Greedy Adaptive Local Recovery of Functions in Sobolev Spaces, 2024, arxiv pdf
Geometric Analysis of Vision Cues for the Flare, 2021, pdf
Modelling Recovered Cases and Death Probabilities for the COVID-19
Outbreak, March 26th, 2020, pdf
arxiv
update of table and figures using data up to April 7: pdf
A Greedy Method for Solving Classes of PDE problems, 2019, pdf
arxiv
The Mathematics of Fuel-to-Distance-Optimal Flight, 2017, arxiv
later
MATLAB Programming for Kernel-Based Methods, technical report 2011, still
not fully revised, pdf
A strongly improved version of the software, by Oleg Davydov, Gießen, is here.
The Matérn Model: A Journey through Statistics, Numerical Analysis and
Machine Learning,
(with Emilio Porcu, Moreno Bevilacqua, and Chris J. Oates),
Statistical Science 39 (2024), 469-492, https://dx.doi.org/10.1214/24-STS923,
arxiv, pdf,
Introducing the evaluation condition number: A novel assessment of
conditioning in radial basis function methods (with A. Noorizadegan),
Engineering Analysis with Boundary Elements 166, 2024, 105827, https://doi.org/10.1016/j.enganabound.2024.105827,
early version:
https://dx.doi.org/10.2139/ssrn.4776821
Scaling of Radial Basis Functions (with Elisabeth Larsson, Uppsala),
IMA Journal of Numerical Analysis 44 (2024), 1130-1152, https://doi.org/10.1093/imanum/drad035,
pdf, arxiv
H-Sets for Kernel-Based Spaces, https://doi.org/10.1016/j.jat.2023.105942,
Journal of Approximation Theory 294 (2023), Article 105942, arxiv,
pdf
Small Errors imply Large Evaluation Instabilities, https://doi.org/10.1007/s10444-023-10026-2,
Advances in Computational Mathematics 49, 25 (2023), arxiv,
pdf
An Approximation Theorist's View on Solving Operator Equations, - with
special attention to Trefftz, MFS, MPS, and DRM methods - ,
https://doi.org/10.1016/j.camwa.2019.09.005,
Computers and Mathematics with Applications 88 (2021) 70–77 arxiv
pdf
A Meshfree Method for Solving the Monge–Ampère Equation (with Klaus Böhmer, Marburg),
Convergence Analysis of General Spectral Methods (with Maryam Mohammadi,
Kharazmi University, Tehran, Iran),
Journal of Computational and Applied Mathematics 313, 2017, 284–293, https://doi.org/10.1016/j.cam.2016.09.031,
pdf
Approximation of Eigenfunctions in Kernel-based Spaces (with Gabriele
Santin, Padova),
Advances in Computational Mathematics 42, 2016,
973–993, https://doi.org/10.1007/s10444-015-9449-5,
pdf
All Well--Posed Problems have Uniformly Stable and Convergent
Discretizations,
Numerische Mathematik 132, 2016, 597-630 https://doi.org/10.1007/s00211-015-0731-8,
pdf
Error Bounds for Kernel-Based Numerical Differentiation (with Oleg
Davydov, Univ. Gießen),
Numerische Mathematik 132, 2016, 243-269, https://doi.org/10.1007/s00211-015-0722-9,
pdf
Multivariate Approximation, in: Encyclopedia of Applied and Computational
Mathematics,
ISBN 978-3-540-70528-4, 2015, https://doi.org/10.1007/978-3-540-70529-1_370,
Springer-Verlag Berlin Heidelberg pdf
Shift-Invariant Approximation, in: Encyclopedia of Applied and
Computational Mathematics,
ISBN 978-3-540-70528-4, 2015, https://doi.org/10.1007/978-3-540-70529-1_334,
Springer-Verlag Berlin Heidelberg pdf
Radial Kernels via Scale Derivatives (with M. Bozzini, M. Rossini and E.
