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This paper is concerned with the development and testing of advanced time-stepping methods suited for the integration of time-accurate, real-world applications of computational fluid dynamics (CFD). The performance of several time…
We present a new analytical and numerical framework for solution of Partial Differential Equations (PDEs) that is based on an exact transformation that moves the boundary constraints into the dynamics of the corresponding governing…
Physical models with uncertain inputs are commonly represented as parametric partial differential equations (PDEs). That is, PDEs with inputs that are expressed as functions of parameters with an associated probability distribution.…
Three numerical algorithms are proposed to solve the time-dependent elastodynamic equations in elastic solids. All algorithms are based on approximating the solution of the equations, which can be written as a matrix exponential. By…
In this paper we develop an adaptive procedure for the numerical solution of semilinear parabolic problems, with possible singular perturbations. Our approach combines a linearization technique using Newton's method with an adaptive…
In memory-constrained algorithms we have read-only access to the input, and the number of additional variables is limited. In this paper we introduce the compressed stack technique, a method that allows to transform algorithms whose space…
In this article, we develop a systematic approach of the invariant subspace method combined with variable transformation to find the generalized separable exact solutions of the nonlinear two-component system of time-fractional PDEs…
Time-dependent Partial Differential Equations with given initial conditions are considered in this paper. New differentiation techniques of the unknown solution with respect to time variable are proposed. It is shown that the proposed…
Time-spectral solution of ordinary and partial differential equations is often regarded as an inefficient approach. The associated extension of the time domain, as compared to finite difference methods, is believed to result in…
In this paper, we consider a formulation of nonlinear constrained optimization problems. We reformulate it as a time-varying optimization using continuous-time parametric functions and derive a dynamical system for tracking the optimal…
We introduce a family of numerical algorithms for the solution of linear system in higher dimensions with the matrix and right hand side given and the solution sought in the tensor train format. The proposed methods are rank--adaptive and…
We present a framework for recovering/approximating unknown time-dependent partial differential equation (PDE) using its solution data. Instead of identifying the terms in the underlying PDE, we seek to approximate the evolution operator of…
The tracer equations are part of the primitive equations used in ocean modeling and describe the transport of tracers, such as temperature, salinity or chemicals, in the ocean. Depending on the number of tracers considered, several…
This paper introduces a novel Transformed Primal-Dual with variable-metric/preconditioner (TPDv) algorithm, designed to efficiently solve affine constrained optimization problems common in nonlinear partial differential equations (PDEs).…
Time-dependent Maxwell's equations govern electromagnetics. Under certain conditions, we can rewrite these equations into a partial differential equation of second order, which in this case is the vectorial wave equation. For the vectorial…
Ordinary and partial differential equations (ODEs/PDEs) play a paramount role in analyzing and simulating complex dynamic processes across all corners of science and engineering. In recent years machine learning tools are aspiring to…
Matrix equations are omnipresent in (numerical) linear algebra and systems theory. Especially in model order reduction (MOR) they play a key role in many balancing based reduction methods for linear dynamical systems. When these systems…
Many econometric analyses involve spatio--temporal data. A considerable amount of literature has addressed spatio--temporal models, with Spatial Dynamic Panel Data (SDPD) being widely investigated and applied. In real data applications,…
This report discusses two new ideas for using perturbation methods to solve the time-independent Schr\"odinger equation. The first concept begins with rewriting the perturbation equations in a form that is closely related to matrix…
We develop a numerical method to reconstruct systems of ordinary differential equations (ODEs) from time series data without {\it a priori} knowledge of the underlying ODEs using sparse basis learning and sparse function reconstruction. We…