Numerical Analysis Archives for Academic Year 2020

Deterministic numerical methods for the study of stochastic dynamics with small noise

When: Tue, September 3, 2019 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Masha Cameron (Department of Mathematics, University of Maryland, College Park) -
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 Lorenz’63 model perturbed by small white noise and to genetic switch models will be presented.

Bespoke Stochastic Galerkin Approximation of Nearly Incompressible Elasticity

When: Tue, September 10, 2019 - 3:30pm
Where: Kirwan Hall 3206
Speaker: David Silvester (Department of Mathematics, University of Manchester) -
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

Wave Propagation in Inhomogeneous Media: An introduction to Generalized Plane Waves

When: Tue, September 17, 2019 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Lise-Marie Imbert-Gerard (Department of Mathematics, University of Maryland, College Park) -
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.

Structure Exploiting Optimization Algorithms and Applications

When: Tue, September 24, 2019 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Harbir Antil (Department of Mathematical Sciences, George Mason University) -
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.

Low-rank tensor compression algorithms for enabling high-dimensional motion planning

When: Tue, October 15, 2019 - 3:30pm
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.

Spectrally Accurate Methods for Kinetic Electron Plasma Wave Dynamics

When: Wed, October 23, 2019 - 5:00pm
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.

Data-dependent performance modeling of linear solvers for sparse matrices

When: Tue, October 29, 2019 - 3:30pm
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.

Extreme event probability estimation by combining large deviation theory and PDE-constrained optimization, with application to tsunamis

When: Tue, November 5, 2019 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Georg Stadler (Courant Institute, New York) -
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).

Numerical modelling of lateral phase separation on evolving surfaces

When: Thu, November 7, 2019 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Maxim Olshanskii (Department of Mathematics, University of Houston) -
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.

Two applications of PDE constrained optimization

When: Tue, November 19, 2019 - 3:30pm
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.

Evolution of a Scalable 3D Helmholtz Solver with Geophysical Applications

When: Tue, December 3, 2019 - 3:30am
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.

Provable Quantum Advantages for Optimization and Machine Learning

When: Tue, January 28, 2020 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Xiaodi Wu (Department of Computer Science and Institute for Advanced Computer Studies, UMD ) -
Abstract: Recent developments in quantum computation, especially in the emerging topic of “quantum machine learning”, suggest that quantum algorithms might offer significant speed-ups for optimization and machine learning problems. In this talk, I will describe three recent developments on provable quantum advantages for optimization and machine learning from our group.

(1) The first one optimizes a general (dimension n) convex function over a convex body using O(n) quantum queries to oracles that evaluate the objective function and determine membership in the convex body; this gives a quadratic improvement over the best-known classical algorithm.

(2) The second one solves a special class of convex optimization problems, semidefinite programs (SDPs), where we design a quantum algorithm that solves n-dimensional SDPs with m constraints in O(\sqrt{n} +\sqrt{m}), whereas the state-of-the-art classical algorithms run in time at least linear in n and m.

(3) The third one is a sublinear quantum algorithm for training linear and kernel-based classifiers that runs in O(\sqrt{n}+\sqrt{d}) given n data points in R^d, whereas the state-of-the-art (and optimal) classical algorithm runs in O(n +d).

Based on results in arXiv:1809.01731v1 (QIP 2019), arXiv:1710.02581v2 (QIP 2019), and arXiv: 1904.02276 (ICML 2019).

Cells as living liquid crystals and the role of topological defects

When: Tue, February 4, 2020 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Francesca Serra (Department of Physics and Astronomy, Johns Hopkins University) -
Abstract: Certain types of living cells are elongated, they align with each other spontaneously and they achieve long-range orientational order. All these characteristics make them remarkably similar to liquid crystals. However, unlike traditional liquid crystals, cells can deform, move and multiply. In this “special” liquid crystal system we look at topological defects, i.e. the regions where the cells cannot align. We use topographical cues to guide the local orientation of the cells, and we characterize the arrangement of cells near topological defects. From our observations, we intend to extract relevant physical parameters and correlate them with the properties of the cell types.

Nonequilibrium thermodynamic principles and property preserving numerical approximations to PDEs --- A modeling guided approach

When: Tue, February 11, 2020 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Qi Wang (Department of Mathematics, University of South Carolina) -
Abstract: In principle, dynamical models including differential equation models for nonequilibrium phenomena should be derived following some nonequilibrium principles. The prevailent principle one uses today is the thermodynamical second law in the form of Clausius-Duhem inequality or more generally the linear response theory due to Onsager, which we name it the Onsager principle. The Onsager principle is not a physical law, rather it provides a paradigm to derive thermodynamically consistent models satisfying the second law of thermodynamics. In this talk, I will discuss how thermodynamical and hydrodynamical models can be derived using the Onsager principle and what mathematical structure/property the models have when derived this way. We then argue that hydrodynamic models are nothing but constrained thermodynamical models subject to a set of physically motivated constraints such as mass conservation, momentum conservation, energy conservation, etc. Some of these constraints can be effectively cast into gradient flows (relaxation dynamics) so that these hydrodynamical models can be recast into a generalized gradient flow. For these models, we will discuss the energy quadratization strategy to devise energy production/dissipation rate preserving numerical algorithms. Additional strategies to preserve energy production rate through projection or Lagrange multipler methods will be mentioned as well. Some examples will be provided to illustrate the procedures.

