Related papers: Error estimation for the time to a threshold value…
In this work, we derive two-sided a posteriori error estimates for the dual-weighted residual (DWR) method. We consider both single and multiple goal functionals. Using a saturation assumption, we derive lower bounds yielding the efficiency…
We derive an anisotropic a posteriori error estimate for the adaptive conforming Virtual Element approximation of a paradigmatic two-dimensional elliptic problem. In particular, we introduce a quasi-interpolant operator and exploit its…
A method for numerical approximation of a new class of fractional parabolic stochastic evolution equations is introduced and analysed. This class of equations has recently been proposed as a space-time extension of the SPDE-method in…
We provide a bound for the error committed when using a Fourier method to price European options when the underlying follows an exponential \levy dynamic. The price of the option is described by a partial integro-differential equation…
A general adaptive refinement strategy for solving linear elliptic partial differential equation with random data is proposed and analysed herein. The adaptive strategy extends the a posteriori error estimation framework introduced by…
Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) are key ingredients in a number of models in physics and financial engineering. In particular, parabolic PDEs and BSDEs are fundamental…
We study an asymptotic preserving scheme for the temporal discretization of a system of parabolic semilinear SPDEs with two time scales. Owing to the averaging principle, when the time scale separation $\epsilon$ vanishes, the slow…
Operator learning for partial differential equations (PDEs) aims to learn solution operators on infinite-dimensional function spaces from finite-resolution data. In this setting, it is important for the learned model to be…
In the recent article [Kopteva, N., Numer. Math., 137, 607--642 (2017)] the author obtained residual-type a posteriori error estimates in the energy norm for singularly perturbed semilinear reaction-diffusion equations on anisotropic…
We present the first systematic work for deriving a posteriori error estimates for general non-polynomial basis functions in an interior penalty discontinuous Galerkin (DG) formulation for solving eigenvalue problems associated with second…
Space-time finite element discretizations of time-optimal control problems governed by linear parabolic PDEs and subject to pointwise control constraints are considered. Optimal a priori error estimates are obtained for the control variable…
We consider the problem of numerically approximating the solutions to a partial differential equation (PDE) when there is insufficient information to determine a unique solution. Our main example is the Poisson boundary value problem, when…
We introduce a method for the fast numerical approximation of linear, second-order parabolic partial differential equations (PDEs for short) with time-independent coefficients based on model order reduction techniques and the Laplace…
We propose a novel a-posteriori error estimation technique where the target quantities of interest are ratios of high-dimensional integrals, as occur e.g. in PDE constrained Bayesian inversion and PDE constrained optimal control subject to…
We derive a posteriori error estimates for the hybridizable discontinuous Galerkin (HDG) methods, including both the primal and mixed formulations, for the approximation of a linear second-order elliptic problem on conforming simplicial…
Recovery type a posteriori error estimators are popular, particularly in the engineering community, for their computationally inexpensive, easy to implement, and generally asymptotically exactness. Unlike the residual type error estimators,…
In this paper, we develop a numerical scheme for the space-time fractional parabolic equation, i.e., an equation involving a fractional time derivative and a fractional spatial operator. Both the initial value problem and the…
The first order by time partial differential equations are used as models in applications such as fluid flow, heat transfer, solid deformation, electromagnetic waves, and others. In this paper we propose the new numerical method to solve a…
Ordinary differential equations (ODE's) are widespread models in physics, chemistry and biology. In particular, this mathematical formalism is used for describing the evolution of complex systems and it might consist of high-dimensional…
A multilevel adaptive refinement strategy for solving linear elliptic partial differential equations with random data is recalled in this work. The strategy extends the a posteriori error estimation framework introduced by Guignard and…