Subgradient-based Lavrentiev regularisation of monotone ill-posed problems
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| Format: | Preprint |
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2020
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| _version_ | 1866910624485736448 |
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| author | Grasmair, Markus Hildrum, Fredrik |
| author_facet | Grasmair, Markus Hildrum, Fredrik |
| contents | We introduce subgradient-based Lavrentiev regularisation of the form \begin{equation*} \mathcal{A}(u) + α\partial \mathcal{R}(u) \ni f^δ\end{equation*} for linear and nonlinear ill-posed problems with monotone operators $\mathcal{A}$ and general regularisation functionals $\mathcal{R}$. In contrast to Tikhonov regularisation, this approach perturbs the equation itself and avoids the use of the adjoint of the derivative of $\mathcal{A}$. It is therefore especially suitable for time-causal problems that only depend on information in the past and allows for real-time computation of regularised solutions. We establish a general well-posedness theory in Banach spaces and prove convergence-rate results with variational source conditions. Furthermore, we demonstrate its application in total-variation denoising in linear Volterra integral operators of the first kind and parameter-identification problems in semilinear parabolic PDEs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2005_08917 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Subgradient-based Lavrentiev regularisation of monotone ill-posed problems Grasmair, Markus Hildrum, Fredrik Optimization and Control Numerical Analysis 47H05, 47J06, 45Q05, 65J20, 65R30 We introduce subgradient-based Lavrentiev regularisation of the form \begin{equation*} \mathcal{A}(u) + α\partial \mathcal{R}(u) \ni f^δ\end{equation*} for linear and nonlinear ill-posed problems with monotone operators $\mathcal{A}$ and general regularisation functionals $\mathcal{R}$. In contrast to Tikhonov regularisation, this approach perturbs the equation itself and avoids the use of the adjoint of the derivative of $\mathcal{A}$. It is therefore especially suitable for time-causal problems that only depend on information in the past and allows for real-time computation of regularised solutions. We establish a general well-posedness theory in Banach spaces and prove convergence-rate results with variational source conditions. Furthermore, we demonstrate its application in total-variation denoising in linear Volterra integral operators of the first kind and parameter-identification problems in semilinear parabolic PDEs. |
| title | Subgradient-based Lavrentiev regularisation of monotone ill-posed problems |
| topic | Optimization and Control Numerical Analysis 47H05, 47J06, 45Q05, 65J20, 65R30 |
| url | https://arxiv.org/abs/2005.08917 |