Baylor Applied Math Seminar

Title: Polynomial Preconditioning for Indefinite Matrices

Abstract: Indefinite spectra occur for operators in many applications, for example the wave equation.  Indefinite problems can be very difficult for solving the associated linear equations using Krylov iterative methods.  We investigate adding polynomial preconditioning for such problems and show it can give tremendous improvement.  Several difficulties can arise in polynomial preconditioning for indefinite matrices.  These are mentioned along with some algorithmic solutions. 

Title: "Recent Developments in the PDE Theory of Solutions in Poro-elasticity"

Abstract: Poro-elastic systems---bulk 3D elasticity as well as 2D plate models---have received recent attention in their applications to biological systems. As most biological tissues are both porous and elastic, there is immense applicability in studying Biot systems for which the permeability depends on the local fluid content. This translates to a quasilinear elliptic-parabolic (or hyperbolic-parabolic) system, with solution-dependent permeability. In the quasi-static case, the system of interest can be rewritten as a single implicit degenerate evolution equation, for which a general theory has been developed. In this talk, we will present an overview of solution methods for such implicit, degenerate evolution equations, as well as describe recent well-posedness of weak solutions for 2D and 3D Biot systems. We shall focus on cases where coefficients may be time-dependent. It is of particular note that even the linear Biot problem with time-dependent permeability---upon which fixed-point and discretization approaches are based---exhibits unexpected and challenging issues for weak solutions. Time-permitting, we will discuss recent work on the coupling of Biot dynamics to a free Stokes flow.

Title: Differential Game Models of Optimal Debt Management

Abstract: In this talk, I will present recent results on game theoretical formulation of optimal debt management problems in an infinite time horizon with exponential discount, modeled as a noncooperative interaction between a borrower and a pool of risk-neutral lenders. Here, the yearly income of the borrower is governed by a stochastic process and bankruptcy instantly occurs when the debt-to-income ratio reaches a threshold. Since the borrower may go bankrupt in finite time, the risk-neutral lenders will charge a higher interest rate in order to compensate for this possible loss of their investment. Thus, a “solution” must be understood as a Nash equilibrium, where the strategy implemented by the borrower represents the best reply to the strategy adopted by the lenders, and conversely. This leads to highly nonstandard optimization processes.

Title: Composite finite elements for linear elasticity 

Abstract: 

In this talk, we discuss construction of finite element for linear elasticity problems.

By using composite shape functions which are piecewise polynomials on subtetrahedra, simple finite elements can be  constructed.

Title: GPT-PINN: Generative Pre-Trained Physics-Informed Neural Networks toward non-intrusive Meta-learning of parametric PDEs

Abstract:  Physics-Informed Neural Network (PINN) has proven itself a powerful tool to obtain the numerical solutions of nonlinear partial differential equations (PDEs) leveraging the expressivity of deep neural networks and the computing power of modern heterogeneous hardware. However, its training is still time-consuming, especially in the multi-query and real-time simulation settings, and its parameterization often overly excessive.

In this talk, we present the recently proposed Generative Pre-Trained PINN (GPT-PINN). It mitigates both challenges in the setting of parametric PDEs. GPT-PINN represents a brand-new meta-learning paradigm for parametric systems. As a network of networks, its outer-/meta-network is hyper-reduced with only one hidden layer having significantly reduced number of neurons. Moreover, its activation function at each hidden neuron is a (full) PINN pre-trained at a judiciously selected system configuration. The meta-network adaptively “learns” the parametric dependence of the system and “grows” this hidden layer one neuron at a time. In the end, by encompassing a very small number of networks trained at this set of adaptively-selected parameter values, the meta-network is capable of generating surrogate solutions for the parametric system across the entire parameter domain accurately and efficiently. Time permitting, we will discuss the Transformed GPT-PINN, TGPT-PINN, which achieves nonlinear model reduction via the addition of a transformation layer before the pre-trained PINN layer.

