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In this work, we derive a reliable and efficient residual-typed error estimator for the finite element approximation of a 2d cathodic protection problem governed by a steady-state diffusion equation with a nonlinear boundary condition. We…
In this paper, we present a contraction-guided adaptive partitioning algorithm for improving interval-valued robust reachable set estimates in a nonlinear feedback loop with a neural network controller and disturbances. Based on an estimate…
This paper presents a goal-oriented a posteriori error estimation framework for linear functionals in the stabilized finite element discretization of the stationary convection-diffusion-reaction (CDR) equation. The theoretical framework for…
For the model problem of the heat equation discretized by an implicit Euler method in time and a conforming finite element method in space, we prove the efficiency of a posteriori error estimators with respect to the energy norm of the…
The estimation of normalizing constants is a fundamental step in probabilistic model comparison. Sequential Monte Carlo methods may be used for this task and have the advantage of being inherently parallelizable. However, the standard…
In this work, we investigate the numerical reconstruction of inclusions in a semilinear elliptic equation arising in the mathematical modeling of cardiac ischemia. We propose an adaptive finite element method for the resulting constrained…
Bayesian curve fitting plays an important role in inverse problems, and is often addressed using the Reversible Jump Markov Chain Monte Carlo (RJMCMC) algorithm. However, this algorithm can be computationally inefficient without…
A new technique of residual-type a posteriori error analysis is developed for the lowest-order Raviart-Thomas mixed finite element discretizations of convection-diffusion-reaction equations in two- or three-dimension. Both centered mixed…
The convergence analysis for least-squares finite element methods led to various adaptive mesh-refinement strategies: Collective marking algorithms driven by the built-in a posteriori error estimator or an alternative explicit…
We consider unconstrained stochastic optimization problems with no available gradient information. Such problems arise in settings from derivative-free simulation optimization to reinforcement learning. We propose an adaptive sampling…
This paper presents a new parameter estimation algorithm for the adaptive control of a class of time-varying plants. The main feature of this algorithm is a matrix of time-varying learning rates, which enables parameter estimation error…
We propose and analyze reliable and efficient a posteriori error estimators for an optimal control problem that involves a nondifferentiable cost functional, the Poisson problem as state equation and control constraints. To approximate the…
In the context of stochastic continuum-armed bandits, we present an algorithm that adapts to the unknown smoothness of the objective function. We exhibit and compute a polynomial cost of adaptation to the H{\"o}lder regularity for regret…
An explicit and computable error estimator for the $hp$ version of the virtual element method (VEM), together with lower and upper bounds with respect to the exact energy error, is presented. Such error estimator is employed to provide $hp$…
The rigorous convergence analysis of adaptive finite element methods for regularized variational models of quasi-static brittle fracture in strain-limiting elastic solids is presented. This work introduces two novel adaptive mesh refinement…
This paper derives an a posteriori error estimator for the nonlinear first-order optimality conditions associated with the electrically and flexoelectrically coupled Frank-Oseen model of liquid crystals, building on previous results for…
A novel residual-type {\it a posteriori} error analysis technique is developed for multipoint flux mixed finite element methods for flow in porous media in two or three space dimensions. The derived {\it a posteriori} error estimator for…
We consider the a posteriori error analysis of approximations of parabolic problems based on arbitrarily high-order conforming Galerkin spatial discretizations and arbitrarily high-order discontinuous Galerkin temporal discretizations.…
Probabilistic solvers for ordinary differential equations assign a posterior measure to the solution of an initial value problem. The joint covariance of this distribution provides an estimate of the (global) approximation error. The…
The numerical simulation of complex physical processes requires the use of economical discrete models. This lecture presents a general paradigm of deriving a posteriori error estimates for the Galerkin finite element approximation of…