MATHICSE Technical Report : Random time step probabilistic methods for uncertainty quantification in chaotic and geometric numerical integration
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We consider a method to efficiently evaluate in a real-time context an output based on the numerical solution of a partial differential equation depending on a large number of parameters. We state a result allowing to improve the computational performance ...
The 3DVAR filter is prototypical of methods used to combine observed data with a dynamical system, online, in order to improve estimation of the state of the system. Such methods are used for high dimensional data assimilation problems, such as those arisi ...
We present a novel, cell-local shock detector for use with discontinuous Galerkin (DG) methods. The output of this detector is a reliably scaled, element-wise smoothness estimate which is suited as a control input to a shock capture mechanism. Using an art ...
In this project, we study and compare two methods to solve stochastic ordinary differential equations. The first is the Monte Carlo method and the second uses Polynomial Chaos. In the first part, we will solve a stochastic ordinary differential equation by ...
The numerical analysis of a dynamic constrained optimization problem is presented. It consists of a global minimization problem that is coupled with a system of ordinary differential equations. The activation and the deactivation of inequality constraints ...
Multiscale differential equations arise in the modeling of many important problems in the science and engineering. Numerical solvers for such problems have been extensively studied in the deterministic case. Here, we discuss numerical methods for (mean-squ ...
The bounded confidence model of opinion dynamics, introduced by Deffuant et al, is a stochastic model for the evolution of continuous-valued opinions within a finite group of peers. We prove that, as time goes to infinity, the opinions evolve globally into ...
We discuss adaptive sparse grid algorithms for stochastic differential equations with a particular focus on applications to electromagnetic scattering by structures with holes of uncertain size, location, and quantity. Stochastic collocation (SC) methods a ...
We consider the long time behavior of the semidiscrete scheme for the Perona-Malik equation in one dimension. We prove that approximated solutions converge, in a slow time scale, to solutions of a limit problem. This limit problem evolves piecewise constan ...
B-series are a fundamental tool in practical and theoretical aspects of numerical integrators for ordinary differential equations. A composition law for B-series permits an elegant derivation of order conditions, and a substitution law gives much insight i ...
