Using low-rank approximations of gridded data for spline surface fitting. (English) Zbl 1522.65018
Summary: The paper is devoted to the problem of finding bivariate tensor-product spline (or similar) functions that approximate gridded data in the least-squares sense. We propose to apply a low rank approximation of matrices to the data, to find the result by solving a sequence of univariate fitting problems that can be handled efficiently. This can be seen as a generalization of the method proposed by I. Georgieva and C. Hofreither [J. Comput. Appl. Math. 310, 80–91 (2017; Zbl 1348.65051)] that combines cross approximation (which is a particular method for low rank matrix approximation) with spline interpolation.
While the algorithm yields the best least-squares approximation after \(r\) steps, where \(r\) denotes the rank of the data matrix, terminating it earlier yields a low-rank approximation, which often provides a sufficient level of accuracy. We also present a stopping criterion (based on a lower error estimate) that allows to use the method efficiently in the situation when the required number of degrees of freedom is not known in advance.
While the algorithm yields the best least-squares approximation after \(r\) steps, where \(r\) denotes the rank of the data matrix, terminating it earlier yields a low-rank approximation, which often provides a sufficient level of accuracy. We also present a stopping criterion (based on a lower error estimate) that allows to use the method efficiently in the situation when the required number of degrees of freedom is not known in advance.
MSC:
| 65D07 | Numerical computation using splines |
Citations:
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