Volonte, Univ. Milano-Biccocca),
Adv. in Comp. Math. 41 (2015) 277-291, https://doi.org/10.1007/s10444-014-9366-z,
pdf
A meshless method for solving the 2D Brusselator reaction-diffusion
system
(with M. Mohammadi und R. Mokhtari, Isfahan University of Technology), https://doi.org/10.3970/cmes.2014.101.113,
Computer Modeling in Engineering & Sciences, Vol. 101, No. 2, pp.
113-138, 2014 pdf
The Meshless Kernel-Based Method of Lines for Parabolic Equations, with
Y.C. Hon, HongKong City Univ. and Min Zhong, Fudan U. Shanghai,
Computers and Mathematics with Applications 68 (2014), pp. 2057-2067, https://doi.org/10.1016/j.camwa.2014.09.015,
pdf
Greedy Sparse Linear Approximations of Functionals from Nodal Data, https://doi.org/10.1007/s11075-013-9806-8,
Numerical Algorithms: Volume 67, Issue 3 (2014), Page 531-547, (with
examples not in the final version: pdf)
Solving Heat Conduction Problems by the Direct Meshless Local
Petrov-Galerkin (DMLPG) method (with D. Mirzaei, Univ. of Isfahan),
Numerical Algorithms 65 no. 2, 275-291, 2014, https:/doi.org/10.1007/s11075-013-9711-1,
pdf
Interpolation of spatial data - a stochastic or a deterministic problem?,
with Michael Scheuerer and Martin Schlather,
European Journal of Applied Mathematics 24 Nr. 4, 2013, pp. 601-629, https://doi.org/10.1017/S0956792513000016,
pdf
Generalized Whittle--Matern and Polyharmonic Kernels, with Mira
Bozzini and Milvia Rossini, Univ. Milano-Biccocca,
Advances in Computational Mathematics: Volume 39, Issue 1 (2013), Page
129-141, https://doi.org/10.1007/s10444-012-9277-9,
online
version old pdf
Direct Discretizations with Applications to Meshless Methods for PDEs,
Dolomites Research Notes on Approximation,
Proceedings of DWCAA12, Volume 6, 2013, p. 37-51, https://doi.org/10.14658/PUPJ-DRNA-2013-Special_Issue-6,
open
access pdf, local pdf
Publications of Helmut Werner. A commented list, for the website "History of Approximation Theory". pdf
Solving the 3D Laplace Equation by Meshless Collocation via Harmonic Kernels (with Y.C. Hon, Hong Kong City University), The Missing Wendland Functions, Advances in Computational Mathematics,
43 (2010), 67-81,
https://doi.org/10.1007/s10444-009-9142-7,
pdf Maple
A Posteriori Error Estimates for Meshless Methods,
6-page abstract for ECCOMAS 2007, pdf
Why Does MLPG Work?
Accepted paper for a keynote lecture in ICCES 07, 2006, https://doi.org/10.3970/icces.2007.003.081,
pdf
Results on Meshless Collocation Techniques (with L. Ling and R. Opfer),
Engineering Analysis with Boundary Elements, Volume 30, Issue 4, 2006,
247-253,
https://doi.org/10.1016/j.enganabound.2005.08.008,
pdf
Kernel-B-Splines and Interpolation (with M. Bozzini and L.
Lenarduzzi) later pdf older pdf,
Numerical Algorithms 41 (2006) p. 1-16, https://doi.org/10.1007/s11075-005-9000-8
Multivariate Interpolation by Polynomials and Radial Basis
Functions,
Constructive Approximation 21, p. 293-317, 2005, https://doi.org/10.1007/s00365-004-0585-2,
pdf file
Near-optimal Data-independent Point Locations for Radial Basis Function
Interpolation (with St. de Marchi and H. Wendland),
Adv. in Comp. Math. 23, 317- 330, 2005, https://doi.org/10.1007/s10444-004-1829-1,
pdf
On Adaptive Unsymmetric Meshless Collocation (with L. Ling),
Proceedings of the 2004 International Conference on Computational &
Experimental Engineering and Sciences (Forsyth, USA) (S.N. Atluri and
A.J.B. Tadeu, eds.),
CD-ROM, Advances in Computational & Experimental Engineering &
Sciences, Tech Science Press, 2004, paper # 270 pdf
Interpolation by Basis Functions of Different Scales and Shapes (with M.