Title: Equations on Wasserstein space and applications

Abstract: The purpose of this talk is to give an overview of recent work involving differential equations posed on spaces of probability measures and their use in analyzing controlled multi-agent systems. The study of such systems has seen increased interest in recent years, due to their ubiquity in applications coming from macroeconomics, social behavior, and telecommunications. When the number of agents becomes large, the model can be formally replaced by one involving a mean-field description of the population, analogously to similar models in statistical physics. Justifying this continuum limit is often nontrivial and is sensitive to the type of stochastic noise influencing the population, i.e. idiosyncratic or systemic. We will describe settings for which the convergence to mean field stochastic control problems can be resolved through the analysis of a certain Hamilton-Jacobi-Bellman equation posed on Wasserstein spaces. In particular, we develop new stability and regularity results for the equations. These allow for new convergence results for more general problems, for example, zero-sum stochastic differential games of mean-field type. We conclude with a discussion of some further problems for which the techniques for equations on Wasserstein space may be amenable.

Title: Fast high-order solvers on the simplicial de Rham complex via sparsity-promoting bases

Coauthors: Pablo Brubeck (Oxford), Patrick Farrell (Oxford), and Robert Kirby (Baylor)

Abstract

We present new high-order finite elements discretizing the L2 de Rham complex on triangular and tetrahedral meshes. The finite elements discretize the same spaces as usual, but with different basis functions. They allow for fast linear solvers based on static condensation and space decomposition methods. The new elements build upon the definition of degrees of freedom for interpolation given by Demkowicz et al. [1], and consist of integral moments on a symmetric reference simplex with respect to a numerically computed polynomial basis that is orthogonal in both the L2- and H(d)-inner products (d = grad, curl, or div). On the reference symmetric simplex, the resulting stiffness matrix has diagonal interior block, and does not couple together the interior and interface degrees of freedom. Thus, on the reference simplex, the Schur complement resulting from elimination of interior degrees of freedom is simply the interface block itself.

This sparsity is not preserved on arbitrary cells mapped from the reference cell. Nevertheless, the interior-interface coupling is weak because it is only induced by the geometric transformation. We devise a preconditioning strategy by neglecting the interior-interface coupling.  We precondition the interface Schur complement with the interface block, and simply apply point-Jacobi to precondition the interior block. We further precondition the interface block by applying a space decomposition method with small subdomains constructed around vertices, edges, and faces. This allows us to solve the canonical Riesz maps in H(grad), H(curl), and H(div), at very high order.  We empirically demonstrate iteration counts that are robust with respect to the polynomial degree.

References

[1] L. Demkowicz, P. Monk, L. Vardapetyan, and W. Rachowicz. De Rham diagram for hp finite element spaces. Comput. Math. Appl., 39(7-8):29--38, 2000. 

Analysis and numerical approximation of mean field game partial differential inclusions

Joint work with Mr Yohance A. P. Osborne

Mean field games (MFG) are models for differential games involving large numbers of players, where each player is solving an optimal control problem with dynamics governed by a controlled stochastic process. In a standard formulation, the Nash equilibria of the game are characterized by the solutions of a coupled system of partial differential equations, involving the Hamilton—Jacobi—Bellman equation for the value function and the Fokker—Planck equation for the density of players over the state space of the game.

However, in many realistic applications, the optimal feedback controls of the underlying optimal control problem are not necessarily unique, and this leads to the immediate problem of how the players should choose among nonunique optimal controls. Mathematically, this translates into the fact that the Hamiltonian of the MFG system becomes nondifferentiable, and the usual formulation of the MFG system is then no longer well-defined.

In this talk, we consider the central question of how to formulate and analyse MFG in such situations. We will show that a suitable generalization of the problem is provided by relaxing the Fokker—Planck equation to a partial differential inclusion (PDI), and we will explain how our approach can handle more complicated Nash equilibria with division among player choices of controls. Under suitable and rather general conditions on the problem data, we prove the existence of a weak solution of the resulting MFG PDI system via Kakutani’s fixed-point theorem, and we prove uniqueness of the solution under a standard monotonicity condition. We also introduce a stabilized finite element method for the problem, and we prove its convergence.

