P l 1 (R d ) is a low degree d-variate polynomial. The additional conditions in (??) become N a j q(x j ) = 0, q P l 1 (R d ). (4) i=1 5 of 41

7 Extension to R d Going to R d is straightforward: radiality becomes more evident s(x) = N a j φ( x x j 2 ) + p(x), x R d, (3) i=1 where φ : [0, ) R is a univariate fixed function and p P l 1 (R d ) is a low degree d-variate polynomial. The additional conditions in (??) become N a j q(x j ) = 0, q P l 1 (R d ). (4) Obs: i=1 We can omit the side conditions on the coefficients (??) (like using B-splines ) and consider approximants of the form s(x) = N a j φ( x x j 2 ) i=1 and also omit the 2-norm symbol 5 of 41

8 Scattered data fitting:i Setting Given data X = {x j, j = 1,..., N}, with x j Ω R d and values Y = {y j R, j = 1,..., N} (y j = f(x j )), find a (continuous) function P f span{φ( x i ), x i X} s.t. P f = N α j φ( x i ), s.t.. (P f ) X = Y (interpolation) j=1 6 of 41

9 Scattered data fitting:i Setting Given data X = {x j, j = 1,..., N}, with x j Ω R d and values Y = {y j R, j = 1,..., N} (y j = f(x j )), find a (continuous) function P f span{φ( x i ), x i X} s.t. P f = N α j φ( x i ), s.t.. (P f ) X = Y (interpolation) j=1 Figure: Data points, data values and data function Obs: P f = n i=j u j f j, with u j (x i ) = δ ji. 6 of 41

10 Scattered data fitting:ii Problem For which functions φ : [0, ) R such that for all d, N N and all pairwise distrinct x 1,..., x n R d the matrix det(a φ,x ) 0? When the matrix A φ,x := (φ( x i x j 2 ) 1 i,j N, is invertible? 7 of 41

11 Scattered data fitting:ii Problem For which functions φ : [0, ) R such that for all d, N N and all pairwise distrinct x 1,..., x n R d the matrix det(a φ,x ) 0? When the matrix A φ,x := (φ( x i x j 2 ) 1 i,j N, is invertible? Answer The functions φ should be positive definite or conditionally positive definite of some order. 7 of 41

12 General kernel-based approximation φ, Conditionally Positive Definite (CPD) of order l or Strictly Positive Definite (SPD) and radial globally supported: locally supported: name φ l Gaussian C (GA) e ε2 r 2 0 Generalized Multiquadrics C (GM) (1 + r 2 /ε 2 ) 3/2 2 name φ l Wendland C 2 (W2) (1 εr) 4 + (4εr + 1) 0 Buhmann C 2 (B2) 2r 4 log r 7/2r /3r 3 2r 2 + 1/6 0 we often consider φ(ε ), with ε called shape parameter 8 of 41

13 General kernel-based approximation φ, Conditionally Positive Definite (CPD) of order l or Strictly Positive Definite (SPD) and radial globally supported: locally supported: name φ l Gaussian C (GA) e ε2 r 2 0 Generalized Multiquadrics C (GM) (1 + r 2 /ε 2 ) 3/2 2 name φ l Wendland C 2 (W2) (1 εr) 4 + (4εr + 1) 0 Buhmann C 2 (B2) 2r 4 log r 7/2r /3r 3 2r 2 + 1/6 0 we often consider φ(ε ), with ε called shape parameter kernel notation K(x, y)(= K ε (x, y)) = Φ ε (x y) = φ(ε x y 2 ) 8 of 41