Bozzini, L. Lenarduzzi, M. Rossini),
CALCOLO 41, 77-87 (2004), https://doi.org/10.1007/BF02637256,
pdf
On the Versatility of Meshless Kernel Methods,
Advances in Computational and Experimental Engineering and Sciences,
eds. S.N. Atluri, D.E. Beskos, D. Polyzos, CD ROM, ICCES proceedings paper
#428, 2003 pdf
An Adaptive Greedy Algorithm for Solving Large RBF Collocation Problems
(with Y.C. Hon and X. Zhou),
Numerical Algorithms 32 (2003), 13-25, https://doi.org/10.1023/A:1022253303343,
pdf
Mathematical Results Concerning Kernel Techniques,
Proceedings of the 13th IFAC Symposium on System Identification, 2003
pdf
Recent Developments in the Dual Reciprocity Method Using Compactly
Supported Radial Basis Functions (with C.S. Chen and M. A. Golberg),
in:
Transformation of Domain Effects to the Boundary, ed. Y.F. Rashed and C.A.
Brebbia,, 2003,
Advances in Boundary Elements 14,WIT Press, Southampton, Boston,
138-225, pdf
Approximation by Positive Definite Kernels (with H. Wendland), in:
Advanced Problems in Constructive Approximation
(M.D. Buhmann and D.H. Mache, eds.), International Series in Numerical
Mathematics 142 (2002) 203-221,
https://doi.org/10.1007/978-3-0348-7600-1_15,
pdf
Stability of Radial Basis Function Interpolants, Approximation Theory X:
Wavelets, Splines, and Applications (C.K. Chui, L.L. Schumaker, and J.
Stöckler, eds.)
Vanderbilt Univ. Press, Nashville 2002, p. 433-440 pdf
Inverse and saturation theorems for radial basis function interpolation
(with H. Wendlland), Mathematics of Computation 71 (2002), 669-681,
http://www.jstor.org/stable/2698839,
early pdf later pdf
Adaptive Interpolation by Scaled Multiquadrics (with M. Bozzini and
L. Lenarduzzi),
Adv. in Comp. Math.16 (2002), 375-387, https://doi.org/10.1023/A:1014584220418,
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Approximation in Sobolev Spaces by Kernel Expansions (with F.J. Narcowich
and J.D. Ward),
J. of Approx. Th. 114 (2002) 70-83, https://doi.org/10.1006/jath.2001.3637,
pdf
Optimal Stability Results for Interpolation by Kernel Functions (with F.J. Narcowich and J.D. Ward), preprint 2001 pdf
Characterization and Construction of Radial Basis Functions (with H.
Wendland),
in: Multivariate Approximation and Applications (N. Dyn, D. Leviatan, D.