Title: The Moment-Closure Problem, Numerical Methods, and Applications

Abstract:

Kinetic models are widely used to simulate the dynamics of plasmas and rarefied gases in many application areas, including astrophysics, magnetically confined fusion, high-altitude aircraft, various propulsion mechanisms for spacecraft, and microfluidic devices. Kinetic models arise from a statistical mechanics approximation of the underlying particle motion and typically manifest as nonlinear integro-differential equations defined in a high-dimensional phase space. The challenge of reducing this phase space to a lower dimensional space is called the moment-closure problem. The goal is to replace the full kinetic distribution function with a (hopefully small) set of moments (e.g., mass, momentum, and energy); unfortunately, with only a finite set of moments, the underlying evolution equations are not closed, and additional assumptions must be made. In this talk, I will describe the moment-closure problem and some attempts to solve it. I will also discuss some issues developing high-order numerical methods to solve the resulting systems.

Various parts of the talk are joint work with Pierson Guthrey, Yifan Hu, Preeti Sar, and Christine Vaughan.

Title: High-order bounds-satisfying approximation of partial differential equations via finite element variational inequalities

Abstract: Solutions to many important partial differential equations satisfy bounds constraints, but approximations computed by finite element or finite difference methods typically fail to respect the same conditions.  Chang and Nakshatrala enforce such bounds in finite element methods through the solution of variational inequalities rather than linear variational problems.  Here, we provide a theoretical justification for this method, including higher-order discretizations.  We prove an abstract best approximation result for the linear variational inequality and estimates showing that bounds-constrained polynomials provide comparable approximation power to standard spaces.  For any unconstrained approximation to a function, there exists a constrained approximation which is comparable in the $W^{1,p}$ norm.

In practice, one cannot efficiently represent and manipulate the entire family of bounds-constrained polynomials, but applying bounds constraints to the coefficients of a polynomial in the Bernstein basis guarantees those constraints on the polynomial.  Although our theoretical results do not guaruntee high accuracy for this subset of bounds-constrained polynomials, numerical results indicate optimal orders of accuracy for smooth solutions and sharp resolution of features in convection-diffusion problems, all subject to bounds constraints.

Title: Accumulation Properties of an Iterated Mean Field Game

Abstract: We look at a particular Mean Field Game that gives players incentive to congregate. With a small time horizon, this type of game has a unique Nash Equilibrium given an initial distribution of players. Since the game is played only for a short time, we iterate the game, each new iteration starting at the final distribution of the previous game. We prove that after sufficiently many iterations, the players do congregate in tighter and tighter clusters and show where these clusters form.

Title: OPTIMAL CONTROL OF MOVING SETS

We consider a controlled reaction-diffusion equation, modeling the spreading of an invasive population, and derive a simpler model describing the controlled evolution of a contaminated set. We first analyze the optimal control of 1-dimensional traveling wave profiles. Using Stokes’ formula, explicit solutions are obtained, which in some cases require measure-valued optimal controls. Then we introduce a family of optimization problems for a moving set and show how these can be derived from the original parabolic problems, by taking a sharp interface limit. Assuming that the initial contaminated set is convex, we prove that an eradication strategy is optimal if an only if at each given time the control is active along the portion of the boundary where the curvature is maximal. 

This is a joint work with Stefano Bianchini, Alberto Bressan and Najmeh Salehi.

Title:

Best approximation results and essential boundary conditions for novel types of weak adversarial network discretization for PDEs

Abstract:

This talk presents our in-depth analysis of the cutting-edge Weakly Adversarial Networks (WAN) method and its variations, spotlighting its applicability in approximating high-dimensional and non-linear partial differential equations. Our three primary highlights are:

1. **Existence & Stability of WAN Solutions**: A meticulous validation of discrete solutions, interpreted in the weak context, complemented by an approximation principle reminiscent of Cea's lemma, a cornerstone in finite element analyses.

2. **Optimized WAN Formulations & Dirichlet Boundary Problem**: Introduction of two novel WAN stabilization strategies that sidestep direct normalization, paired with a critical treatment of WAN's efficacy for the Dirichlet boundary challenge.

3. **The pseudo-time XNODE Neural Network **: Unveiling the pseudo-time XNODE neural network paradigm, signifying a transformative leap with accelerated convergence rates over conventional DNN architectures.