14 General kernel-based approximation φ, Conditionally Positive Definite (CPD) of order l or Strictly Positive Definite (SPD) and radial globally supported: locally supported: name φ l Gaussian C (GA) e ε2 r 2 0 Generalized Multiquadrics C (GM) (1 + r 2 /ε 2 ) 3/2 2 name φ l Wendland C 2 (W2) (1 εr) 4 + (4εr + 1) 0 Buhmann C 2 (B2) 2r 4 log r 7/2r /3r 3 2r 2 + 1/6 0 we often consider φ(ε ), with ε called shape parameter kernel notation K(x, y)(= K ε (x, y)) = Φ ε (x y) = φ(ε x y 2 ) native space N K (Ω) (where K is the reproducing kernel) finite subspace N K (X) = span{k(, x) : x X} N K (Ω). 8 of 41

15 Error estimates, conditioning and stability 9 of 41

16 Separation distance, fill-distance and power function q X := 1 2 min x i x j 2, (separation distance) h X,Ω := sup min x x j 2, (fill distance) i j x Ω x j X P Φε,X (x) := Φ ε(0) (u (x)) T Au (x), (power function) u vector of cardinal functions Figure: The fill-distance of 25 Halton points h Figure: Power function for the Gaussian kernel with ε = 6 on a grid of 81 uniform, Chebyshev and Halton points, respectively. 10 of 41

17 and C K (x) = max β =2κ max w,z Ω B(x,c2 h X,Ω ) D β 2 Φ(w, z). 11 of 41 Pointwise error estimates Theorem Let Ω R d and K C(Ω Ω) be PD on R d. Let X = {x 1,...,, x n } be a set of distinct points. Take a function f N Φ (Ω) and denote with P f its interpolant on X. Then, for every x Ω f(x) P f (x) P Φε,X(x) f NK (Ω). (5) Theorem Let Ω R d and K C 2κ (Ω Ω) be symmetric and positive definite, X = {x 1,...,, x N } a set of distinct points. Consider f N K (Ω) and its interpolant P f on X. Then, there exist positive constants h 0 and C (independent of x, f and Φ), with h X,Ω h 0, such that f(x) P f (x) C h κ X,Ω C K (x) f NK (Ω). (6)

18 Reducing the interpolation error Obs: The choice of the shape parameter ε in order to get the smallest (possible) interpolation error is crucial. Trial and Error Power function minimization Leave One Out Cross Validation (LOOCV) 12 of 41

19 Reducing the interpolation error Obs: The choice of the shape parameter ε in order to get the smallest (possible) interpolation error is crucial. Trial and Error Power function minimization Leave One Out Cross Validation (LOOCV) Trial and error strategy: interpolation of the 1-d sinc function with Gaussian for ε [0, 20], taking 100 values of ε and different equispaced data points. 12 of 41

20 Trade-off principle Consider the errors (RMSE and MAXERR) and the condition number µ 2 of A = (Φ ε ( x i x j )) N i,j=1 Figure: RMSE, MAXERR and Condition Number, µ 2, with 30 values of ε [0.1, 20], for interpolation of the Franke function on a grid of Chebyshev points Trade-off or uncertainty principle [Schaback 1995] Accuracy vs Stability Accuracy vs Efficiency Accuracy and stability vs Problem size 13 of 41

21 Accuracy vs stability The error bounds for non-stationary interpolation using infinitely smooth basis functions show that the error decreases (exponentially) as the fill distance decreases. For well-distributed data a decrease in the fill distance also implies a decrease of the separation distance But now the condition estimates above imply that the condition number of A grows exponentially This leads to numerical instabilities which make it virtually impossible to obtain the highly accurate results promised by the theoretical error bounds. 14 of 41

22 Stable basis approaches 15 of 41

23 Problem setting and questions Obs: the standard basis of translates (data-dependent) of N K (X) is unstable and not flexible 16 of 41

24 Problem setting and questions Obs: the standard basis of translates (data-dependent) of N K (X) is unstable and not flexible Question 1 Is it possible to find a better basis U of N K (X)? Question 2 How to embed information about K and Ω in the basis U? Question 3 Can we extract U U s.t. s f is as good as s f? 16 of 41

