TY - JOUR

T1 - Algorithms defined by Nash iteration

T2 - Some implementations via multilevel collocation and smoothing

AU - Fasshauer, Gregory E.

AU - Gartland, Eugene C.

AU - Jerome, Joseph W.

N1 - Funding Information:
The second author is supported by the National Science Foundation under grant DMS-9870420. The third author is supported by the National Science Foundation under grant DMS-9704458. We thank the referee for a number of helpful comments. This paper is dedicated to Professor Larry Schumaker on the occasion of his sixtieth birthday.

PY - 2000/7/1

Y1 - 2000/7/1

N2 - We describe the general algorithms of Nash iteration in numerical analysis. We make a particular choice of algorithm involving multilevel collocation and smoothing. Our test case is that of a linear differential equation, although the theory allows for the approximate solution of nonlinear differential equations. We describe the general situation completely, and employ an adaptation involving a splitting of the inversion and the smoothing into two separate steps. We had earlier shown how these ideas apply to scattered data approximation, but in this work we are interested in the application of the ideas to the numerical solution of differential equations. We make use of approximate smoothers, involving the solution of evolution equations with calibrated time steps.

AB - We describe the general algorithms of Nash iteration in numerical analysis. We make a particular choice of algorithm involving multilevel collocation and smoothing. Our test case is that of a linear differential equation, although the theory allows for the approximate solution of nonlinear differential equations. We describe the general situation completely, and employ an adaptation involving a splitting of the inversion and the smoothing into two separate steps. We had earlier shown how these ideas apply to scattered data approximation, but in this work we are interested in the application of the ideas to the numerical solution of differential equations. We make use of approximate smoothers, involving the solution of evolution equations with calibrated time steps.

KW - Approximate smoothing

KW - Multilevel collocation

KW - Nash iteration

KW - Newton iteration

KW - Radial basis functions

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U2 - 10.1016/S0377-0427(00)00377-0

DO - 10.1016/S0377-0427(00)00377-0

M3 - Article

AN - SCOPUS:0033691073

VL - 119

SP - 161

EP - 183

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 1-2

ER -