Title: The master equation for mean  field games with fractional and nonlocal diffusions

Abstract: https://acrobat.adobe.com/link/review?uri=urn:aaid:scds:US:09d3d82b-0c10-310f-a6bc-738b3f063a7e

Title: Exact domain truncation for scattering problems

Abstract: While scattering problems are posed on unbounded domains, volumetric discretizations typically require truncating the domain at a finite distance, closing the system with some sort of boundary condition.  These conditions typically suffer from some deficiency, such as perturbing the boundary value problem to be solved or changing the character of the operator so that the discrete system is difficult to solve with iterative methods.

We introduce a new technique for the Helmholtz problem, based on using the Green formula representation of the solution at the artificial boundary.  Finite element discretization of the resulting system gives optimal convergence estimates.  The resulting algebraic system can be solved effectively with a matrix-free GMRES implementation, preconditioned with the local part of the operator.  Extensions to the Morse-Ingard problem, a coupled system of pressure/temperature equations arising in modeling trace gas sensors, will also be given.

Title: Answering Rob Kirby’s question on the coercivity of the Maxwell system

Abstract: I will show that the bilinear form naturally arising while adopting exact

(nonlocal) domain truncation boundary conditions for the Maxwell system of

electromagnetism is coercive. This crucial ingredient in implementing numerical methods

answers a question posed by Rob Kirby, and extends work done in the scalar case, for the

Helmholtz operator in acoustic scattering.

Title: Immersed Finite Element Methods for Three-Dimensional Interface Problems

Abstract: Interface problems arise in many applications in science and engineering. Partial differential equations (PDEs) are often used to model interface problems. Solutions to these PDE interface problems often involves kinks, singularities, discontinuities, and other non-smooth behaviors. The immersed finite element method (IFEM) is a class of numerical methods for solving PDE interface problems with unfitted meshes.

In this talk, I will some recent advances on development and analysis of several IFEMs for solving three-dimensional interface problems. The proposed method can be utilized on interface-unfitted tetrahedral and cuboidal meshes even if the 3D interface surface possesses an arbitrary shape. Fundamental estimates such as the trace inequalities, inversed inequalities, and the approximation capabilities will be established. Optimal a priori error estimates are proved in both energy and L2 norms. Numerical examples will be provided not only to verify our theoretical results but also to demonstrate the applicability of this method in tackling some real-world 3D interface models.

Title: Finite element methods for the Stokes–Onsager–Stefan–Maxwell equations of multicomponent flow

Abstract: The Onsager framework for linear irreversible thermodynamics provides a thermodynamically consistent model of mass transport in a phase consisting of multiple species, via the Stefan–Maxwell equations, but a complete description of the overall transport problem necessitates also solving the momentum equations for the flow velocity of the medium. We derive a novel nonlinear variational formulation of this coupling, called the (Navier–)Stokes–Onsager–Stefan–Maxwell system, which governs molecular diffusion and convection within a non-ideal, single-phase fluid composed of multiple species, in the regime of low Reynolds number in the steady state. We propose an appropriate Picard linearisation posed in a novel Sobolev space relating to the diffusional driving forces, and prove convergence of a structure-preserving finite element discretisation. This represents some of the first rigorous numerics for the coupling of multicomponent molecular diffusion with compressible convective flow. The broad applicability of our theory is illustrated with simulations of the centrifugal separation of noble gases and the microfluidic mixing of hydrocarbons.

Title:  Monte Carlo for the Trace of the Inverse of a Large Matrix

Abstract:  Monte Carlo is a really bad way to do a calculation, unless it is the only practical way to do it.  We will discuss a Monte Carlo method for calculating the trace of the inverse of a large matrix from lattice QCD (quantum chromodynamics).  The Monte Carlo sampling can be fairly simple: calculate b^T A^{-1}b for many vectors b that are made of random 1’s and -1’s, and average these quantities.  However, solving many large systems of linear equations to find the A^{-1}b vectors is very time consuming.   We have a method of speeding this up that has a multilevel Monte Carlo using different degree polynomials as the levels.  This method is improved by eigenvalue deflation (in three different ways) and by polynomial preconditioning.

The plan is to discuss some of the basics of Monte Carlo along with polynomial approximation of the inverse of a large matrix, before going into some of the details of the new method.  Hopefully we will run out of time before hitting too many of the details. 

