相关论文: Partial inversion of the elliptic operator to spee…

We study the inverse problem of recovering the order and the diffusion coefficient of an elliptic fractional partial differential equation from a finite number of noisy observations of the solution. We work in a Bayesian framework and show…

We present a novel algorithm based on the ensemble Kalman filter to solve inverse problems involving multiscale elliptic partial differential equations. Our method is based on numerical homogenization and finite element discretization and…

This paper tackles efficient methods for Bayesian inverse problems with priors based on Whittle--Mat\'ern Gaussian random fields. The Whittle--Mat\'ern prior is characterized by a mean function and a covariance operator that is taken as a…

In computational inverse problems, it is common that a detailed and accurate forward model is approximated by a computationally less challenging substitute. The model reduction may be necessary to meet constraints in computing time when…

We consider the statistical linear inverse problem of making inference on an unknown source function in an elliptic partial differential equation from noisy observations of its solution. We employ nonparametric Bayesian procedures based on…

This paper develops a probabilistic numerical method for solution of partial differential equations (PDEs) and studies application of that method to PDE-constrained inverse problems. This approach enables the solution of challenging inverse…

This paper investigates the formulation and implementation of Bayesian inverse problems to learn input parameters of partial differential equations (PDEs) defined on manifolds. Specifically, we study the inverse problem of determining the…

A common task in experimental sciences is to fit mathematical models to real-world measurements to improve understanding of natural phenomenon (reverse-engineering or inverse modeling). When complex dynamical systems are considered, such as…

We consider Bayesian inference for large scale inverse problems, where computational challenges arise from the need for repeated evaluations of an expensive forward model. This renders most Markov chain Monte Carlo approaches infeasible,…

We study Bayesian inference methods for solving linear inverse problems, focusing on hierarchical formulations where the prior or the likelihood function depend on unspecified hyperparameters. In practice, these hyperparameters are often…

Bayesian methods are useful for statistical inference. However, real-world problems can be challenging using Bayesian methods when the data analyst has only limited prior knowledge. In this paper we consider a class of problems, called…

A wide variety of (fixed-point) iterative methods for the solution of nonlinear equations (in Hilbert spaces) exists. In many cases, such schemes can be interpreted as iterative local linearization methods, which, as will be shown, can be…

In this paper, we present a theoretical and computational workflow for the non-parametric Bayesian inference of drift and diffusion functions of autonomous diffusion processes. We base the inference on the partial differential equations…

Fractional diffusion equations have been an effective tool for modeling anomalous diffusion in complicated systems. However, traditional numerical methods require expensive computation cost and storage resources because of the memory effect…

Inverse problems are ubiquitous because they formalize the integration of data with mathematical models. In many scientific applications the forward model is expensive to evaluate, and adjoint computations are difficult to employ; in this…

Bayesian methods have been widely used in the last two decades to infer statistical properties of spatially variable coefficients in partial differential equations from measurements of the solutions of these equations. Yet, in many cases…

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…

Likelihood-free (a.k.a. simulation-based) inference problems are inverse problems with expensive, or intractable, forward models. ODE inverse problems are commonly treated as likelihood-free, as their forward map has to be numerically…

In distributed optimization and distributed numerical linear algebra, we often encounter an inversion bias: if we want to compute a quantity that depends on the inverse of a sum of distributed matrices, then the sum of the inverses does not…

We present a novel numerical scheme to approximate the solution map $s\mapsto u(s) := \mathcal{L}^{-s}f$ to partial differential equations involving fractional elliptic operators. Reinterpreting $\mathcal{L}^{-s}$ as interpolation operator…