Quasi-interpolation for high-dimensional function approximation
The paper proposes a general quasi-interpolation scheme for high-dimensional function approximation. To facilitate error analysis, we view our quasi-interpolation as a two-step procedure. In the first step, we approximate a target function by a purpose-built convolution operator (with an error term...
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| Published in | Numerische Mathematik Vol. 156; no. 5; pp. 1855 - 1885 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.10.2024
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0029-599X 0945-3245 |
| DOI | 10.1007/s00211-024-01435-6 |
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| Abstract | The paper proposes a general quasi-interpolation scheme for high-dimensional function approximation. To facilitate error analysis, we view our quasi-interpolation as a two-step procedure. In the first step, we approximate a target function by a purpose-built convolution operator (with an error term referred to as convolution error). In the second step, we discretize the underlying convolution operator using certain quadrature rule at the given sampling data sites (with an error term called discretization error). The final approximation error is obtained as an optimally balanced sum of these two errors, which in turn views our quasi-interpolation as a regularization technique that balances convolution error and discretization error. As a concrete example, we construct a sparse grid quasi-interpolation scheme for high-dimensional function approximation. Both theoretical analysis and numerical implementations provide evidence that our quasi-interpolation scheme is robust and is capable of mitigating the curse of dimensionality for approximating high-dimensional functions. |
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| AbstractList | The paper proposes a general quasi-interpolation scheme for high-dimensional function approximation. To facilitate error analysis, we view our quasi-interpolation as a two-step procedure. In the first step, we approximate a target function by a purpose-built convolution operator (with an error term referred to as convolution error). In the second step, we discretize the underlying convolution operator using certain quadrature rule at the given sampling data sites (with an error term called discretization error). The final approximation error is obtained as an optimally balanced sum of these two errors, which in turn views our quasi-interpolation as a regularization technique that balances convolution error and discretization error. As a concrete example, we construct a sparse grid quasi-interpolation scheme for high-dimensional function approximation. Both theoretical analysis and numerical implementations provide evidence that our quasi-interpolation scheme is robust and is capable of mitigating the curse of dimensionality for approximating high-dimensional functions. |
| Author | Gao, Wenwu Sun, Zhengjie Fasshauer, Gregory E. Wang, Jiecheng |
| Author_xml | – sequence: 1 givenname: Wenwu surname: Gao fullname: Gao, Wenwu organization: School of Big Data and Statistics, Anhui University – sequence: 2 givenname: Jiecheng surname: Wang fullname: Wang, Jiecheng organization: School of Economic, Management and Law, University of South China – sequence: 3 givenname: Zhengjie surname: Sun fullname: Sun, Zhengjie email: sunzhengjie1218@163.com organization: School of Mathematics and Statistics, Nanjing University of Science and Technology – sequence: 4 givenname: Gregory E. surname: Fasshauer fullname: Fasshauer, Gregory E. organization: Department of Applied Mathematics and Statistics, Colorado School of Mines |
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| References_xml | – reference: BeatsonRPowellMUnivariate multiquadric approximation: quasi-interpolation to scattered dataConstr. Approx.199282752881164070 – reference: LemieuxCMonte Carlo and Quasi-Monte Carlo sampling2009New YorkSpringer – reference: PottsDSchmischkeMApproximation of high-dimensional periodic functions with Fourier-based methodsSIAM J. Numer. Anal.202159239324294313847 – reference: GaoWWFasshauerGEFisherNDivergence-free quasi-interpolationAppl. Comput. Harmon. Anal.2022604714884423813 – reference: Garcke, J.: Sparse grid tutorial, pp. 1–26 (2006). https://www.researchgate.net/publication/228357801 – reference: GaoWWFasshauerGESunXPZhouXOptimality and regularization properties of quasi-interpolation: deterministic and stochastic approachesSIAM J. Numer. Anal.202058205920784119331 – reference: KolomoitsevYLomakoTTikhonovSSparse grid approximation in weighted Wiener spacesJ. Four. Anal. Appl.20232919514552583 – reference: Tan, P.N., Steinbach, M., Vipin,K.: Introduction to Data Mining. Pearson Addison-Wesley (2006) – reference: BuhmannMDaiFPointwise approximation with quasi-interpolation by radial basis functionsJ. Approx. Theroy20151921561923313479 – reference: BellmannRAdaptive Control Processes: A Guided Tour1961New YorkPrinceton Press – reference: SunZGaoWWYangRA convergent iterated quasi-interpolation for periodic domain and its applications to surface PDEsJ. Sci. Comput.2022932374483528 – reference: SloanIHJoeSLattice Methods for Multiple Integration1994New YorkOxford University Press – reference: DũngDSampling and cubature on sparse grids based on a B-spline quasi-interpolationFound. Comput. Math.201616119312403552844 – reference: BackusGGilbertFThe resolving power of gross earth dataGeophys. J. R. Astr. Soc.196816169205 – reference: DũngDB-spline quasi-interpolant representations and sampling recovery of functions with mixed smoothnessJ. Complexity2011275415672846705 – reference: SpeleersHManniCEffortless quasi-interpolation in hierarchical spacesNumer. Math.20161321551843439218 – reference: BuhmannMOn quasi-interpolation with radial basis functionsJ. Approx. Theory1993721031301198376 – reference: KuoFYSchwabChSloanIHQuasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficientSIAM J. Numer. Anal.201250335133743024159 – reference: HennigPOsborneMAGirolamiMProbabilistic numerics and uncertainty in computationsProc. Roy. Soc.2015472179 – reference: UstaFLevesleyJMultilevel quasi-interpolation on a sparse grid with the GaussianNumer. Algorithms2018177938083766595 – reference: GrohsPQuasi-interpolation in Riemannian manifoldsIMA J. Numer. Anal.2013333849874308148610.1093/imanum/drs026 – reference: PottsDVolkmerTSparse high-dimensional FFT based on rank-1 lattice samplingAppl. Comput. Harmon. 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| SubjectTerms | Approximation Convolution Dimensional analysis Discretization Error analysis Interpolation Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Numerical Analysis Numerical and Computational Physics Operators (mathematics) Quadratures Regularization Simulation Theoretical |
| Title | Quasi-interpolation for high-dimensional function approximation |
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