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Maximum Likelihood Estimation and Uncertainty Quantification for Gaussian Process approximation of Deterministic Functions

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SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION 2020年第3期8卷 926-958页

作者： Karvonen, Toni Wynne, George Tronarp, Filip Oates, Chris Sarkka, Simo Aalto Univ Dept Elect Engn & Automat Espoo Finland Alan Turing Inst London NW1 2DB England Imperial Coll London Dept Math London SW7 2AZ England Univ Tubingen Tubingen Germany Newcastle Univ Sch Math Stat & Phys Newcastle Upon Tyne NE1 7RU NE England

Despite the ubiquity of the Gaussian process regression model, few theoretical results are available that account for the fact that parameters of the covariance kernel typically need to be estimated from the data set. This article provides one of the first theoretical analyses in the context of Gaussian process regression with a noiseless data set. Specifically, we consider the scenario where the scale parameter of a Sobolev kernel (such as a Matern kernel) is estimated by maximum likelihood. We show that the maximum likelihood estimation of the scale parameter alone provides significant adaptation against misspecification of the Gaussian process model in the sense that the model can become "slowly" overconfident at worst, regardless of the difference between the smoothness of the data-generating function and that expected by the model. The analysis is based on a combination of techniques from nonparametric regression and scattered data interpolation. Empirical results are provided in support of the theoretical findings.

关键词： nonparametric regression scattered data approximation credible sets Bayesian cubature model misspecification

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Kernel based approximation in Sobolev spaces with radial basis functions

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APPLIED MATHEMATICS AND COMPUTATION 2009年第6期215卷 2229-2237页

作者： Ma, Limin Wu, Zongmin Zhejiang Gongshang Univ Dept Informat & Comp Sci Hangzhou 310018 Zhejiang Peoples R China Fudan Univ Shanghai Key Lab Contemporary Appl Math Sch Math Sci Shanghai 200433 Peoples R China

In this paper, we study several radial basis function approximation schemes in Sobolev spaces. We obtain an optional error estimate by using a class of smoothing operators. We also discussed sufficient conditions for the smoothing operators to attain the desired approximation order. We then construct the smoothing operators by some compactly supported radial kernels, and use them to approximate Sobolev space functions with optimal convergence order. These kernels can be simply constructed and readily applied to practical problems. The results show that the approximation power depends on the precision of the sampling instrument and the density of the available data. (C) 2009 Published by Elsevier Inc.

关键词： scattered data approximation Smoothing by convolution Reproducing kernel Hilbert space Radial basis function

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An efficient ADI-solver for scattered data problems with global smoothing

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JOURNAL OF COMPUTATIONAL PHYSICS 1998年第2期139卷 343-358页

作者： Arge, E Kunoth, A Rhein Westfal TH Aachen Inst Geometrie & Prakt Math D-52056 Aachen Germany Numer Objects AS N-0314 Oslo Norway

For the approximate representation of large data sets over a parameter domain in R-2, many geological and other applications require the construction of surfaces which have minimal energy, i.e., minimal curvature. One way to achieve this is by solving a fourth-order elliptic partial differential equation. Its discretization by a difference scheme makes it particularly easy to incorporate (appropriate approximations of) known data points. In this paper, we investigate the performance of different solution methods for the resulting symmetric Linear system of equations since this is the most CPU-demanding step in the scattered data approximation procedure. Specifically, we test first the performance of a preconditioned conjugate gradient method with an SSOR and an RILU preconditioner. However, since the partial differential operator does not contain mixed derivatives, using an alternating-direction-implicit scheme (ADI method) which has been employed successfully in the past for second-order problems, together with a Cholesky factorization of the corresponding one-dimensional operators has also been tried for the problem at hand. The computational studies that we have performed here show that for our problem an instationary ADI method is superior to the above-mentioned preconditioned conjugate gradient solvers both with respect to work load and accuracy of the solution. For the fourth-order model problem considered in this paper, the instationary ADI method with Wachspress parameters results in a number of iterations that is essentially independent of the number of variables. (C) 1998 Academic Press.

