An posteriori error analysis for the virtual element method (VEM) applied to general elliptic
problems is presented. The resulting error estimator is of residual-type and applies on very …

In recent years operator networks have emerged as promising deep learning tools for
approximating the solution to partial differential equations (PDEs). These networks map …

We introduce a level set based approach to Bayesian geometric inverse problems. In these
problems the interface between different domains is the key unknown, and is realized as the …

For Bayesian inverse problems with input-to-response maps given by well-posed partial
differential equations and subject to uncertain parametric or function space input, we …

Multiscale partial differential equations (PDEs) arise in various applications, and several
schemes have been developed to solve them efficiently. Homogenization theory is a …

We present convergence estimates of two types of greedy algorithms in terms of the metric
entropy of underlying compact sets. In the first part, we measure the error of a standard …

This is a survey on the theory of adaptive finite element methods (AFEMs), which are
fundamental in modern computational science and engineering but whose mathematical …

This paper considers the Dirichlet problem \rm-div(a∇u_a)=f\,\,on\,\D,\,u_a=0\,\,on\,\,∂D, for
a Lipschitz domain D⊂R^d, where a is a scalar diffusion function. For a fixed f, we discuss …

We design an adaptive virtual element method (AVEM) of lowest order over triangular
meshes with hanging nodes in 2d, which are treated as polygons. AVEM hinges on the …

For Bayesian inverse problems with input-to-response maps given by well-posed partial
differential equations (PDEs) and subject to uncertain parametric or function space input, we …