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We develop an efficient, unconditionally stable, variable step second order exponential time differencing scheme for the incompressible Navier Stokes equations in two and three spatial dimensions under periodic boundary conditions, together…
Stochastic gradient descent (SGD) algorithm and its variations have been effectively used to optimize neural network models. However, with the rapid growth of big data and deep learning, SGD is no longer the most suitable choice due to its…
We compute the first order correction of the effective viscosity for a suspension containing solid particles with arbitrary shapes. We rewrite the computation as an homogenization problem for the Stokes equations in a perforated domain.…
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…
Implicit time-stepping for advection is applied locally in space and time where Courant numbers are large, but standard explicit time-stepping is used for the remaining solution which is typically the majority. This adaptively implicit…
In this paper, a new family of very-high-order TENO schemes with adaptive accuracy order and adaptive dissipation control (TENO-AA) is proposed. The new framework allows for constructing arbitrarily high-order TENO schemes in a unified…
In wave propagation theories, many problems of multi-sensor systems utilize time delay in their solution in signal processing. This technique finds great utility in seismic exploration and static correction (low-velocity weathering), which…
We present a fully adaptive multiresolution scheme for spatially two-dimensional, possibly degenerate reaction-diffusion systems, focusing on combustion models and models of pattern formation and chemotaxis in mathematical biology.…
We present a numerical method for rigorous over-approximation of a reachable set of differential inclusions. The method gives high-order error bounds for single step approximations and a uniform bound on the error over the finite time…
A variational framework, initially developed for high-order mesh optimisation, is being extended for r-adaptation. The method is based on the minimisation of a functional of the mesh deformation. To achieve adaptation, elements of the…
Selecting an effective step-size is a fundamental challenge in first-order optimization, especially for problems with non-Euclidean geometries. This paper presents a novel adaptive step-size strategy for optimization algorithms that rely on…
In this work we analyze the convergence properties of the Spectral Deferred Correction (SDC) method originally proposed by Dutt et al. [BIT, 40 (2000), pp. 241--266]. The framework for this high-order ordinary differential equation (ODE)…
We numerically investigate a time-delayed reservoir computer architecture based on a single mode laser diode with optical injection and optical feedback. Through a high-resolution parametric analysis, we reveal unforeseen regions of high…
We consider the setting where a master wants to run a distributed stochastic gradient descent (SGD) algorithm on $n$ workers each having a subset of the data. Distributed SGD may suffer from the effect of stragglers, i.e., slow or…
We compare some first order well-balanced numerical schemes for shallow water system with special interest in applications where there are abrupt variations of the topography. We show that the space step required to obtain a prescribed…
We develop an adaptive control architecture to achieve stabilization and command following of uncertain dynamical systems with improved transient performance. Our framework consists of a new reference system and an adaptive controller. The…
This paper presents a sequence of deferred correction (DC) schemes built recursively from the implicit midpoint scheme for the numerical solution of general first order ordinary differential equations (ODEs). It is proven that each scheme…
We consider a class of relaxation problems mixing slow and fast variations which can describe population dynamics models or hyperbolic systems, with varying stiffness (from non-stiff to strongly dissipative), and develop a multi-scale…
We consider suspensions of rigid bodies in a two-dimensional viscous fluid. Even with high-fidelity numerical methods, unphysical contact between particles occurs because of spatial and temporal discretization errors. We apply the method of…
Integral deferred correction (IDC) methods have been shown to be an efficient way to achieve arbitrary high order accuracy and possess good stability properties. In this paper, we construct high order operator splitting schemes using the…