25 The natural basis The natural (data-independent) basis for Hilbert spaces (Mercer s theorem,1909) Let K be a continuous, positive definite kernel on a bounded Ω. Then K has an eigenfunction expansion with non-negative coefficients, the eigenvalues, s.t. K(x, y) = λ j ϕ j (x)ϕ j (y), x, y Ω. j=0 Moreover, λ j ϕ j (x) = K(x, y)ϕ j (y)dy := T [ϕ j ](x), x Ω, j 0 Ω {ϕ j } j>0 orthonormal N K (Ω) {ϕ j } j>0 orthogonal L 2 (Ω), ϕ j 2 = λ L 2 (Ω) j 0, λ j = K(0, 0) Ω, (the operator is of trace-class) j>0 Notice: the functions ϕ i are explicitely in very few cases [Fasshauer,McCourt 2012, for GRBF]. 17 of 41

26 Notation T X = {K(, x i ), x i X}: the standard basis of translates; U = {u i N K (Ω), i = 1,..., N}: another basis s.t. span(u) = span(t X ). 18 of 41

27 Notation T X = {K(, x i ), x i X}: the standard basis of translates; U = {u i N K (Ω), i = 1,..., N}: another basis s.t. span(u) = span(t X ). Change of basis [Pazouki,Schaback 2011] Let A ij = K(x i, x j ) R N N. Any other basis U arises from a factorization A C U = V U or A = V U C U 1, where V U = (u j (x i )) 1 i,j N and C U is the matrix of change of basis. Each N K (Ω)-orthonormal basis U arises from an orthonormal decomposition A = B T B with B = C U 1, V U = B T = (C U 1 ) T. Each l 2 (X)-orthonormal basis U arises from a decomposition A = Q B with Q = V U, Q T Q = I, B = C U 1 = Q T A. Notice: the best bases in terms of stability are the N K (Ω)-orthonormal ones! 18 of 41

28 Notation T X = {K(, x i ), x i X}: the standard basis of translates; U = {u i N K (Ω), i = 1,..., N}: another basis s.t. span(u) = span(t X ). Change of basis [Pazouki,Schaback 2011] Let A ij = K(x i, x j ) R N N. Any other basis U arises from a factorization A C U = V U or A = V U C U 1, where V U = (u j (x i )) 1 i,j N and C U is the matrix of change of basis. Each N K (Ω)-orthonormal basis U arises from an orthonormal decomposition A = B T B with B = C U 1, V U = B T = (C U 1 ) T. Each l 2 (X)-orthonormal basis U arises from a decomposition A = Q B with Q = V U, Q T Q = I, B = C U 1 = Q T A. Notice: the best bases in terms of stability are the N K (Ω)-orthonormal ones! Q1: Yes, we can! 18 of 41

29 WSVD Basis construction Q2: How to embed information on K and Ω in U? Symmetric Nyström method [Atkinson,Han 2001] Idea: discretize the natural basis (Mercer s theorem) by a convergent cubature rule (X, W), with X = {x j } N Ω and positive weights j=1 W = {w j } N j=1 19 of 41

30 WSVD Basis construction Q2: How to embed information on K and Ω in U? Symmetric Nyström method [Atkinson,Han 2001] Idea: discretize the natural basis (Mercer s theorem) by a convergent cubature rule (X, W), with X = {x j } N Ω and positive weights j=1 W = {w j } N j=1 λ j ϕ j (x i ) = applying the cubature rule λ j ϕ j (x i ) Ω K(x i, y)ϕ j (y)dy i = 1,..., N, j > 0, N K(x i, x h )ϕ j (x h )w h i, j = 1,..., N. (7) h=1 Letting W = diag(w j ), we reduce to solve the eigen-problem λv = (A W)v (8) 19 of 41

