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We derive an efficient stochastic algorithm for inverse problems that present an unknown linear forcing term and a set of nonlinear parameters to be recovered. It is assumed that the data is noisy and that the linear part of the problem is…
Most inverse problems from physical sciences are formulated as PDE-constrained optimization problems. This involves identifying unknown parameters in equations by optimizing the model to generate PDE solutions that closely match measured…
A comprehensive convergence and stability analysis of some probabilistic numerical methods designed to solve Cauchy-type inverse problems is performed in this study. Such inverse problems aim at solving an elliptic partial differential…
This paper presents new results allowing an unknown non-Gaussian positive matrix-valued random field to be identified through a stochastic elliptic boundary value problem, solving a statistical inverse problem. A new general class of…
In this work, we investigate the inverse problem of recovering a potential coefficient in an elliptic partial differential equation from the observations at deterministic sampling points in the domain subject to random noise. We employ a…
A recently developed measure-theoretic framework solves a stochastic inverse problem (SIP) for models where uncertainties in model output data are predominantly due to aleatoric (i.e., irreducible) uncertainties in model inputs (i.e.,…
Inverse problems are paramount in Science and Engineering. In this paper, we consider the setup of Statistical Inverse Problem (SIP) and demonstrate how Stochastic Gradient Descent (SGD) algorithms can be used in the linear SIP setting. We…
We present a method for obtaining efficient probabilistic solutions to geostatistical and linear inverse problems in spherical geometry. Our Spherical Direct Sequential Simulation (SDSSIM) framework combines information from possibly noisy…
In this work, we present a novel approach for solving stochastic shape optimization problems. Our method is the extension of the classical stochastic gradient method to infinite-dimensional shape manifolds. We prove convergence of the…
This paper introduces a statistical treatment of inverse problems constrained by models with stochastic terms. The solution of the forward problem is given by a distribution represented numerically by an ensemble of simulations. The goal is…
This paper concerns models and convergence principles for dealing with stochasticity in a wide range of algorithms arising in nonlinear analysis and optimization in Hilbert spaces. It proposes a flexible geometric framework within which…
In this work the LHC inverse problem is quantified in the Bayesian context by clarifying the relation between the mapping from the theory parameter space to experimental signature space and the inverse map. We demonstrate that, after…
We present a computational framework for estimating the uncertainty in the numerical solution of linearized infinite-dimensional statistical inverse problems. We adopt the Bayesian inference formulation: given observational data and their…
Consider a probability measure supported by a regular geodesic ball in a manifold. For any p larger than or equal to 1 we define a stochastic algorithm which converges almost surely to the p-mean of the measure. Assuming furthermore that…
We formulate, and present a numerical method for solving, an inverse problem for inferring parameters of a deterministic model from stochastic observational data (quantities of interest). The solution, given as a probability measure, is…
We prove a general theorem providing smoothed analysis estimates for conic condition numbers of problems of numerical analysis. Our probability estimates depend only on geometric invariants of the corresponding sets of ill-posed inputs.…
Stochastic inverse problems considered in this article consist of estimating the probability distributions of intrinsically random inputs of computer models. These estimations are based on observable outputs affected by model noise, and…
Predictions for physical systems often rely upon knowledge acquired from ensembles of entities, e.g., ensembles of cells in biological sciences. For qualitative and quantitative analysis, these ensembles are simulated with parametric…
Self-organizing systems demonstrate how simple local rules can generate complex stochastic patterns. Many natural systems rely on such dynamics, making self-organization central to understanding natural complexity. A fundamental challenge…
The inverse potential problem consists in determining the density of the volume potential from measurements outside the sources. Its ill-posedness is due both to the non-uniqueness of the solution and to the instability of the solution with…