关键词： scattered data approximation fourth-order elliptic problems difference methods preconditioned conjugate gradient methods ADI methods

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Robust approximation Algorithms for the Detection of Attraction Basins in Dynamical Systems

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JOURNAL OF SCIENTIFIC COMPUTING 2016年第1期68卷 395-415页

作者： Cavoretto, Roberto De Rossi, Alessandra Perracchione, Emma Venturino, Ezio Univ Turin Dept Math G Peano Via Carlo Alberto 10 I-10123 Turin Italy

A particular solution of a dynamical system is completely determined by its initial condition. When the omega limit set reduces to a point, the solution settles at steady state. The possible steady states of the system are completely determined by its parameters. However, with the same parameter set, it is possible that several steady states can originate from different initial conditions (multi-stability). In that case the outcome depends on the chosen initial condition. Therefore, it is important to assess the domain of attraction for each possible attractor. The algorithms presented here are general and robust enough so as to solve the problem of reconstructing the basin of attraction of each stable equilibrium point. In order to have a graphical representation of the separatrix manifolds, we focus on systems of two and three ordinary differential equations exhibiting bi- or tri-stability. For this purpose we have implemented several Matlab functions for the approximation of the points lying on the curves or on the surfaces determining the basins of attraction and for the reconstruction of such curves and surfaces. We approximate the latter with the implicit partition of unity method using radial basis functions as local approximants. Numerical results, obtained with a Matlab package made available to the scientific community, support our findings.

关键词： scattered data approximation Partition of unity method Radial basis functions Dynamical systems Competition population models Basins of attraction

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Error saturation in Gaussian radial basis functions on a finite interval

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2010年第5期234卷 1435-1441页

作者： Boyd, John P. Univ Michigan Dept Atmospher Ocean & Space Sci Ann Arbor MI 48109 USA

Radial basis function (RBF) interpolation is a "meshless" strategy with great promise for adaptive approximation. One restriction is "error saturation" which occurs for many types of RBFs including Gaussian RBFs of the form phi(x;alpha, h) = exp(alpha(2)(x/h)(2)): in the limit h -> 0 for fixed alpha, the error does not converge to zero, but rather to epsilon(S)(alpha). Previous studies have theoretically determined the saturation error for Gaussian RBF on an infinite, uniform interval and for the same with a single point omitted. (The gap enormously increases epsilon(S) (alpha).) We show experimentally that the saturation error on the unit interval, X is an element of[-1, 1] is about 0.06 exp(-0.47/alpha(2))vertical bar vertical bar f vertical bar vertical bar infinity - huge compared to the 0(2 pi/alpha(2)) exp (-pi/[4 alpha(2)]) saturation error for a grid with one point omitted. We show that the reason the saturation is so large on a finite interval is that it is equivalent to an infinite grid which is uniform except for a gap of many points. The saturation error can be avoided by choosing alpha << 1, the "flat limit", but the condition number of the interpolation matrix explodes as O(exp(pi(2)/[4 alpha(2)])). The best strategy is to choose the largest a which yields an acceptably small saturation error: If the user chooses an error tolerance delta, then alpha(optimum)(delta) = 1/root-2 log(delta/0.06). (C) 2010 Elsevier By. All rights reserved.

关键词： Radial basis functions Interpolation scattered data approximation

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Reverse engineering with subdivision surfaces

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COMPUTING 2007年第2-4期79卷 119-129页

作者： Keller, P. Bertram, M. Hagen, H. TU Kaiserslautern FB Informat D-67653 Kaiserslautern Germany

Reverse engineering is concerned with the reconstruction of surfaces from three-dimensional point clouds originating from laser-scanned objects. We present an adaptive surface reconstruction method providing a hierarchy of quadrilateral meshes adapting surface topology when a mesh is refined. This way, a user can choose a model with proper resolution and topology from the hierarchy without having to run the algorithm multiple times with different parameters. The multiresolution mesh representation can be used subsequently for view-dependent rendering and wavelet compression.

