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In this paper, we analyze posterior consistency of a Bayesian data assimilation problem under discretization. We prove convergence rates for the discrete posterior to ground truth solution under both conforming discretization and finite…
This paper focuses on an inverse problem associated with the plate equation which is derived from models in fluid mechanics and elasticity. We establish the unique identifying results in simultaneously determining both the unknown density…
Recent findings suggest that abnormal operating conditions of equipment in the oil and gas supply chain represent a large fraction of anthropogenic methane emissions. Thus, effective mitigation of emissions necessitates rapid identification…
The Inverse Problem for the estimation of a point-wise approximation error occurring at the discretization and solving of the system of partial differential equations is addressed. The set of the differences between the numerical solutions…
In inverse problems, one attempts to infer spatially variable functions from indirect measurements of a system. To practitioners of inverse problems, the concept of "information" is familiar when discussing key questions such as which parts…
We consider a finite element discretization for the reconstruction of the final state of the heat equation, when the initial data is unknown, but additional data is given in a sub domain in the space time. For the discretization in space we…
We present an approach for synthesising observational data with elastodynamic finite element models by extending the statistical finite element method (statFEM) framework. The proposed formulation adopts a Bayesian filtering approach to…
In multi-phase fluid flow, fluid-structure interaction, and other applications, partial differential equations (PDEs) often arise with discontinuous coefficients and singular sources (e.g., Dirac delta functions). These complexities arise…
In target tracking and sensor fusion contexts it is not unusual to deal with a large number of Gaussian densities that encode the available information (multiple hypotheses), as in applications where many sensors, affected by clutter or…
A fundamental problem of statistical data analysis, distribution density estimation by experimental data, is considered. A new method with optimal asymptotic behavior, the root density estimator, is developed. The method proposed may be…
Using piezoelectric impedance/admittance sensing for structural health monitoring is promising, owing to the simplicity in circuitry design as well as the high-frequency interrogation capability. The actual identification of fault location…
The inverse conductivity problem aims at determining the unknown conductivity inside a bounded domain from boundary measurements. In practical applications, algorithms based on minimizing a regularized residual functional subject to PDE…
We present a framework for fine-tuning flow-matching generative models to enforce physical constraints and solve inverse problems in scientific systems. Starting from a model trained on low-fidelity or observational data, we apply a…
We investigate an optimization problem that arises when working within the paradigm of Data-Driven Computational Mechanics. In the context of the diffusion-reaction problem, such an optimization problem seeks for the continuous primal…
Continuous data assimilation addresses time-dependent problems with unknown initial conditions by incorporating observations of the solution into a nudging term. For the prototypical heat equation with variable conductivity and the Neumann…
We address the initial source identification problem for the heat equation, a notably ill-posed inverse problem characterized by exponential instability. Departing from classical Tikhonov regularization, we propose a novel approach based on…
We consider an inverse problem involving the reconstruction of the solution to a nonlinear partial differential equation (PDE) with unknown boundary conditions. Instead of direct boundary data, we are provided with a large dataset of…
We study the problem $-\Delta u = \gamma$, where $\gamma$ is a singular measure, with support on a curve or a point. We prove that optimal rates of convergence for the finite element method can be obtained using properly graded meshes. In…
We introduce Rectified Point Flow, a unified parameterization that formulates pairwise point cloud registration and multi-part shape assembly as a single conditional generative problem. Given unposed point clouds, our method learns a…
Over the last decade, a series of applied mathematics papers have explored a type of inverse problem--called by a variety of names including "inverse sensitivity", "pushforward based inference", "consistent Bayesian inference", or…