Related papers: High-order combined Multi-step Scheme for solving …
Backward stochastic differential equations (BSDEs) appear in numeruous applications. Classical approximation methods suffer from the curse of dimensionality and deep learning-based approximation methods are not known to converge to the BSDE…
We develop summation by parts (SBP) approach for generating high-order finite-difference schemes on the interval and propose new sets of schemes up to the 12th order. The coefficients of the schemes are governed by values of grid spacing…
We introduce new multilevel methods for solving large-scale unconstrained optimization problems. Specifically, the philosophy of multilevel methods is applied to Newton-type methods that regularize the Newton sub-problem using second order…
We present a stepwise approach to estimate high dimensional Gaussian graphical models. We exploit the relation between the partial correlation coefficients and the distribution of the prediction errors, and parametrize the model in terms of…
Ordinary differential equations are arguably the most popular and useful mathematical tool for describing physical and biological processes in the real world. Often, these physical and biological processes are observed with errors, in which…
We present a novel approach for high-order accurate numerical differentiation on unstructured meshes of quadrilateral elements. To differentiate a given function, an auxiliary function with greater smoothness properties is defined which…
We introduce a new class of "filtered" schemes for some first order non-linear Hamilton-Jacobi-Bellman equations. The work follows recent ideas of Froese and Oberman (SIAM J. Numer. Anal., Vol 51, pp.423-444, 2013). The proposed schemes are…
We investigate a family of approximate multi-step proximal point methods, framed as implicit linear discretizations of gradient flow. The resulting methods are multi-step proximal point methods, with similar computational cost in each…
This study presents an efficient, accurate, effective and unconditionally stable time stepping scheme for the Darcy-Brinkman equations in double-diffusive convection. The stabilization within the proposed method uses the idea of stabilizing…
Relying on the classical connection between Backward Stochastic Differential Equations (BSDEs) and non-linear parabolic partial differential equations (PDEs), we propose a new probabilistic learning scheme for solving high-dimensional…
When integrating the radiative transfer equation for polarized light, the necessity of high-order numerical methods is well known. In fact, well-performing high-order formal solvers enable higher accuracy and the use of coarser spatial…
We derive a new high-order compact finite difference scheme for option pricing in stochastic volatility models. The scheme is fourth-order accurate in space and second-order accurate in time. Under some restrictions, theoretical results…
We explore the utilization of higher-order discretization techniques in optimizing the gate count needed for quantum computer based solutions of partial differential equations. To accomplish this, we present an efficient approach for…
Synchronizations of processing elements (PEs) in massively parallel simulations, which arise due to communication or load imbalances between PEs, significantly affect the scalability of scientific applications. We have recently proposed a…
In the present work, a high order finite element type residual distribution scheme is designed in the framework of multidimensional compressible Euler equations of gas dynamics. The strengths of the proposed approximation rely on the…
Integration of Ordinary Differential Equations (ODEs) using Backward Difference formula (BDF) methods with p backward steps achieves order p accuracy if specific conditions are met. This work extends the composition technique with complex…
We introduce a higher order phase averaging method for nonlinear oscillatory systems. Phase averaging is a technique to filter fast motions from the dynamics whilst still accounting for their effect on the slow dynamics. Phase averaging is…
This work deals with the problem of choosing a time step for the numerical solution of boundary value problems for parabolic equations. The problem solution is derived using the fully implicit scheme, whereas a time step is selected via…
The single-step explicit time integration methods have long been valuable for solving large-scale nonlinear structural dynamic problems, classified into single-solve and multi-sub-step approaches. However, no existing explicit single-solve…
We introduce a class of unconditionally energy stable, high order accurate schemes for gradient flows in a very general setting. The new schemes are a high order analogue of the minimizing movements approach for generating a time discrete…