Robert Schaback's research page

Old teaching handouts and lecture notes

Available preprints, sorted backwards in time, with links to pdf files.
These files are early preprints, not in the finally published form!

Paths of Stochastic Processes: a Sudden Turnaround (with Emilio Porcu), 2024, pdf
Optimal Compactly Supported Functions in Sobolev Spaces, 2024, arxiv, later: pdf
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. 

• pdf files are those of submitted versions, see the journals for the final form.
• Handouts and lecture notes are elsewhere.

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

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 

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,  pdf

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,   earlier pdf     pdf

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 link,   pdf

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,   pdf

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