Homogenization of nonlinear stochastic partial differential equations in a general ergodic environment

In this paper, we show that the concept of sigma-convergence associated to stochastic processes can tackle the homogenization of stochastic partial differential equations. In this regard, the homogenization problem for a stochastic nonlinear partial differential equation is studied. Using some deep compactness results such as the Prokhorov and Skorokhod theorems, we prove that the sequence of solutions of this problem converges in probability towards the solution of an equation of the same type. To proceed with, we use a suitable version of sigma-convergence method, the sigma-convergence for stochastic processes, which takes into account both the deterministic and random behaviours of the solutions of the problem. We apply the homogenization result to some concrete physical situations such as the periodicity, the almost periodicity, the weak almost periodicity, and others.Comment: To appear in: Stochastic Analysis and Application

On the generalized almost periodic homogenization of stochastic conservation laws with multiplicative noise

We consider the generalized almost periodic homogenization problem for two types of stochastic conservation laws with oscillatory coefficients and a multiplicative noise, namely, the nonlinear transport equation and the equation with a stiff oscillating external force. We use the notion of homogenization by noise-approximation, introduced here, which amounts to the homogenization of the equations with an approximate noise as well as showing that the solutions of the approximate equations with an artificial viscosity term converge, as the noise-approximation parameter goes to zero, to the solutions of the original equation with artificial viscosity, whose solutions are shown to converge to the solutions of the original equation as the viscosity parameter goes to zero. Besides, the homogenization limits of the approximate equations are themselves limits of a two-parameter sequence, when one of these parameters, representing viscosity, goes to zero, and whose counterpart limits, obtained when the other parameter, representing the noise-approximation, goes to zero, converge to a well determined limit which we call the homogenization limit by noise-approximation (b.n.a.). In both cases the multiplicative noise is prescribed so that the corresponding stochastic equation has special solutions that play the role of steady-state solutions in the deterministic case and are crucial elements in the homogenization analysis. As a byproduct, our prescription of the multiplicative noise provides a way to justify the noise perturbation of the corresponding deterministic equation

New Directions in Rough Path Theory (online meeting)

Rough path theory emerged as novel approach for dealing with interactions in complex random systems. It settled significant questions and provided an effective deterministic alternative to Itô calculus, itself a major contribution to 20$^{\mathrm{th}}$ century mathematics. Its impact has grown substantially in recent years: most prominently, rough paths ideas are at the core of Martin Hairer's Fields Medal-winning work on regularity structures, but there are also original and successful applications in other areas. The workshop focused on three areas that have been strongly influenced by the core ideas in rough path theory and which have witnessed considerable activity over the past few years: applications to data science, algebraic aspects and connections with stochastic analysis

A high order solver for signature kernels

Signature kernels are at the core of several machine learning algorithms for analysing multivariate time series. The kernel of two bounded variation paths (such as piecewise linear interpolations of time series data) is typically computed by solving a Goursat problem for a hyperbolic partial differential equation (PDE) in two independent time variables. However, this approach becomes considerably less practical for highly oscillatory input paths, as they have to be resolved at a fine enough scale to accurately recover their signature kernel, resulting in significant time and memory complexities. To mitigate this issue, we first show that the signature kernel of a broader class of paths, known as smooth rough paths, also satisfies a PDE, albeit in the form of a system of coupled equations.We then use this result to introduce new algorithms for the numerical approximation of signature kernels. As bounded variation paths (and more generally geometric p-rough paths) can be approximated by piecewise smooth rough paths, one can replace the PDE with rapidly varying coefficients in the original Goursat problem by an explicit system of coupled equations with piecewise constant coefficients derived from the first few iterated integrals of the original input paths. While this approach requires solving more equations, they do not require looking back at the complex and fine structure of the initial paths, which significantly reduces the computational complexity associated with the analysis of highly oscillatory time series

Disordered Systems: Random Schrödinger Operators and Random Matrices

[no abstract available

Internationales Kolloquium über Anwendungen der Informatik und Mathematik in Architektur und Bauwesen : 04. bis 06.07. 2012, Bauhaus-Universität Weimar

The 19th International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering will be held at the Bauhaus University Weimar from 4th till 6th July 2012. Architects, computer scientists, mathematicians, and engineers from all over the world will meet in Weimar for an interdisciplinary exchange of experiences, to report on their results in research, development and practice and to discuss. The conference covers a broad range of research areas: numerical analysis, function theoretic methods, partial differential equations, continuum mechanics, engineering applications, coupled problems, computer sciences, and related topics. Several plenary lectures in aforementioned areas will take place during the conference. We invite architects, engineers, designers, computer scientists, mathematicians, planners, project managers, and software developers from business, science and research to participate in the conference

Internationales Kolloquium über Anwendungen der Informatik und Mathematik in Architektur und Bauwesen : 04. bis 06.07. 2012, Bauhaus-Universität Weimar

The 19th International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering will be held at the Bauhaus University Weimar from 4th till 6th July 2012. Architects, computer scientists, mathematicians, and engineers from all over the world will meet in Weimar for an interdisciplinary exchange of experiences, to report on their results in research, development and practice and to discuss. The conference covers a broad range of research areas: numerical analysis, function theoretic methods, partial differential equations, continuum mechanics, engineering applications, coupled problems, computer sciences, and related topics. Several plenary lectures in aforementioned areas will take place during the conference. We invite architects, engineers, designers, computer scientists, mathematicians, planners, project managers, and software developers from business, science and research to participate in the conference