Levin, and A. Pinkus, eds.),
Cambridge University Press, 2001, p.1-24, Cambridge
University Press link, pdf
Adaptive greedy techniques for approximate solution of large rbf
systems (with H. Wendland) ,
Numerical Algorithms 24 (2000), 239-254, https://doi.org/10.1023/A:1019105612985,
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Remarks on Meshless Local Construction of Surfaces, invited survey, The
mathematics of surfaces IX (eds: R. Cipolla et. al. )
Proceedings of the Ninth IMA Conference on the Mathematics of Surfaces ,
Springer, https://doi.org/10.1007/978-1-4471-0495-7_3,
2000, pdf
Numerical Techniques Based on Radial Basis Functions (with H. Wendland),
in: Curve and Surface Fitting: Saint-Malo 1999 (invited lecture) ,
Albert Cohen, Christophe Rabut, and Larry L. Schumaker (eds.),
Vanderbilt University Press, 2000, p. 359-374, publisher
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A Unified Theory of Radial Basis Functions (Native Hilbert Spaces for
Radial Basis Functions II),
J. of Computational and Applied Mathematics 121 (2000) 165--177, https://doi.org/10.1016/S0377-0427(00)00345-9,
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On Unsymmetric Collocation by Radial Basis Functions (with. Y.C. Hon),
Journal of Applied Mathematics and Computation 119, 177-186, 2001, https://doi.org/10.1016/S0096-3003(99)00255-6,
pdf
Using compactly supported radial basis functions to solve partial
differential equations (with H. Wendland),
in Boundary Element Technology XIII (invited lecture) ,
Chen, C. S. and Brebbia, C. A. and Pepper, D. W. (eds), WitPress,
Southampton, Boston, 1999, 311--324, WITpress
link, pdf
Native Hilbert Spaces for Radial Basis Functions I,
International Series of Numerical Mathematics 132 (1999), Birkhäuser
Verlag, 255-282,
---> pdf file (71 Kb)
Multilevel Interpolation and Approximation (with F. Narcowich
& J. Ward, Texas A&M University),
Appl. Comput. Harmonic Anal., 7 (1999) 243-261 pdf
Mathematical model of a LIF-fibre optic sensor head for trace detection
in soil
(with U. Bünting, F. Lewitzka, and P. Karlitschek),
Applied Spectroscopy 53 ,49-56 (1999)
Improved Error Bounds for Scattered Data Interpolation by Radial Basis
Functions,
Math. Comp. 68 (1999), 201-216 ---> pdf
Optimal Geometric Hermite Interpolation of Curves,
in: Mathematical methods for curves and surfaces II,
Innov. Appl. Math. p. 417-428, Vanderbilt U. Press, Nashville TN,
1998 pdf
Convergence Order Estimates of Meshless Collocation Methods using Radial
Basis Functions (with C. Franke),
Advances in Computational Mathematics, Volume 8, Issue 4, June 1998, pp.
381-399 pdf
Grundlagen der Informatik für das Nebenfachstudium, Vieweg, 1998
Solving Partial Differential Equations by Collocation using Radial Basis
Functions (with C. Franke),
Applied Mathematics and Computation, Volume 93, Issue 1, June 1998, pp.
73-82 pdf
Radial Basis Functions Viewed from Cubic Splines,
in: Multivariate Approximation and Splines, G. Nürnberger, J.W. Schmidt,
and G. Walz (eds.),
Birkhäuser 1997, pp. 245-258 ---> pdf
Construction Techniques for Highly Accurate Quasi-Interpolation Operators
(with Z. Wu),
J. of Approx. Th. 91 (1997) pp. 320-331 pdf
Optimal Recovery of Functions in Translation-invariant Spaces of
Functions,
Annals of Numerical Mathematics 4 (1997) pp. 547-555 pdf
On the Efficiency of Interpolation by Radial Basis Functions,
in: A. LeMéhauté, C. Rabut, and L.L Schumaker (eds.):
Proceedings of Chamonix 1996, pp. 309-318 pdf
Multivariate Interpolation and Approximation by Translates of a Basis
Function,
Approximation Theory VIII (invited lecture), C.K. Chui , L.L. Schumaker
(eds),
Vanderbilt University Press 1996, 25 pages, survey
---> pdf
file (85 Kb)
Operators on radial functions (with Z. Wu),
