Stochastic data-driven Bouligand–Landweber method for solving non-smooth inverse problems
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Harshit Bajpai
Abstract
In this study, we present and analyze a novel variant of the stochastic gradient descent method, referred as Stochastic data-driven Bouligand–Landweber iteration tailored for addressing the system of non-smooth ill-posed inverse problems. Our method incorporates the utilization of training data, using a bounded linear operator, which guides the iterative procedure. At each iteration step, the method randomly chooses one equation from the nonlinear system with data-driven term. When dealing with the precise or exact data, it has been established that mean square iteration error converges to zero. However, when confronted with the noisy data, we employ our approach in conjunction with a predefined stopping criterion, which we refer to as an a priori stopping rule. We provide a comprehensive theoretical foundation, establishing convergence and stability for this scheme within the realm of infinite-dimensional Hilbert spaces. These theoretical underpinnings are further bolstered by a numerical experiment on a system of linearly ill-posed problems and by discussing an example that fulfills the assumptions of the paper.
Funding statement: Harshit Bajpai would like to acknowledge Science and Engineering Research Board (SERB), India for providing a PhD fellowship through grant CRG/2022/005491. Ankik Kumar Giri would also like to thank SERB for supporting his research work as a part of the project “Fast and efficient iterative methods for solving nonlinear inverse problems” under the grant CRG/2022/005491.
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© 2025 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Local convergence of the error-reduction algorithm for real-valued objects
- A rotation total variation regularization for full waveform inversion
- Stochastic data-driven Bouligand–Landweber method for solving non-smooth inverse problems
- On determining the fractional exponent of the subdiffusion equation
- Tow-parameter quasi-boundary value method for a backward abstract time-degenerate fractional parabolic problem
- Determining both leading coefficient and source in a nonlocal elliptic equation
- Discrete dynamical systems: Inverse problems and related topics
- Stability estimates for an inverse problem of determining time-dependent coefficients in a system of parabolic equations
- Numerical solutions to inverse nodal problems for the Sturm–Liouville operator and their applications
- Revisiting linear machine learning through the perspective of inverse problems
Artikel in diesem Heft
- Frontmatter
- Local convergence of the error-reduction algorithm for real-valued objects
- A rotation total variation regularization for full waveform inversion
- Stochastic data-driven Bouligand–Landweber method for solving non-smooth inverse problems
- On determining the fractional exponent of the subdiffusion equation
- Tow-parameter quasi-boundary value method for a backward abstract time-degenerate fractional parabolic problem
- Determining both leading coefficient and source in a nonlocal elliptic equation
- Discrete dynamical systems: Inverse problems and related topics
- Stability estimates for an inverse problem of determining time-dependent coefficients in a system of parabolic equations
- Numerical solutions to inverse nodal problems for the Sturm–Liouville operator and their applications
- Revisiting linear machine learning through the perspective of inverse problems
