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In this paper, we study temporal splitting algorithms for multiscale problems. The exact fine-grid spatial problems typically require some reduction in degrees of freedom. Multiscale algorithms are designed to represent the fine-scale…
Successive quadratic approximations, or second-order proximal methods, are useful for minimizing functions that are a sum of a smooth part and a convex, possibly nonsmooth part that promotes regularization. Most analyses of iteration…
Symplectic integration methods based on operator splitting are well established in many branches of science. For Hamiltonian systems which split in more than two parts, symplectic methods of higher order have been studied in detail only for…
We describe a novel algorithm for solving general parametric (nonlinear) eigenvalue problems. Our method has two steps: first, high-accuracy solutions of non-parametric versions of the problem are gathered at some values of the parameters;…
Two-step predictor/corrector methods are provided to solve three classes of problems that present themselves as systems of ordinary differential equations (ODEs). In the first class, velocities are given from which displacements are to be…
An interior-point algorithm framework is proposed, analyzed, and tested for solving nonlinearly constrained continuous optimization problems. The main setting of interest is when the objective and constraint functions may be nonlinear…
In this paper, a new implicit-explicit local method with an arbitrary order is produced for stiff initial value problems. Here, a general method for one-step time integrations has been created, considering a direction free approach for…
We present a new time-stepping algorithm for nonlinear PDEs that exhibit scale separation in time. Our scheme combines asymptotic techniques (which are inexpensive but can have insufficient accuracy) with parallel-in-time methods (which,…
Higher-order nonlinear time-evolution equations have widespread applications in science and engineering, such as in solid mechanics, materials science, and fluid mechanics. This paper mainly studies a direct time-parallel algorithm for…
This paper is devoted to the study of an inertial accelerated primal-dual algorithm, which is based on a second-order differential system with time scaling, for solving a non-smooth convex optimization problem with linear equality…
The nonlinear gyrokinetic equations describe plasma turbulence in laboratory and astrophysical plasmas. To solve these equations, massively parallel codes have been developed and run on present-day supercomputers. This paper describes…
High-order implicit shock tracking is a new class of numerical methods to approximate solutions of conservation laws with non-smooth features. These methods align elements of the computational mesh with non-smooth features to represent them…
In this work we present a new simple but efficient scheme - Subsquares approach - for development of algorithms for enclosing the solution set of overdetermined interval linear systems. We are going to show two algorithms based on this…
We develop and analyze a highly efficient, second-order time-marching scheme for infinite-dimensional nonlinear geophysical fluid models, designed to accurately approximate invariant measures-that is, the stationary statistical properties…
We present an algorithm for the simulation of the exact real-time dynamics of classical many-body systems with discrete energy levels. In the same spirit of kinetic Monte Carlo methods, a stochastic solution of the master equation is found,…
A second order explicit one-step numerical method for the initial value problem of the general ordinary differential equation is proposed. It is obtained by natural modifications of the well-known leapfrog method, which is a second order,…
We show that, even for extremely stiff systems, explicit integration may compete in both accuracy and speed with implicit methods if algebraic methods are used to stabilize the numerical integration. The required stabilizing algebra depends…
This paper investigates a subgradient-based algorithm to solve the system identification problem for linear time-invariant systems with non-smooth objectives. This is essential for robust system identification in safety-critical…
This paper presents three main contributions to the field of multi-step system identification. First, drawing inspiration from Neural Network (NN) training, it introduces a tool for solving identification problems by leveraging first-order…
Temporal-Difference (TD) learning with nonlinear smooth function approximation for policy evaluation has achieved great success in modern reinforcement learning. It is shown that such a problem can be reformulated as a stochastic…