On deconvolution methods
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2003
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| _version_ | 1866912655440084992 |
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| author | Ramm, Alexander G. Galstian, A. |
| author_facet | Ramm, Alexander G. Galstian, A. |
| contents | Several methods for solving efficiently the one-dimensional deconvolution problem are proposed. The problem is to solve the Volterra equation ${\mathbf k} u:=\int_0^t k(t-s)u(s)ds=g(t),\quad 0\leq t\leq T$. The data, $g(t)$, are noisy. Of special practical interest is the case when the data are noisy and known at a discrete set of times. A general approach to the deconvolution problem is proposed: represent ${\mathbf k}=A(I+S)$, where a method for a stable inversion of $A$ is known, $S$ is a compact operator, and $I+S$ is injective. This method is illustrated by examples: smooth kernels $k(t)$, and weakly singular kernels, corresponding to Abel-type of integral equations, are considered. A recursive estimation scheme for solving deconvolution problem with noisy discrete data is justified mathematically, its convergence is proved, and error estimates are obtained for the proposed deconvolution method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_0301382 |
| institution | arXiv |
| publishDate | 2003 |
| record_format | arxiv |
| spellingShingle | On deconvolution methods Ramm, Alexander G. Galstian, A. Numerical Analysis 45D05, 45L05, 45P05, 65R20, 65R30 Several methods for solving efficiently the one-dimensional deconvolution problem are proposed. The problem is to solve the Volterra equation ${\mathbf k} u:=\int_0^t k(t-s)u(s)ds=g(t),\quad 0\leq t\leq T$. The data, $g(t)$, are noisy. Of special practical interest is the case when the data are noisy and known at a discrete set of times. A general approach to the deconvolution problem is proposed: represent ${\mathbf k}=A(I+S)$, where a method for a stable inversion of $A$ is known, $S$ is a compact operator, and $I+S$ is injective. This method is illustrated by examples: smooth kernels $k(t)$, and weakly singular kernels, corresponding to Abel-type of integral equations, are considered. A recursive estimation scheme for solving deconvolution problem with noisy discrete data is justified mathematically, its convergence is proved, and error estimates are obtained for the proposed deconvolution method. |
| title | On deconvolution methods |
| topic | Numerical Analysis 45D05, 45L05, 45P05, 65R20, 65R30 |
| url | https://arxiv.org/abs/math/0301382 |