关键词： 3D surface reconstruction surface fitting shape recovery scattered data approximation

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approximation to the k-th derivatives by multiquadric quasi-interpolation method

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2009年第2期231卷 925-932页

作者： Ma, Limin Wu, Zongmin Fudan Univ Sch Math Sci Shanghai Key Lab Contemporary Appl Math Shanghai 200433 Peoples R China

Quasi-interpolation is very useful in the study of approximation theory and its applications, since it can yield solutions directly without the need to solve any linear system of equations. Based on the good performance (simple to evaluate) of multiquadric functions, the advantages (approximation order, convexity and monotonlcity preserving) of multiquadric quasi-interpolation have been widely discussed. However, it is usually only for the approximation properties of the function itself and the good properties for derivatives, whereas the high order derivatives have been largely ignored. In this paper, we go further into the approximation properties to the k-th derivatives by using multiquadric quasi-interpolation. Furthermore, we develop two kinds of multiquadric quasi-interpolation schemes on bounded interval [x(0). x(N)], whose derivatives converge to the corresponding derivatives of the approximated functions. Finally, the numerical experiments are presented to confirm the accuracy of the presented scheme. Both theoretical results and numerical examples show this scheme provides good accuracy even if the data points in [x(0), x(N)] are irregularly distributed. (C) 2009 Elsevier B.V. All rights reserved.

关键词： Quasi-interpolation Hardy's multiquadric scattered data approximation Radial basis functions (RBFs) approximation to derivatives

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Local polynomial reproduction and moving least squares approximation

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IMA JOURNAL OF NUMERICAL ANALYSIS 2001年第1期21卷 285-300页

作者： Wendland, H Univ Gottingen Inst Numer & Angew Math D-37083 Gottingen Germany

Local polynomial reproduction is a key ingredient in providing error estimates for several approximation methods. To bound the Lebesgue constants is a hard task especially in a multivariate setting. We provide a result which allows us to bound the Lebesgue constants uniformly and independently of the space dimension by oversampling. We get explicit and small bounds for the Lebesgue constants. Moreover, we use these results to establish error estimates for the moving least squares approximation scheme, also with special emphasis on the involved constants. We discuss the numerical treatment of the method and analyse its effort. Finally, we give large scale examples.

关键词： scattered data approximation approximation orders

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Least squares surface approximation using multiquadrics and parametric domain distortion

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COMPUTER AIDED GEOMETRIC DESIGN 1999年第3期16卷 177-196页

作者： Franke, R Hagen, H USN Postgrad Sch Dept Math Monterey CA 93943 USA Univ Kaiserslautern Fachbereich Informat D-67653 Kaiserslautern Germany

A method for approximating scattered data using multiquadric approximation on a transformed domain region is given. With the imposition of a one-to-one constraint on the domain transformation, ail properties of the approximating space carry over. Such a domain transformation is defined in terms of a biquadratic Bezier representation. The parameters in the transformation, and the location of the base points (or knots) in the multiquadric approximation are simultaneously determined using a nonlinear least squares paradigm. The results of some comparisons with previous approaches are given and discussed. (C) 1999 Elsevier Science B.V. All rights reserved.

关键词： multiquadrics domain distortion scattered data approximation least squares

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Dynamically knots setting in meshless method for solving time dependent propagations equation

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2004年第12-14期193卷 1221-1229页

作者： Wu, ZM Fudan Univ Dept Math Shanghai Key Lab Comtemporary Appl Math Shanghai 200433 Peoples R China

In order to display a function with some finite discrete sampling data efficiently, we require more sampling data where the function is more oscillatory, and less sampling data where the function is more flat. If the function is happen to be a solution of partial differential equation and is solved by finite elements method, then we should construct finer element near the singularity. However, we do not know where the oscillation or even shocks will happen to a function, which is a solution of non-linear partial differential equation. Therefore we cannot preset the finer elements near such points. A trivial method is taking very dense knots or very fine elements everywhere to keep the accuracy. This strategy cost the computation time. Another idea is to move the position of the sampling points according to the varying of the function with the time. We observed that, this approach could be achieved by Lagrangian formulation for finite difference method. However the Lagrangian formulation does not possess shape preserving and variation diminishing properties. It is difficult to achieve the approach for finite elements methods too, because the moving knots will destroy the topology of the mesh. Recently the meshless method becomes to topic to solve partial differential equation numerically. The meshless method does not require any mesh or any structure of the knots (sampling points);therefore we can move the knots freely to simulate the problem. The only restriction is to keep the knots no overlapping. This paper is a test of our approach for the Burger's equation with radial basis quasi-interpolation. (C) 2004 Elsevier B.V. All rights reserved.

关键词： moving knots scattered data approximation numerical solution of PDE meshless method Quasi-interpolation

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