Structure preservation properties of the active flux scheme for multi-dimensional hyperbolic equations

When: Tue, February 18, 2020 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Wasilij Barsukow (University of Zurich, Switzerland) -
Abstract: The Active Flux scheme is a finite volume scheme with additional, pointwise degrees of freedom distributed along the cell boundary. For their time evolution the data is reconstructed continuously and an initial value problem is solved (exactly or approximately). The thus obtained values along the boundary yield the intercell flux required for the update of the average. The evolution of the pointwise degrees of freedom ensures the correct direction of information propagation and provides stability. The Active Flux scheme is third order accurate. I will show that this scheme is stationarity preserving and low Mach number compliant for multi-dimensional linear acoustics without the need for any fix. I will also show progress towards generalizing this for the compressible inviscid multi-dimensional Euler equations.

Convergent finite element methods in micromagnetics

When: Tue, February 25, 2020 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Michele Ruggeri (Institute for Analysis and Scientific Computing, TU Vienna, Austria) -
Abstract: Micromagnetics is the study of magnetic processes on a submicrometer length scale.
The magnetic state of a ferromagnetic body is described in terms of a unit-length
vector field, usually referred to as magnetization. Admissible magnetization
equilibrium configurations are those which minimize an appropriate energy functional.
A well-established model for the dynamics of nonequilibrium magnetization configuration
is the Landau-Lifshitz-Gilbert equation (LLG). The numerical approximation of LLG poses
several challenges: strong nonlinearities, a nonconvex pointwise constraint, an intrinsic
energy law combining conservative and dissipative effects, and the presence of nonlocal
field contributions. In this talk, we discuss numerical schemes for LLG, based on
lowest-order finite elements in space, that are proven to be (unconditionally) convergent
towards a weak solution of the problem.

Reynolds-robust preconditioning for the incompressible Navier-Stokes equations

When: Tue, March 3, 2020 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Florian Wechsung (Courant Institute) -
Abstract: We consider finite element approximations of the stationary incompressible Navier-Stokes equations. An ideal preconditioner for the linear systems arising from these equations yields convergence that is algorithmically optimal and parameter robust, i.e. the number of Krylov iterations required to solve the linear system to a given accuracy does not grow substantially as the mesh or problem parameters are changed.

It has proven challenging to develop solvers that exhibit both properties; matrix factorisations are robust to Reynolds number but scale badly with dof count, whereas Schur complement based algorithms such as PCD and LSC scale linearly in the dof count but their performance decreases as the Reynolds number is increased

Building on the ideas of Schöberl, Benzi, and Olshanskii, we present an augmented Lagrangian based preconditioner with linear complexity and iteration counts that only grow mildly with respect to the Reynolds number. The key ingredient is a tailored multigrid scheme for the exactly divergence-free Scott-Vogelius discretisation consisting of custom smoothing and prolongation operators.

This work has been done in collaboration with Patrick Farrell, Lawrence Mitchell, and Ridgway Scott.

Convex relaxation approaches for strictly correlated density functional theory

When: Tue, March 31, 2020 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Lexing Ying (Department of Mathematics, Stanford University) -
Abstract: In this talk, we introduce methods from convex optimization to solve the multi-marginal
transport-type problems that arise in the context of density functional theory. Convex relaxations are used to provide outer approximation to the set of N-representable 2-marginals and 3-marginals, which in turn provide lower bounds to the energy. We further propose rounding schemes to obtain upper bounds to the energy.

Solving inverse problems with machine learning

When: Wed, April 1, 2020 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Lexing Ying (Department of Mathematics, Stanford University) -
Abstract: This talk is about some recent progress on solving inverse problems using deep learning. Compared to traditional machine learning problems, inverse problems are often limited by the size of the training data set. We show how to overcome this issue by incorporating mathematical analysis and physics into the design of neural network architectures. We first describe neural network representations of pseudodifferential operators and Fourier integral operators. We then continue to discuss applications including electric impedance tomography, optical tomography, inverse acoustic/EM scattering, seismic imaging, and travel-time tomography.

Geometric stochastic PDEs

When: Tue, April 21, 2020 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Martin Hairer (Department of Mathematics, Imperial College London, UK) -
We give a quick review of Parisi and Wu's stochastic quantisation procedure and apply it to the 1D non-linear sigma model as well as the Yang-Mills model. We then review a number of recent results on the resulting equations.

Taming Infinities

When: Wed, April 22, 2020 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Martin Hairer (Department of Mathematics, Imperial College London, UK) -
Abstract: Some physical and mathematical theories have the unfortunate feature that if one takes them at face value, many quantities of interest appear to be infinite! What's worse, this doesn't just happen for some exotic theories, but in the standard theories describing some of the most fundamental aspects of nature. Various techniques, usually going under the common name of “renormalisation” have been developed over the years to address this, allowing mathematicians and physicists to tame these infinities. We will tip our toes into some of the conceptual and mathematical aspects of these techniques and we will see how they have recently been used in probability theory to study equations whose meaning was not even clear until now.