Abstract: Nongradient Stochastic Differential Equations with small noise often arise in modeling biological and ecological systems.
An effective description of the dynamics of such systems can be given by comparing the stability of different attractors, finding a low-dimensional manifold to which the dynamics is virtually restricted (if any), finding transition rates between different attractors and the maximum likelihood transition paths. Addressing these questions by means of direct simulations may be difficult or impossible due to long waiting times. Alternatively, one can use asymptotic analysis tools for the vanishing noise limit offered by the Large Deviation Theory (Freidlin and Wentzell, 1970s). The key function of the Large Deviation Theory is the quasi-potential that is somewhat analogous to the potential for gradient systems. It gives estimates for transition rates, transition paths, and the invariant probability measure.
In this talk, I will introduce a family of Dijkstra-like Ordered Line Integral Methods (OLIMs) for computing the quasi-potential on 2D and 3D meshes. A number of technical innovations allowed us to make them accurate and fast. I will demonstrate what one can find out about stochastic systems once the quasi-potential is computed. Application to the Lorenzâ63 model perturbed by small white noise and to genetic switch models will be presented.
Abstract: We discuss the key role that bespoke linear algebra plays in modelling PDEs with random coefficients using stochastic Galerkin approximation methods. As a specific example, we consider nearly incompressible linear elasticity problems with an uncertain spatially varying Young's modulus. The uncertainty is modelled with a finite set of parameters with prescribed probability distribution. We introduce a novel three-field mixed variational formulation of the PDE model and focus on the efficient solution of the associated high-dimensional indefinite linear system of equations. Eigenvalue bounds for the preconditioned system are established and shown to be independent of the discretisation parameters and the Poisson ratio. If time permits we will also discuss the solution of poroelasticity problems modelled by Biot consolidation with uncertain coefficients.
This is joint work with Arbaz Khan and Catherine Powell
Abstract: Trefftz methods rely, in broad terms, on the idea of approximating
solutions to PDEs using basis functions which are exact solutions of the
Partial Differential Equation (PDE), making explicit use of information
about the ambient medium. But wave propagation problems in inhomogeneous
media is modeled by PDEs with variable coefficients, and in general no
exact solutions are available. Generalized Plane Waves (GPWs) are
functions that have been introduced, in the case of the Helmholtz
equation with variable coefficients, to address this problem: they are
not exact solutions to the PDE but are instead constructed locally as
high order approximate solutions. We will discuss the origin, the
construction, and the properties of GPWs. The construction process
introduces a consistency error, requiring a specific analysis.
Abstract: We present a novel algorithm to handle both equality and inequality constraints in
infinite dimensional optimization problems. The inequality constraints are tackled
via a nonstandard penalty. On the other hand, the equality constraints are handled
using trust region methods. The latter permits inexact PDE solves. As applications,
we consider PDE constrained optimization (PDECO) problems with contact type
constraints and topology optimization problems.
We will also introduce novel optimal control concepts within the realm of PDECO
problems with fractional/nonlocal PDEs as constraints and discuss their applications
in geophysics and imaging sciences. We will further illustrate the role of fractional
operators as a regularizer in machine learning.
We conclude this talk by introducing a general framework based on Gibbs posterior
to update the belief distributions for inverse problems governed by PDEs. Contrary to
traditional Bayesian analysis, noise model is not assumed to be known.
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