31 WSVD basis Cont Rewrite (??) (using the fact that the weights are positive) as λ j ( w i ϕ j (x i )) = N ( w i K(x i, x h ) w h )( w h ϕ j (x h )) i, j = 1,..., N, h=1 and then to consider the corresponding scaled eigenvalue problem λ ( W v ) = ( W A W ) ( W v ) which is equivalent to the previous one, now involving the symmetric and positive definite matrix (9) A W := W A W (10) 20 of 41

32 WSVD Basis Definition {λ j, ϕ j } j>0 are then approximated by eigenvalues/eigenvectors of A W. This matrix is normal, then a singular value decomposition of A W is a unitary diagonalization. Definition A weighted SVD basis U is a basis for N K (X) s.t. V U = W 1 Q Σ, C U = W Q Σ 1 since A = V U C U 1, then Q Σ 2 Q T is the SVD of A W. Here Σ jj = σ j, j = 1,..., N and σ 2 1 σ2 > 0 are the singular values N of A W. 21 of 41

33 WSVD Basis Properties This basis is in fact an approximation of the natural one (provided w i > 0, N i=1 w i = Ω ) 22 of 41

34 WSVD Basis Properties This basis is in fact an approximation of the natural one (provided w i > 0, N i=1 w i = Ω ) Properties of the new basis U (cf. [De Marchi-Santin 2013]) T N [u j ](x) = σ j u j (x), 1 j N, x Ω; N K (Ω)-orthonormal; l w 2 (X)-orthogonal, u j 2 l w 2 (X) = σ2 j, u j U; N j=1 σ 2 j = K(0, 0) Ω. 22 of 41

35 WSVD Basis Approximation Interpolant: s f (x) = N j=1 (f, u j) K u j (x) x Ω { } WDLS: s M := argmin f g : g span{u f l w 1,..., u (X) M } 2 Weighted Discrete Least Squares as truncation Let f N K (Ω), 1 M N. Then x Ω s M f (x) = M j=1 (f, u j ) l w 2 (X) (u j, u j ) l w 2 (X) u j (x) = M (f, u j ) l w j=1 σ 2 j 2 (X) u j (x) = M (f, u j ) K u j (x) j=1 23 of 41

36 WSVD Basis Approximation Interpolant: s f (x) = N j=1 (f, u j) K u j (x) x Ω { } WDLS: s M := argmin f g : g span{u f l w 1,..., u (X) M } 2 Weighted Discrete Least Squares as truncation Let f N K (Ω), 1 M N. Then x Ω s M f (x) = M j=1 (f, u j ) l w 2 (X) (u j, u j ) l w 2 (X) u j (x) = M (f, u j ) l w j=1 σ 2 j 2 (X) u j (x) = M (f, u j ) K u j (x) j=1 Q3: Can we extract U U s.t. s f is as good as s f? Yes we can! Take U = {u 1,..., u M }. 23 of 41

37 WSVD Basis Approximation II If we define the pseudo-cardinal functions as l i = s M, we get l i N M u s M j (x i ) (x) = f(x f i ) l i (x), l i (x) = u j (x). i=1 j=1 Generalized Power Function and Lebesgue constant If f N K (Ω), f(x) s M (x) f P (M) (x) f x Ω, where K,X N K (Ω) [ (M) P (x)] 2 M K,X = K(0, 0) [u j (x)] 2, (generalized PF) j=1 s M f Λ X f X, σ 2 j where Λ X = max x Ω N l i (x) is the pseudo-lebesgue constant. i=1 24 of 41

38 WSVD Basis Sub-basis How can we extract U U s.t. s f is as good as s f? ε = 9 ε = 4 ε = recall that u j l w 2 (X) = σ 2 j 0 we can choose M s.t. σ 2 M+1 < τ we don t need u j, j > M Figure: Gaussian kernel, σ 2 j points of the square on equispaced 25 of 41

39 WSVD Basis Example ε = 1 ε = 4 ε = 9 Gaussian IMQ Matern Table: Optimal M for different kernels and shape parameter that correspond to the indexes such that the weighted least-squares approximant s M provides the best approximation of the function f f(x, y) = cos(20(x + y)) on the disk with center C = (1/2, 1/2) and radius R = 1/2 26 of 41

