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.
Abstract: Autonomous systems that operate in real time pose unique challenges for reliable, accurate, and optimal uncertainty quantification and control. Many algorithms with strong optimality guarantees encounter the curse-of-dimensionality; their computational expense grows exponentially with the size of the state space. In this talk, we describe new developments in low-rank multilinear algebra that enable foundational algorithms within autonomy for high-dimensional systems. We demonstrate how compression techniques based on a continuous extension of tensor decompositions can be used to solve Markov decisions processes (MDPs) that arise in systems described by stochastic differential equations. The resulting dynamic programming algorithms scale polynomially with dimension with guaranteed convergence. Applications to stochastic optimal control, differential games, and linear temporal logic are discussed. Experimental results are shown for an agile quadcopter system, where we achieve 7 orders of magnitude compression of a discretized space with $10^12$ states.
Abstract: We present two numerical methods for computing solutions of the Vlasov-Fokker-Planck-Poisson equations that are spectrally accurate in all three variables (time, space and velocity).
The first is a Chebyshev collocation method for solving the Volterra/Penrose integral equation for the space-time evolution of the plasma density in the linearized, collisionless problem. The distribution function is represented in physical space using Fourier modes and in velocity space using a reconstruction formula that can be computed rapidly at any desired set of velocities once the time evolution of the plasma density is known. We also show how to use this framework to efficiently represent the velocity distribution in Case-van Kampen normal modes.
The second is an arbitrary-order exponential time differencing scheme that makes use of the Duhamel principle to fold in the effects of collisions and nonlinearity. We investigate the emergence of a continuous spectrum in the collisionless limit, reaching the opposite conclusion to Ng, Bhattacharjee and Skiff in their 1999 PRL paper. In our approach, only self-adjoint operators are diagonalized, which avoids the extreme ill-conditioning that arises in this problem when the full non-selfadjoint operator is truncated in a Hermite basis and diagonalized directly.
With the two methods, we resolve the effects of filamentation, temporal echoes, Landau damping, collisional damping, and unstable background electron velocity distribution functions while maintaining arbitrarily high accuracy.
Abstract: Performance of scientific codes is increasingly dependent on the input problem,
its data representation and the underlying hardware with the increase in code
and architectural complexity. This makes the task of identifying the fastest
algorithm for solving a problem more challenging. In this talk, I will focus on
modeling the performance of numerical libraries used to solve a sparse linear
system. We use machine learning to develop data-driven models of performance of linear solver implementations. These models can be used by a novice user to
identify the fastest preconditioner and solver for a given input matrix. We use a
variety of features that represent the matrix structure, numerical properties
of the matrix and the underlying mesh or input problem as input to the model.
We model the performance of nine linear solvers and thirteen preconditioners
available in Trilinos using 1240 sparse matrices obtained from two different
sources. Our prediction models perform significantly better than a blind
classifier and black-box SVM and k-NN classifiers.
Abstract: Tsunami waves are caused by a sudden change of ocean depth
(bathymetry) after an earthquake below the ocean floor. Since large
tsunami waves are extreme events, they correspond to the tail part of
a probability distribution, whose exploration would require
impractically many samples of a Monte Carlo method. We propose an
alternative method to estimate extreme probabilities using large
deviation theory, which relates the probabilities of extreme events to
the solutions of a one-parameter family of optimization problems. To
model tsunami waves, we use the shallow water equations, which thus
appear as PDE-constraints in this optimization problem. The
optimization objective includes a term that measures how extreme the
event is, and a term corresponding to the likelihood of bathymetry
changes, which are modeled as a Gaussian random field. Preliminary
numerical results with the 1D inviscid shallow water equation are
presented. This is joint work with Shanyin Tong and Eric
Vanden-Eijnden (both NYU).
Abstract: We discuss a model of lateral phase separation in a two-component thin material layer, a prototypical problem for understanding spinodal decomposition and pattern formation observed in biological membranes, e.g., lipid bilayers. The modelling part leads to a fourth order nonlinear PDE that can be seen as the Cahn-Hilliard equation posed on a time-dependent surface. Elementary tangential calculus and the embedding of the surface in R^3 are used to formulate the model, thereby facilitating the development of a fully Eulerian discretization method to solve the problem numerically. We discuss a numerical approach based on geometrically unfitted finite element spaces. The method avoids triangulation of the surface and uses a surface-independent ambient mesh to discretize the equation, and so the method is capable to handle implicitly defined geometry and surfaces undergoing topological transitions. The talk will be illustrated with animated computations. We discuss a model of lateral phase separation in a two-component thin material layer, a prototypical problem for understanding spinodal decomposition and pattern formation observed in biological membranes, e.g., lipid bilayers. The modelling part leads to a fourth order nonlinear PDE that can be seen as the Cahn-Hilliard equation posed on a time-dependent surface. Elementary tangential calculus and the embedding of the surface in $R^3$ are used to formulate the model, thereby facilitating the development of a fully Eulerian discretization method to solve the problem numerically. We discuss a numerical approach based on geometrically unfitted finite element spaces. The method avoids triangulation of the surface and uses a surface-independent ambient mesh to discretize the equation, and so the method is capable to handle implicitly defined geometry and surfaces undergoing topological transitions. The talk will be illustrated with animated computations.
Abstract: Optimization problems with partial differential equations (PDEs) as
constraints is known as PDE constrained optimization. In this talk, we
will discuss an abstract formulation of the problem as well as methods
for solving such problems. We then present two specific problems. One
application involves elastic waves propagating through a piezoelectric
solid where the PDE constraints take the form of a coupled PDE
system.The other application involves fractional (nonlocal) PDE
constraints, which have various applications including image denoising.
4176 Campus Drive - William E. Kirwan Hall
College Park, MD 20742-4015
P: 301.405.5047 | F: 301.314.0827