Related papers: On improving the efficiency of ADER methods
We present a new data-driven reduced-order modeling approach to efficiently solve parametrized partial differential equations (PDEs) for many-query problems. This work is inspired by the concept of implicit neural representation (INR),…
Reducing the variance of the gradient estimator is known to improve the convergence rate of stochastic gradient-based optimization and sampling algorithms. One way of achieving variance reduction is to design importance sampling strategies.…
In this paper, we present a deep learning-based numerical method for approximating high dimensional stochastic partial differential equations (SPDEs). At each time step, our method relies on a predictor-corrector procedure. More precisely,…
The semidefinite programming (SDP) relaxation has proven to be extremely strong for many hard discrete optimization problems. This is in particular true for the quadratic assignment problem (QAP), arguably one of the hardest NP-hard…
We investigate the application of a posteriori error estimates to a fractional optimal control problem with pointwise control constraints. Specifically, we address a problem in which the state equation is formulated as an integral form of…
This paper studies efficient distributed optimization methods for multi-agent networks. Specifically, we consider a convex optimization problem with a globally coupled linear equality constraint and local polyhedra constraints, and develop…
This manuscript presents an adaptive high order discretization technique for elliptic boundary value problems. The technique is applied to an updated version of the Hierarchical Poincar\'e-Steklov (HPS) method. Roughly speaking, the HPS…
We present a new family of high order accurate fully discrete one-step Discontinuous Galerkin (DG) finite element schemes on moving unstructured meshes for the solution of nonlinear hyperbolic PDE in multiple space dimensions, which may…
This work introduces a new method to efficiently solve optimization problems constrained by partial differential equations (PDEs) with uncertain coefficients. The method leverages two sources of inexactness that trade accuracy for speed:…
A numerical method for coupled 3D-1D problems with discontinuous solutions at the interfaces is derived and discussed. This extends a previous work on the subject where only continuous solutions were considered. Thanks to properly defined…
At present, deep learning based methods are being employed to resolve the computational challenges of high-dimensional partial differential equations (PDEs). But the computation of the high order derivatives of neural networks is costly,…
Regression on medical image sequences can capture temporal image pattern changes and predict images at missing or future time points. However, existing geodesic regression methods limit their regression performance by a strong underlying…
Algorithmic differentiation (AD) has become increasingly capable and straightforward to use. However, AD is inefficient when applied directly to solvers, a feature of most engineering analyses. We can leverage implicit differentiation to…
This paper explores a fully discrete approximation for a nonlinear hyperbolic PDE-constrained optimization problem (P) with applications in acoustic full waveform inversion. The optimization problem is primarily complicated by the…
In this paper, we develop and present an arbitrary high order well-balanced finite volume WENO method combined with the modified Patankar Deferred Correction (mPDeC) time integration method for the shallow water equations. Due to the…
We analyze adaptive mesh-refining algorithms for conforming finite element discretizations of certain non-linear second-order partial differential equations. We allow continuous polynomials of arbitrary, but fixed polynomial order. The…
The solutions of fractional differential equations (FDEs) have a natural singularity at the initial point. The accuracy of their numerical solutions is lower than the accuracy of the numerical solutions of FDEs whose solutions are…
We present a mixed-precision implementation of the high-order discontinuous Galerkin method with ADER time stepping (ADER-DG) for solving hyperbolic systems of partial differential equations (PDEs) in the hyperbolic PDE engine ExaHyPE. The…
This paper presents a high-order deferred correction algorithm combined with penalty iteration for solving free and moving boundary problems, using a fourth-order finite difference method. Typically, when free boundary problems are solved…
Verification of solutions is crucial for establishing the reliability of simulations. A central challenge is to find an accurate and reliable estimate of the discretization error. Current approaches to this estimation rely on the observed…