40 WSVD Basis Example, cont Figure: Franke s test function, lens, IMQ Kernel, ε = 1 and RMSE. Left: complete basis. Right: σ 2 M+1 < of 41

41 WSVD Basis Example, cont Figure: Franke s test function, lens, IMQ Kernel, ε = 1 and RMSE. Left: complete basis. Right: σ 2 M+1 < Problem: We have to compute the whole basis before truncation! Solution: Krylov methods, approximate SVD [Novati, Russo 2013] 27 of 41

42 New fast basis [De Marchi-Santin, BIT15]: a comparison N M RMSE new RMSE WSVD Time new Time WSVD Table: Comparison of the WSVD basis and the new basis. Computational time in seconds and corresponding RMSE for the example consisting of interpolation of a function sum of gaussians by GRBF, restricted to N = 15 2, 23 2, 31 2, 39 2 equally spaced points on [ 1, 1] 2. Notice: for each N we computed the optimal M using the new algorithm with tolerance τ = of 41

43 Rescaled, Rational RBF and some applications 29 of 41

44 Rescaled RBF Interpolation Classical interpolant: P f (x) = N α k K(x, x k ), x Ω, x k X. k=1 [Hardy and Gofert 1975] used multiquadrics K(x, y) = 1 + ɛ 2 x y of 41

45 Rescaled RBF Interpolation Classical interpolant: P f (x) = N α k K(x, x k ), x Ω, x k X. k=1 [Hardy and Gofert 1975] used multiquadrics K(x, y) = 1 + ɛ 2 x y 2. Rescaled interpolant: ˆPf (x) = P N f(x) P g (x) = k=1 α k K(x, x k ) N k=1 β k K(x, x k ) where P g is the kernel interpolant of g(x) = 1, x Ω. Obs: Rescaled Localized RBF (RL-RBF) based on CSRBF introduced in [Deparis et al, SISC 2014]: they are smoother even for small radii of the support In [DeM et al 2017] it is shown that it is a Shepard s PU method: exacteness on constants. Linear convergence of localized rescaled interpolants is still an open problem [DeM and Wendland, draft 2017]. 30 of 41

46 Example Take, f(x) = x on [0, 1] by using W2 at the points set X = {0, 1/6, 1/3, 1/2, 2/3, 5/6, 1}, ε = 5 (ɛ = 1/r). Figure: (left) interpolants and (right) the abs error For more results see [Deparis et al 14, Idda s master thesis 2015]. 31 of 41

47 RBF- PU interpolation Ω = s j=1 Ω i (can overlap): that is, each x Ω belongs to a limited number of subdomains, say s 0 < s. Then we consider, W j non-negative functions on Ω j, s.t. s j=1 W j = 1 is a partition of unity. A possibile choice is W j (x) = W j (x) s W, j = 1,..., s (11) k=1 k (x) where Wj are compactly supported functions on Ω j. RBF-PU interpolant s I(x) = R j (x)w j (x), x Ω (12) j=1 N j where R j is a RBF interpolant on Ω j i.e. R j (x) = α j K(x, i xj), i N j = Ω j x j k X N j, k = 1,..., N j. 32 of 41 i=1

48 Rescaling gives a Shepard method We start by writing the interpolant of a function f N K using the N cardinals u j (x i ) = δ i,j, P f = f(x j )u j, so that for g 1 we get N P g = u j. j=1 j=1 33 of 41

49 Rescaling gives a Shepard method We start by writing the interpolant of a function f N K using the N cardinals u j (x i ) = δ i,j, P f = f(x j )u j, so that for g 1 we get j=1 N P g = u j. The rescaled interpolant is then j=1 ˆP f = N j=1 f(x j)u j N k=1 u k = N u j f(x j ) N k=1 u k j=1 N =: f(x j )û j, j=1 where we introduced the (new) cardinal functions û j := u j N k=1 u. k 33 of 41

