Related papers: Learning nonlocal regularization operators
Regularization techniques are widely employed in optimization-based approaches for solving ill-posed inverse problems in data analysis and scientific computing. These methods are based on augmenting the objective with a penalty function,…
This paper introduces a novel approach to learning sparsity-promoting regularizers for solving linear inverse problems. We develop a bilevel optimization framework to select an optimal synthesis operator, denoted as $B$, which regularizes…
We consider solving a probably infinite dimensional operator equation, where the operator is not modeled by physical laws but is specified indirectly via training pairs of the input-output relation of the operator. Neural operators have…
Motivated by problems in machine learning, we study a class of variational problems characterized by nonlocal operators. These operators are characterized by power-type weights, which are singular at a portion of the boundary. We identify a…
Regularization plays a pivotal role in integrating prior information into inverse problems. While many deep learning methods have been proposed to solve inverse problems, determining where to apply regularization remains a crucial…
These lecture notes for a graduate class present the regularization theory for linear and nonlinear ill-posed operator equations in Hilbert spaces. Covered are the general framework of regularization methods and their analysis via spectral…
Learning kernels in operators from data lies at the intersection of inverse problems and statistical learning, providing a powerful framework for capturing non-local dependencies in function spaces and high-dimensional settings. In contrast…
Variational regularization is commonly used to solve linear inverse problems, and involves augmenting a data fidelity by a regularizer. The regularizer is used to promote a priori information and is weighted by a regularization parameter.…
Neural networks have emerged as effective tools for solving ill-posed inverse problems. In many scientific applications, however, observational training data are insufficient, and learned inverse operators must instead be trained on…
We consider a statistical inverse learning problem, where we observe the image of a function $f$ through a linear operator $A$ at i.i.d. random design points $X_i$, superposed with an additive noise. The distribution of the design points is…
A learning approach to selecting regularization parameters in multi-penalty Tikhonov regularization is investigated. It leads to a bilevel optimization problem, where the lower level problem is a Tikhonov regularized problem parameterized…
In this work, we investigate various approaches that use learning from training data to solve inverse problems, following a bi-level learning approach. We consider a general framework for optimal inversion design, where training data can be…
We prove higher Sobolev regularity for bounded weak solutions to a class of nonlinear nonlocal integro-differential equations. The leading operator exhibits nonuniform growth, switching between two different fractional elliptic ``phases"…
In this article, we study the regularity of solutions to inhomogeneous time-fractional evolution equations involving anisotropic non-local operators in mixed-norm Sobolev spaces of variable order, with non-trivial initial conditions. The…
In this chapter we provide a theoretically founded investigation of state-of-the-art learning approaches for inverse problems from the point of view of spectral reconstruction operators. We give an extended definition of regularization…
This study investigates the impact of Sobolev Training on operator learning frameworks for improving model performance. Our research reveals that integrating derivative information into the loss function enhances the training process, and…
We consider linear and obstacle problems driven by a nonlocal integral operator, for which nonlocal interactions are restricted to a ball of finite radius. These type of operators are used to model anomalous diffusion and, for a special…
We introduce and study a mathematical framework for a broad class of regularization functionals for ill-posed inverse problems: Regularization Graphs. Regularization graphs allow to construct functionals using as building blocks linear…
We study the local solvability of a class of operators with multiple characteristics. The class considered here complements and extends the one studied in [9], in that in this paper we consider some cases of operators with complex…
In this work we consider a generalized bilevel optimization framework for solving inverse problems. We introduce fractional Laplacian as a regularizer to improve the reconstruction quality, and compare it with the total variation…