Related papers: On Ambiguity in Linear Inverse Problems: Entrywise…
We consider the ill-posed inverse problem of identifying a nonlinearity in a time-dependent PDE model. The nonlinearity is approximated by a neural network, and needs to be determined alongside other unknown physical parameters and the…
Many inverse problems arising in engineering and applied sciences involve unknown quantities with pronounced spatial inhomogeneity, such as localized defects or spatially varying material properties, making reliable uncertainty…
In this work we develop a new numerical approach for recovering a spatially dependent source component in a standard parabolic equation from partial interior measurements. We establish novel conditional Lipschitz stability and H\"{o}lder…
In this work, a method for obtaining pixel-wise error bounds in Bayesian regularization of inverse imaging problems is introduced. The proposed method employs estimates of the posterior variance together with techniques from conformal…
A novel framework is introduced to formalize identifiability in well-specified but ill-posed linear regression models. The framework is distribution-free and accommodates highly correlated features that may or may not relate to the…
The Bayesian approach has proved to be a coherent approach to handle ill posed Inverse problems. However, the Bayesian calculations need either an optimization or an integral calculation. The maximum a posteriori (MAP) estimation requires…
In this work, we develop a new complexity metric for an important class of low-rank matrix optimization problems in both symmetric and asymmetric cases, where the metric aims to quantify the complexity of the nonconvex optimization…
The real-life data have a complex and non-linear structure due to their nature. These non-linearities and the large number of features can usually cause problems such as the empty-space phenomenon and the well-known curse of dimensionality.…
The \(L_1/L_2\) norm ratio has gained significant attention as a measure of sparsity due to three merits: sharper approximation to the \(L_0\) norm compared to the \(L_1\) norm, being parameter-free and scale-invariant, and exceptional…
To quantify uncertainties in inverse problems of partial differential equations (PDEs), we formulate them into statistical inference problems using Bayes' formula. Recently, well-justified infinite-dimensional Bayesian analysis methods have…
Classically, to solve differential equation problems, it is necessary to specify sufficient initial and/or boundary conditions so as to allow the existence of a unique solution. Well-posedness of differential equation problems thus involves…
Conditional stability estimates allow us to characterize the degree of ill-posedness of many inverse problems, but without further assumptions they are not sufficient for the stable solution in the presence of data perturbations. We here…
Low-rank tensor approximation error bounds are proposed for the case of noisy input data that depend on low-rank representation type, rank and the dimensionality of the tensor. The bounds show that high-dimensional low-rank structured…
The inverse problem is studied in multi-body systems with nonlinear dynamics representing, e.g., phase-locked wave systems, standard multimode and random lasers. Using a general model for four-body interacting complex-valued variables we…
Recently, inverse problems have attracted more and more attention in computational mathematics and become increasingly important in engineering applications. After the discretization, many of inverse problems are reduced to linear systems.…
In this paper, we introduce a general framework for analyzing the numerical conditioning of minimal problems in multiple view geometry, using tools from computational algebra and Riemannian geometry. Special motivation comes from the fact…
Interior-point methods for linear programming problems require the repeated solution of a linear system of equations. Solving these linear systems is non-trivial due to the severe ill-conditioning of the matrices towards convergence. This…
Despite a variety of available techniques the issue of the proper regularization parameter choice for inverse problems still remains one of the biggest challenges. The main difficulty lies in constructing a rule, allowing to compute the…
The approximation and convergence properties of implicit neural representations (INRs) are known to be highly sensitive to parameter initialization strategies. While several data-driven initialization methods demonstrate significant…
Perturbation theory plays a crucial role in sensitivity analysis, which is extensively used to assess the robustness of numerical techniques. To quantify the relative sensitivity of any problem, it becomes essential to investigate…