Related papers: New efficient substepping methods for exponential …
Randomized block Krylov subspace methods form a powerful class of algorithms for computing the extreme eigenvalues of a symmetric matrix or the extreme singular values of a general matrix. The purpose of this paper is to develop new…
The main challenge in the analysis of numerical schemes for the Zakharov system originates from the presence of derivatives in the nonlinearity. In this paper a new trigonometric time-integration scheme for the Zakharov system is…
This paper is concerned with the adaptive numerical treatment of stochastic partial differential equations. Our method of choice is Rothe's method. We use the implicit Euler scheme for the time discretization. Consequently, in each step, an…
Calculating the long term solution of ordinary differential equations, such as those of the $N$-body problem, is central to understanding a wide range of dynamics in astrophysics, from galaxy formation to planetary chaos. Because generally…
We derive a second-order ordinary differential equation (ODE) which is the limit of Nesterov's accelerated gradient method. This ODE exhibits approximate equivalence to Nesterov's scheme and thus can serve as a tool for analysis. We show…
We introduce a family of implicit probabilistic integrators for initial value problems (IVPs), taking as a starting point the multistep Adams-Moulton method. The implicit construction allows for dynamic feedback from the forthcoming…
Since being analyzed by Rokhlin, Szlam, and Tygert and popularized by Halko, Martinsson, and Tropp, randomized Simultaneous Power Iteration has become the method of choice for approximate singular value decomposition. It is more accurate…
Recently, our group developed explicit symplectic methods for curved spacetimes that are not split into several explicitly integrable parts, but are via appropriate time transformations. Such time-transformed explicit symplectic integrators…
In this paper, we present a novel spectral renormalization exponential integrator method for solving gradient flow problems. Our method is specifically designed to simultaneously satisfy discrete analogues of the energy dissipation laws and…
On the basis of additive schemes (splitting schemes) we construct efficient numerical algorithms to solve approximately the initial-boundary value problems for systems of time-dependent partial differential equations (PDEs). In many applied…
In this paper, we introduce the Method of Ellipcenters (ME) for unconstrained minimization. At the cost of two gradients per iteration and a line search, we compute the next iterate by setting it as the center of an elliptical…
We analyze temporal approximation schemes based on overlapping domain decompositions. As such schemes enable computations on parallel and distributed hardware, they are commonly used when integrating large-scale parabolic systems. Our…
We propose implicit integrators for solving stiff differential equations on unit spheres. Our approach extends the standard backward Euler and Crank-Nicolson methods in Cartesian space by incorporating the geometric constraint inherent to…
Molecular Dynamics method is based on solution of Newtonian differential equations of motion. A new very accurate and efficient time-reversible explicit integrator was derived on the basis of second order Tailor expansion of force. There is…
In this paper, we extend several time reversible numerical integrators to solve the Lorentz force equations from second order accuracy to higher order accuracy for relativistic charged particle tracking in electromagnetic fields. A fourth…
In this paper, two multiscale time integrators (MTIs), motivated from two types of multiscale decomposition by either frequency or frequency and amplitude, are proposed and analyzed for solving highly oscillatory second order differential…
We present a simple algorithm to switch between $N$-body time integrators in a reversible way. We apply it to planetary systems undergoing arbitrarily close encounters and highly eccentric orbits, but the potential applications are broader.…
Enlarged Krylov subspace methods and their s-step versions were introduced [7] in the aim of reducing communication when solving systems of linear equations Ax = b. These enlarged CG methods consist of enlarging the Krylov subspace by a…
Nowadays, many fields of study are have to deal with large and sparse data matrixes, but the most important issue is finding the inverse of these matrixes. Thankfully, Krylov subspace methods can be used in solving these types of problem.…
An error analysis of trigonometric integrators (or exponential integrators) applied to spatial semi-discretizations of semilinear wave equations with periodic boundary conditions in one space dimension is given. In particular, optimal…