J. of Computational and Appl. Math. 73 (1996) pp. 257-270
---> pdf file
Approximation by Radial Basis Functions with Finitely Many Centers,
Constructive Approximation 12 (1996) pp. 331-340 ---> pdf file (51 Kb)
Error Estimates and Condition Numbers for Radial Basis Function
Interpolation,
Advances in Computational Mathematics 3 (1995) pp. 251-264
---> pdf file pdf file of Table 2
Creating Surfaces from Scattered Data Using Radial Basis Functions,
Mathematical Methods in Computer Aided Geometric Design III,
M. Dæhlen, T. Lyche and L.L. Schumaker (eds.), Vanderbilt Univ. Press,
1995, pp. 477-496
---> pdf file
Lower Bounds for Norms of Inverses of Interpolation Matrices for Radial
Basis Functions,
J. of Approx. Th. 79 (1994) pp. 287-306 ---> pdf file
Reproduction of Polynomials by Radial Basis Functions,
Wavelets, Images, and Surface Fitting, P.J. Laurent, A. LeMéhauté, and
L.L. Schumaker (eds.),
AKPeters, 1994, pp. 459-466 ---> pdf
file
Limits of Bernstein-Bézier Curves for Periodic Control Nets (with Dong-Xu
Qi),
J. of Approximation Theory and Applications 10 (1994) pp. 5-16
---> pdf file
Shape preserving properties and convergence of univariate multiquadric
quasi-interpolation (with Z. Wu),
Acta Math. Appl. Sin., Engl. Ser. 10, No. 4, pp. 441-446 (1994)
---> pdf file
Local Error Estimates for Radial Basis Function Interpolation of
Scattered Data (with Z. Wu),
IMA J. Numer. Anal. 13 (1993) pp. 13-27 ---> pdf file
Comparison of Radial Basis Function Interpolants,
in K. Jetter, L.L. Schumaker and F. Utreras (eds.):
``Multivariate Approximations: From CAGD to Wavelets'',
World Scientific, 1993, pp. 293-305 ---> pdf
Planar Curve Interpolation by Piecewise Conics of Arbitrary Type,
Constr. Approx. 9, 1993, pp. 373-389 ---> pdf
Error Estimates for Approximations from Control Nets,
CAGD 10, 1993, pp. 57-66 ---> pdf
A Parallel Multistage Method for Surface-Surface Intersection (with H.
Bürger)
in R.E. Barnhill, W. Boehm, and J. Hoschek (eds.):
Freiformkurven und -flächen, Proceedings of an Oberwolfach conference,
CAGD 10, 1993, pp. 277-291 ---> pdf
file 1 pdf file 2
A Multi-Parameter Method for Nonlinear Least-Squares Approximation,
in D. Braess and L.L Schumaker (eds.): Numerical Methods of Approximation
Theory 9,
ISNM 105, Birkhäuser, 1992, pp. 269-283 ---> pdf
Rational Geometric Curve Interpolation,
in T. Lyche and L.L. Schumaker (eds.):
``Mathematical Methods in Computer Aided Geometric Design II'',
Academic Press, 1992, pp. 517-535, survey ---> pdf
Numerische Mathematik (with H. Werner),
4. edition, completely rewritten and in one volume,
Springer Verlag, 1992, 326 pages
Geometrical differentiation and high-accuracy curve interpolation,
in S.P. Singh (ed.): Approximation Theory, Spline Functions, and
Applications,
NATO ASI Series C, Vol. 356, Kluwer, 1992, pp. 445-462
---> pdf
An Extended Continuous Newton Method (with I. Diener),
Journal of Optimization Theory and Applications 67, 1990, pp. 87-107
---> pdf
Remarks on High Accuracy Planar Curve Interpolation,
short preprint 1990, not submitted ---> pdf
Adaptive Rational Splines,
Constructive Approximation 6, 1990, pp. 167-179 --->
pdf
Interpolation in IR2 by piecewise quadratic visually C2
Bézier polynomials,
Computer Aided Geometric Design 6, 1989, pp. 219-233
---> pdf
Convergence of Planar Curve Interpolation Schemes,
in C.K. Chui, L.L. Schumaker and J.D. Ward (eds): Approximation Theory VI,
1989, pp. 581-584
---> pdf
On Global GC2 Convexity Preserving Interpolation of
Planar Curves by Piecewise Bézier Polynomials,
in T. Lyche and L.L. Schumaker (eds.): Mathematical Methods in Computer
Aided Geometric Design,
Academic Press 1989, 539-547 ---> pdf
On the expected sublinearity of the Boyer-Moore algorithm,
SIAM J. of Computing 17, 1988, pp. 648-658 link
pdf