Where: Kirwan Hall 3206

Speaker: Masha Cameron (Department of Mathematics, University of Maryland, College Park) - https://www.math.umd.edu/~mariakc

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 Lorenzâ€™63 model perturbed by small white noise and to genetic switch models will be presented.

Where: Kirwan Hall 3206

Speaker: David Silvester (Department of Mathematics, University of Manchester) - https://personalpages.manchester.ac.uk/staff/david.silvester/

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

Where: Kirwan Hall 3206

Speaker: Lise-Marie Imbert-Gerard (Department of Mathematics, University of Maryland, College Park) - https://www-math.umd.edu/people/faculty/item/1314-limbert.html

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.

Where: Kirwan Hall 3206

Speaker: Harbir Antil (Department of Mathematical Sciences, George Mason University) - http://math.gmu.edu/~hantil/

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.

Where: Kirwan Hall 3206

Speaker: Alex Gorodetsky (Aerospace Engineering University of Michigan Ann Arbor, Michigan) -

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.

Where: Kirwan Hall 3206

Speaker: Jon Wilkening (Department of Mathematics, University of Berkeley) -

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.

Where: Kirwan Hall 3206

Speaker: Abhinav Bhatele (Computer Science Department University of Maryland, College Park) -

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.

Where: Kirwan Hall 3206

Speaker: Georg Stadler (Courant Institute, New York) - https://math.nyu.edu/~stadler/

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).

Where: Kirwan Hall 3206

Speaker: Maxim Olshanskii (Department of Mathematics, University of Houston) - https://www.math.uh.edu/~molshan/

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.

Where: Kirwan Hall 3206

Speaker: Tom Brown (George Mason University, Fairfax, Virginia) -

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.

Where: Kirwan Hall 3206

Speaker: Russell E. Hewett (Mathematics and Computational Modeling and Data Analytics) -

Abstract: While it is theoretically and computationally advantageous to pose the inverse

problem of subsurface recovery in the frequency domain, efficiently solving

time-harmonic wave equations, the associated forward problem, in parallel,

remains challenging for large 3D problems at high-frequency and in

heterogeneous media. In this talk, I present an overview of our development

of parallel implementations of effective solvers for this regime, built using

the method of polarized traces. In particular, I will focus on the challenges

of extending the already efficient 2D method to 3D while maintaining parallel

scalability. I will show some recent results on applications to highly

heterogeneous media in 3D. I will also introduce our variant of polarized

traces, the L-sweeps method, which allows us to solve the 3D Helmholtz

equation in parallel with sub-linear scaling.