Related papers: Efficient iterative arbitrary high order methods: …
We consider high-order splitting schemes for large-scale differential Riccati equations. Such equations arise in many different areas and are especially important within the field of optimal control. In the large-scale case, it is critical…
It is common practice to apply gradient-based optimization algorithms to numerically solve large-scale ODE constrained optimal control problems. Gradients of the objective function are most efficiently computed by approximate adjoint…
In this study we propose a-posteriori error estimation results to approximate the precision loss in quantities of interests computed using reduced order models. To generate the surrogate models we employ Proper Orthogonal Decomposition and…
In this paper, we propose a new unified first order hyperbolic model of Newtonian continuum mechanics coupled with electro-dynamics. The model is able to describe the behavior of moving elasto-plastic dielectric solids as well as viscous…
The discrete gradient methods are integrators designed to preserve invariants of ordinary differential equations. From a formal series expansion of a subclass of these methods, we derive conditions for arbitrarily high order. We derive…
In this paper, we consider the composition of two independent processes : one process corresponds to position and the other one to time. Such processes will be called iterated processes. We first propose an algorithm based on the Euler…
An inexact framework for high-order adaptive regularization methods is presented, in which approximations may be used for the $p$th-order tensor, based on lower-order derivatives. Between each recalculation of the $p$th-order derivative…
We explore a reinforcement learning strategy to automate and accelerate h/p-multigrid methods in high-order solvers. Multigrid methods are very efficient but require fine-tuning of numerical parameters, such as the number of smoothing…
This paper aims to devise an adaptive neural network basis method for numerically solving a second-order semilinear partial differential equation (PDE) with low-regular solutions in two/three dimensions. The method is obtained by combining…
First-order methods with momentum such as Nesterov's fast gradient method are very useful for convex optimization problems, but can exhibit undesirable oscillations yielding slow convergence rates for some applications. An adaptive…
This paper is concerned with the theory, construction and application of variable-stepsize implicit Peer two-step methods that are super-convergent for variable stepsizes, i.e., preserve their classical order achieved for uniform stepsizes…
In this paper we develop a systematic reduction procedure for determining intermediate integrals of second order hyperbolic equations so that exact solutions of the second order PDEs under interest can be obtained by solving first order…
Many important physical problems, such as fluid structure interaction or conjugate heat transfer, require numerical methods that compute boundary derivatives or fluxes to high accuracy. This paper proposes a novel alternative to calculating…
We present a novel framework for PDE-constrained $r$-adaptivity of high-order meshes. The proposed method formulates mesh movement as an optimization problem, with an objective function defined as a convex combination of a mesh quality…
Within recent years, considerable progress has been made regarding high-performance solvers for Partial Differential Equations (PDEs), yielding potential gains in efficiency compared to industry standard tools. However, the latter largely…
The combination of reduced basis and collocation methods enables efficient and accurate evaluation of the solutions to parameterized PDEs. In this paper, we study the stochastic collocation methods that can be combined with reduced basis…
Finding the sparset solution of an underdetermined system of linear equations $y=Ax$ has attracted considerable attention in recent years. Among a large number of algorithms, iterative thresholding algorithms are recognized as one of the…
In this work, we concern with the high order numerical methods for coupled forward-backward stochastic differential equations (FBSDEs). Based on the FBSDEs theory, we derive two reference ordinary differential equations (ODEs) from the…
Following the seminal work of Nesterov, accelerated optimization methods have been used to powerfully boost the performance of first-order, gradient-based parameter estimation in scenarios where second-order optimization strategies are…
Arbitrary high order numerical methods for time-harmonic acoustic scattering problems originally defined on unbounded domains are constructed. This is done by coupling recently developed high order local absorbing boundary conditions (ABCs)…