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Differential equations are important tools to portray dynamic problems, and are widely used in finance, engineering and biology. Here, multiple dynamic differential models were built innovatively, and discretized with the Runge-Kutta…
Classical and new numerical schemes are generated using evolutionary computing. Differential Evolution is used to find the coefficients of finite difference approximations of function derivatives, and of single and multi-step integration…
We study the convergence of the new family of mimetic finite difference schemes for linear diffusion problems recently proposed in [38]. In contrast to the conventional approach, the diffusion coefficient enters both the primary mimetic…
This paper illuminates the derivation, the applicability, and the efficiency of the Multiplicative Runge-Kutta Method, derived in the frame- work of geometric multiplicative calculus. The removal of the restrictions of geometric…
We study Runge-Kutta methods for rough differential equations which can be used to calculate solutions to stochastic differential equations driven by processes that are rougher than a Brownian motion. We use a Taylor series representation…
This work deals with two groups of spectral analysis results for matrices arising in fully implicit Runge-Kutta methods used for linear time-dependent partial differential equations. These were applied for different formulations of the same…
We study distributed stochastic convex optimization under the delayed gradient model where the server nodes perform parameter updates, while the worker nodes compute stochastic gradients. We discuss, analyze, and experiment with a setup…
We study a discrete-time random feature method for nonlinear, time-dependent partial differential equations. In contrast to continuous-time formulations that treat time as an additional input variable, the method advances the solution step…
This work studies the problem of distributed optimization in heterogeneous linear multi-agent systems. Instead of relying on a perfect communication network as in many existing distributed optimization approaches, we considered two…
We consider deep multivariate models for heterogeneous collections of random variables. In the context of computer vision, such collections may e.g. consist of images, segmentations, image attributes, and latent variables. When developing…
Convection-diffusion-reaction equations are a class of second-order partial differential equations widely used to model phenomena involving the change of concentration/population of one or more substances/species distributed in space.…
Some of recent developments, including recent results, ideas, techniques, and approaches, in the study of degenerate partial differential equations are surveyed and analyzed. Several examples of nonlinear degenerate, even mixed, partial…
Multirate integration is an increasingly relevant tool that enables scientists to simulate multiphysics systems. Existing multirate methods are designed for equations whose fast and slow variables can be linearly separated using additive or…
In this paper we discuss a framework for the polynomial approximation to the solution of initial value problems for differential equations. The framework, initially devised for the approximation of ordinary differential equations, is…
System performance for networks composed of interconnected subsystems can be increased if the traditionally separated subsystems are jointly optimized. Recently, parallel and distributed optimization methods have emerged as a powerful tool…
Phase transitions constitute fundamental mechanisms underlying abrupt or qualitative changes in the collective dynamics of interacting units across a wide range of natural and engineered systems. In dynamical networks, such transitions lead…
Although the theory of density evolution in maps and ordinary differential equations is well developed, the situation is far from satisfactory in continuous time systems with delay. This paper reviews some of the work that has been done…
One of main obstacles in verifying the energy dissipation laws of implicit-explicit Runge-Kutta (IERK) methods for phase field equations is to establish the uniform boundedness of stage solutions without the global Lipschitz continuity…
Semi-discrete Runge-Kutta schemes for nonlinear diffusion equations of parabolic type are analyzed. Conditions are determined under which the schemes dissipate the discrete entropy locally. The dissipation property is a consequence of the…
In this paper, stability analysis of time delay systems is considered based on decomposition of the systems to subsystems. The decomposition process needs matrices of these systems to be simultaneously block triangularize. We show that a…