Related papers: A variational approach to the sum splitting scheme
By using a variational principle we find a necessary and sufficient condition for an operator to majorise the parallel sum of two positive definite operators. This result is then used as a vehicle to create new operator inequalities…
The present work provides a comprehensive study of symmetric-conjugate operator splitting methods in the context of linear parabolic problems and demonstrates their additional benefits compared to symmetric splitting methods. Relevant…
The simulation of systems that act on multiple time scales is challenging. A stable integration of the fast dynamics requires a highly accurate approximation whereas for the simulation of the slow part, a coarser approximation is accurate…
We propose a variational symplectic numerical method for the time integration of dynamical systems issued from the least action principle. We assume a quadratic internal interpolation of the state between two time steps and we approximate…
The non-stationary evolution of observable quantities in complex systems can frequently be described as a juxtaposition of quasi-stationary spells. Given that standard theoretical and data analysis approaches usually rely on the assumption…
The reduction of computational costs in the numerical solution of nonstationary problems is achieved through splitting schemes. In this case, solving a set of less computationally complex problems provides the transition to a new level in…
We propose a new asynchronous parallel block-descent algorithmic framework for the minimization of the sum of a smooth nonconvex function and a nonsmooth convex one, subject to both convex and nonconvex constraints. The proposed framework…
Computational methods for fractional differential equations exhibit essential instability. Even a minor modification of the coefficients or other entry data may switch good results to the divergent. The goal of this paper is to suggest the…
Parametric linear programming is a central operation for polyhedral computations, as well as in certain control applications.Here we propose a task-based scheme for parallelizing it, with quasi-linear speedup over large problems.This type…
Conventional finite-difference schemes for solving partial differential equations are based on approximating derivatives by finite-differences. In this work, an alternative theory is proposed which view finite-difference schemes as…
This paper proposes a general formulation for temporal parallelisation of dynamic programming for optimal control problems. We derive the elements and associative operators to be able to use parallel scans to solve these problems with…
In this work, we consider the coupled systems of linear unsteady partial differential equations, which arise in the modeling of poroelasticity processes. Stability estimates of weighted difference schemes for the coupled system of equations…
Standard explicit schemes for parabolic equations are not very convenient for computing practice due to the fact that they have strong restrictions on a time step. More promising explicit schemes are associated with explicit-implicit…
The aim of this paper is to develop a general method for constructing approximation schemes for viscosity solutions of fully nonlinear pathwise stochastic partial differential equations, and for proving their convergence. Our results apply…
We propose a decomposition framework for the parallel optimization of the sum of a differentiable function and a (block) separable nonsmooth, convex one. The latter term is typically used to enforce structure in the solution as, for…
The Parareal algorithm is used to solve time-dependent problems considering multiple solvers that may work in parallel. The key feature is a initial rough approximation of the solution that is iteratively refined by the parallel solvers. We…
We investigate mathematically a nonlinear approximation type approach recently introduced in [A. Ammar et al., J. Non-Newtonian Fluid Mech., 2006] to solve high dimensional partial differential equations. We show the link between the…
We consider a system of differential equations and obtain its solutions with exponential asymptotics and analyticity with respect to the spectral parameter. Solutions of such type have importance in studying spectral properties of…
Current algorithms for large-scale industrial optimization problems typically face a trade-off: they either require exponential time to reach optimal solutions, or employ problem-specific heuristics. To overcome these limitations, we…
We describe a proof-of-concept development and application of a phase averaging technique to the nonlinear rotating shallow water equations on the sphere, discretised using compatible finite element methods. Phase averaging consists of…