Title: Conservative Low-Rank Approximations to Nonlinear Vlasov Dynamics

Abstract: The Vlasov system is known as a fundamental model in plasma physics which describes the dynamics of dilute charged particles due to self-induced electrostatic forces. The main numerical challenges lie in the high dimensionality of the phase space, multi-scale feature, and the inherent conservation property of the solutions. In this talk, I will introduce a novel conservative adaptive low-rank tensor method for the Vlasov system. The approach takes advantage of the fact that the differential operators in the Vlasov equation are tensor friendly, based on which we propose to build up the low-rank solution basis dynamically and adaptively by exploring the intrinsic low-rank structure of Vlasov dynamics. We further develop a novel low-rank scheme with local mass, momentum, and energy conservation by considering the corresponding macroscopic equations. The mass and momentum conservation are achieved by a conservative low-rank truncation, while the energy conservation is achieved by replacing the energy component of the Vlasov solution by the one obtained from a conservative scheme for the macroscopic energy equation. The algorithm is extended to high-dimensional problems with the hierarchical Tucker tensor decomposition of Vlasov solutions, overcoming the curse of dimensionality. An extensive set of nonlinear Vlasov examples are performed to demonstrate the effectiveness and conservation property of the proposed conservative low-rank tensor approach.

Title:

A posteriori error estimate of Darcy flow with low permeability faults/membranes

Abstract:

In this work we consider porous media flow models with low permeability fault/membrane structures.

The solutions of these models have low regularities because fluid flows are hindered by the faults.

We consider a posteriori error estimates of the models for efficient computation of numerical solutions.

Title: A mean field games system with common noise and degenerate idiosyncratic noise

Abstract: I will describe the forward-backward system of stochastic partial differential equations describing a mean field game for a large population of small players subject to both idiosyncratic and common noise. The unique feature of the problem is that the idiosyncratic noise coefficient may be degenerate, so that the system does not admit smooth solutions in general. A new notion of weak solutions for backward stochastic Hamilton-Jacobi-Bellman equations must be developed, and this is used to build probabilistically weak solutions of the mean field game system. The uniqueness of a strong solution can be proved under additional structural assumptions. This is joint work with P. Cardaliaguet and P. Souganidis. 

Title: A polynomial approximation to the inverse of a matrix 

Abstract: In this work, we develop a surprisingly accurate polynomial approximation to the inverse of a large matrix, i.e. $p(A)\approx A^{-1}$. This polynomial is implemented using the roots of the GMRES polynomial, or harmonic ritz values, and can be used to solve multiple right-hand sides as well as trace estimates to the inverse of a matrix. Issues with stability that arise with applying a high-degree polynomial will be discussed. In addition, we give bounds on the error of our approximate inverse.

Title: Multigrid preconditioning for PDE-Constrained optimization: two applications

Abstract: 

We present two applications of a multigrid preconditioning technique that was originally developed for certain classes of inverse problems, and then applied successfully to optimal control of partial differential equations.

The first part of the talk will focus on optimal control problems constrained by elliptic equations with stochastic coefficients. Assuming a generalized polynomial chaos expansion for the stochastic components, our approach uses a stochastic Galerkin finite element discretization for the PDE, thus leading to a discrete optimization problem. The key aspect is solving the potentially very large linear systems arising when solving the system representing the first-order optimality conditions. We show that the multilevel preconditioning technique from the optimal control of deterministic elliptic PDEs has a natural extension to the stochastic case, and exhibits a similar optimal behavior with respect to the mesh size, namely the quality of the preconditioner increases with decreasing mesh-size at the optimal rate. Moreover, under certain assumptions, we show that the quality is robust also with respect to the two additional parameters that influence the dimension of the problem radically: polynomial degree and stochastic dimension.

In the second part of the talk we apply a similar technique to an optimization-based non-overlapping domain decomposition method for elliptic partial differential equations developed by Gunzburger, Heinkenschloss, and Lee (2000). While it is not surprising that, for a fixed partition in subdomains, the preconditioner leads to the expected behavior of increasing quality (lower number of iterations) as the resolution increases, it is remarkable that the quality of the preconditioner is relatively robust with respect to the number and configuration of subdomains.