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We consider a discrete-time model of continuous-time distributed optimization over dynamic directed-graphs (digraphs) with applications to distributed learning. Our optimization algorithm works over general strongly connected dynamic…
In this paper we consider various splitting schemes for unsteady problems containing the grad-div operator. The fully implicit discretization of such problems would yield at each time step a linear problem that couples all components of the…
Stochastic dynamical systems are fundamental in state estimation, system identification and control. System models are often provided in continuous time, while a major part of the applied theory is developed for discrete-time systems.…
We present a methodology for predictable and safe covariance steering control of uncertain nonlinear stochastic processes. The systems under consideration are subject to general uncertainties, which include unbounded random disturbances…
This paper presents a numerical method based on the variational quantum algorithm to solve potential and Stokes flow problems. In this method, the governing equations for potential and Stokes flows can be respectively written in the form of…
The Swift-Hohenberg equation as a central nonlinear model in modern physics has a gradient flow structure. Here we introduce fully discrete discontinuous Galerkin (DG) schemes for a class of fourth order gradient flow problems, including…
Combining the characteristic method and the local discontinuous Galerkin method with carefully constructing numerical fluxes, we design the variational formulations for the time-dependent convection-dominated Navier-Stokes equations in…
We present a general framework of designing efficient dynamic approximate algorithms for optimization on undirected graphs. In particular, we develop a technique that, given any problem that admits a certain notion of vertex sparsifiers,…
Identification of the parameters of stable linear dynamical systems is a well-studied problem in the literature, both in the low and high-dimensional settings. However, there are hardly any results for the unstable case, especially…
A computational tool for coarse-graining nonlinear systems of ordinary differential equations in time is discussed. Three illustrative model examples are worked out that demonstrate the range of capability of the method. This includes the…
In this paper, we propose a unified numerical framework for the time-dependent incompressible Navier--Stokes equation which yields the $H^1$-, $H(\text{div})$-conforming, and discontinuous Galerkin methods with the use of different viscous…
This work aims to construct and analyze a discontinuous Galerkin method on polytopal grids (PolydG) to solve the pseudo-stress formulation of the unsteady Stokes problem. The pseudo-stress variable is introduced due to the growing interest…
The ever increasing complexity of real-time control systems results in significant deviations in the timing of sensing and actuation, which may lead to degraded performance or even instability. In this paper we present a method to analyze…
This study proposes an algorithm for modeling compressible flows in spherical shells in nearly incompressible and weakly compressible regimes based on an implicit direction splitting approach. The method retains theoretically expected…
We propose a new simple and explicit numerical scheme for time-homogeneous stochastic differential equations. The scheme is based on sampling increments at each time step from a skew-symmetric probability distribution, with the level of…
In this paper, we introduce a fourth-order accurate finite element method for incompressible variable density flow. The method is implicit in time and constructed with the Taylor series technique, and uses standard high-order Lagrange basis…
We study the problem of estimating a sequence of evolving probability distributions from historical data, where the underlying distribution changes over time in a nonstationary and nonparametric manner. To capture gradual changes, we…
This work establishes a rigorous connection between stability properties of discrete-time algorithms (DTAs) and corresponding continuous-time dynamical systems derived through $ O(s^r) $-resolution ordinary differential equations (ODEs). We…
Optimization algorithms have a rich and fundamental relationship with ordinary differential equations given by its continuous-time limit. When the cost function varies with time -- typically in response to a dynamically changing environment…
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…