相关论文: Error Inhibiting Block One-Step Schemes for Ordina…

Efficient high order numerical methods for evolving the solution of an ordinary differential equation are widely used. The popular Runge--Kutta methods, linear multi-step methods, and more broadly general linear methods, all have a global…

High order implicit-explicit (IMEX) methods are often desired when evolving the solution of an ordinary differential equation that has a stiff part that is linear and a non-stiff part that is nonlinear. This situation often arises in…

High order methods are often desired for the evolution of ordinary differential equations, in particular those arising from the semi-discretization of partial differential equations. In prior work in we investigated the interplay between…

Modern large-scale statistical models require to estimate thousands to millions of parameters. This is often accomplished by iterative algorithms such as gradient descent, projected gradient descent or their accelerated versions. What are…

This paper investigates online algorithms for smooth time-varying optimization problems, focusing first on methods with constant step-size, momentum, and extrapolation-length. Assuming strong convexity, precise results for the tracking…

The filtering-clustering models, including trend filtering and convex clustering, have become an important source of ideas and modeling tools in machine learning and related fields. The statistical guarantee of optimal solutions in these…

This study introduces new time-stepping strategies with built-in global error estimators. The new methods propagate the defect along with the numerical solution much like solving for the correction or Zadunaisky's procedure; however, the…

In this work, we determine the full expression for the global truncation error of hyperbolic partial differential equations (PDEs). In particular, we use theoretical analysis and symbolic algebra to find exact expressions for the…

Error bound analysis, which estimates the distance of a point to the solution set of an optimization problem using the optimality residual, is a powerful tool for the analysis of first-order optimization algorithms. In this paper, we use…

Splitting methods constitute a widely used class of numerical integrators for ordinary and partial differential equations, particularly well suited to problems that can be decomposed into simpler subproblems. High-order splitting schemes…

In this paper, we study a class of deterministically constrained stochastic optimization problems. Existing methods typically aim to find an $\epsilon$-stochastic stationary point, where the expected violations of both constraints and…

We develop a one step matrix method in order to obtain approximate solutions of first order systems and non-linear ordinary differential equations, reducible to first order systems. We find a sequence of such solutions that converge to the…

Finite Difference (FD) schemes are widely used in science and engineering for approximating solutions of partial differential equations (PDEs). Error analysis of FD schemes relies on estimating the truncation error at each time step. This…

In this paper an asymptotic expansion of the global error on the stepsize for partitioned linear multistep methods is proved. This provides a tool to analyse the behaviour of these integrators with respect to error growth with time and…

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…

Numerical analysis for linear constant-coefficients Finite Difference schemes was developed approximately fifty years ago. It relies on the assumption of scheme stability and in particular -- for the $L^2$ setting -- on the absence of…

We propose a new family of multilevel methods for unconstrained minimization. The resulting strategies are multilevel extensions of high-order optimization methods based on q-order Taylor models (with q >= 1) that have been recently…

We report on a novel algorithm for controlling global error in a step-by-step (stepwise) sense, in the numerical solution of a scalar, autonomous, nonstiff or weakly stiff problem. The algorithm exploits the remainder term of a Taylor…

To facilitate the numerical analysis of particle methods, we derive truncation error estimates for the approximate operators in a generalized particle method. Here, a generalized particle method is defined as a meshfree numerical method…

We consider a class of stochastic gradient optimization schemes. Assuming that the objective function is strongly convex, we prove weak error estimates which are uniform in time for the error between the solution of the numerical scheme,…