50 Rescaling gives a Shepard method We start by writing the interpolant of a function f N K using the N cardinals u j (x i ) = δ i,j, P f = f(x j )u j, so that for g 1 we get j=1 N P g = u j. The rescaled interpolant is then j=1 ˆP f = N j=1 f(x j)u j N k=1 u k = N u j f(x j ) N k=1 u k j=1 N =: f(x j )û j, j=1 where we introduced the (new) cardinal functions û j := Corollary u j N k=1 u. k The rescaled interpolation method is a Shepard-PU method, where the weight functions are defined as û j = u j / ( N k=1 u k ), {uj } j being the cardinal basis of span{k(, x), x X}. 33 of 41

51 Variably Scaled Kernels (VSK) Definition Let ψ : R d (0, ) be a given scale function. A Variably Scaled Kernel (VSK) K ψ on R d is K ψ (x, y) := K((x, ψ(x)), (y, ψ(y))), x, y R d. where K is a kernel on R d of 41

52 Variably Scaled Kernels (VSK) Definition Let ψ : R d (0, ) be a given scale function. A Variably Scaled Kernel (VSK) K ψ on R d is K ψ (x, y) := K((x, ψ(x)), (y, ψ(y))), x, y R d. where K is a kernel on R d+1. Given X and the scale functions ψ j : R d (0, ), j = 1,..., s, the RVSK-PU is s I ψ (x) = R ψj (x) W j (x), x Ω, (13) j=1 with R ψj (x) the RVSK-PU on Ω j. 34 of 41

53 Variably Scaled Kernels (VSK) Definition Let ψ : R d (0, ) be a given scale function. A Variably Scaled Kernel (VSK) K ψ on R d is K ψ (x, y) := K((x, ψ(x)), (y, ψ(y))), x, y R d. where K is a kernel on R d+1. Given X and the scale functions ψ j : R d (0, ), j = 1,..., s, the RVSK-PU is s I ψ (x) = R ψj (x) W j (x), x Ω, (13) j=1 with R ψj (x) the RVSK-PU on Ω j. Obs: (if Φ is radial) (A ψj ) ik = Φ( x j i x j k 2 + (ψ j (x j i ) ψ j(x j k ))2 ), i, k = 1,..., N j. 34 of 41

54 Rational RBF (RRBF) definition R(x) = R(1) (x) N R (2) (x) = k=1 α k K(x, x k ) N k=1 β k K(x, x k ) [Jackbsson et al. 2009, Sarra and Bai 2017] = RRBFs well approximate data with steep gradients or discontinuites [rational with PU+VSK in DeM et al. 2017]. 35 of 41

55 Rational RBF Learning from rational functions, d = 1 polynomial case. r(x) = p 1(x) p 2 (x) = a mx m + + a 0 x 0 x n + b n 1 x n 1 + b 0. M = m + n + 1 unknowns (Padé approximation). If M < N to get the coefficients we may solve the LS problem N min f(x k ) r(x k ) 2. p 1 Π 1 m,p 2 Π 1 n k=1 36 of 41

56 Rational RBF Learning from rational functions, d = 1 polynomial case. r(x) = p 1(x) p 2 (x) = a mx m + + a 0 x 0 x n + b n 1 x n 1 + b 0. M = m + n + 1 unknowns (Padé approximation). If M < N to get the coefficients we may solve the LS problem N min f(x k ) r(x k ) 2. p 1 Π 1 m,p 2 Π 1 n k=1 RBF case. Let X m = {x k,..., x k+m 1 }, X n = {x j,..., x j+n 1 } X be non empty, such that m + n N k+m 1 R(x) = R(1) (x) R (2) (x) = i 1 α =k i1 K(x, x i1 ) j+n 1 i 2 β =j i2 K(x, x i2 ), (14) provided R (2) (x) 0, for all x Ω. 36 of 41

57 Rational RBF Find the coefficients [Jackbsson 2009] show that this is equivalent to solve the following generalized eigenvalue problem Problem 3 with Σq = λθq, Σ = 1 D T A 1 D + A 1, and Θ = 1 D T D + I f 2 f 2 N, 2 2 where I N is the identity matrix. 37 of 41

58 Rational RBF Find the coefficients [Jackbsson 2009] show that this is equivalent to solve the following generalized eigenvalue problem Problem 3 with Σq = λθq, Σ = 1 D T A 1 D + A 1, and Θ = 1 D T D + I f 2 f 2 N, 2 2 where I N is the identity matrix. q is the eigenvector associated to the smallest eigenvalue! 37 of 41

59 New class of rational RBF [Buhmann, DeM, Perracchione 2018] ˆP f (x) = N i=1 α ik(x, x i ) + L m=1 γ mp m (x) N k=1 β K(x, := P g(x) k x k ) P h (x) (15) Ratio of a CPD K of order l and an associate PD K... = two kernel matrices, Φ K and Φ K. 38 of 41

60 New class of rational RBF [Buhmann, DeM, Perracchione 2018] ˆP f (x) = N i=1 α ik(x, x i ) + L m=1 γ mp m (x) N k=1 β K(x, := P g(x) k x k ) P h (x) (15) Ratio of a CPD K of order l and an associate PD K... = two kernel matrices, Φ K and Φ K. Obs: 1 Once we know the function values P h (x i ) = h i, i = 1,..., N, we can construct P g, i.e. it interpolates g = (f 1 h 1,..., f N h N ) T. Hence ˆPf interpolates the given function values f at the nodes X N. 2 If K is PD, we fix K = K so that we deal with the same kernel matrix for both numerator and denominator. 3 We can prove well-posedness, find cardinal functions and give stability analysis. 38 of 41

61 Lectures Notes on RBF 1 Learning from splines 2 Positive definite functions Lectures on Radial Basis Functions by Stefano De Marchi and Emma Perracchione Department of Mathematics Tullio Levi-Civita University of Padua (Italy) February 16, Conditionally positive definite functions 4 Error estimates 5 Stable bases for RBFs 6 Rational RBFs 7 The Partition of unity method 8 Collocation method via RBF 9 Financial applications 10 Exercises 39 of 41

62 Other references R. Cavoretto, S. De Marchi, A. De Rossi, E. Perracchione, G. Santin, Partition of unity interpolation using stable kernel-based techniques, Appl. Numer. Math. 116 (2017), pp R. Cavoretto, A. De Rossi, E. Perracchione, Optimal selection of local approximants in RBF-PU interpolation, to appear on J. Sci. Comput. (2017). S. De Marchi, G. Santin, Fast computation of orthonormal basis for RBF spaces through Krylov space methods, BIT 55 (2015), pp Stefano De Marchi, Andrea Idda and Gabriele Santin: A rescaled method for RBF approximation. Springer Proceedings on Mathematics and Statistics, Vol. 201, pp (2017). S. De Marchi, E. Perracchione, A. Martinez Calomardo and M. Rossini: RBF-based partition of unity method for elliptic PDEs: adaptivity and stability issues via VSKs, submitted to J. Sci. Comput S. De Marchi, E. Perracchione, A. Martinez Calomardo : Rational Radial Basis Functions interpolation, submitted to J. Comput. Appl. Math M. Buhmann, S. De Marchi, E. Perracchione : Analysis of a new class of rational RBF expansions, submitted to IMA Num. Analysis, E. Perracchione: Rational RBF-based partition of unity method for efficiently and accurately approximating 3D objects arxiv preprint arxiv: , To appear in Comput. Appl. Math. 40 of 41

63 grazie per la vostra attenzione! thanks for your attention! and grazie a R.IT.A. 41 of 41