Fall 2022 Seminar
Schedule
Time | Location | Speaker | Title |
---|---|---|---|
Aug 26, 11am | GP346 | Brendan Ames | Non-convex relaxations for the densest submatrix problem |
Sep 09, 11am | GP346 | Denis Aslangil | High-fidelity simulations of multi-physics turbulent flows |
Sep 16, 11am | GP346 | Shibin Dai | Degenerate diffusion and interface motion of single layer and bilayer structures |
Sep 23, 11am | GP346 | Ali Pakniyat | Dualities in optimal control theory |
Oct 07, 11am | Zoom | Songting Luo | Fixed-point iteration methods for some high frequency wave propagation problems |
Oct 28, 3pm (special time) | GP232 | Lorena Bociu | Analysis and control in poroelastic systems with applications to biomedicine |
Nov 02, 11am (special time) | GP346 | Teresa Portone | Quantifying model-form uncertainty with an application to subsurface transport |
Nov 04, 1pm (special time) | Zoom | Abhishek Halder | A distributed algorithm for Wasserstein proximal operator splitting: theory & applications |
Nov 11, 11am | GP346 | Chi-Wang Shu | Explicit-implicit-null (EIN) time-marching for high order PDEs |
Nov 18, 11am | Zoom | Qiwei Feng | Compact finite difference schemes for interface problems |
Abstracts
- Aug 26, 2022
Brendan Ames (Department of Mathematics, The University of Alabama)
Title: Non-convex relaxations for the densest submatrix problem
Abstract: Given a matrix, the densest submatrix problem aims to identify the submatrix of fixed size with fewest zero entries. This problem serves as a model for several important tasks in network analysis, machine learning, and combinatorial optimization, most importantly in community identification and as a model for the maximum clique and biclique problems in the presence of noise. Recent advances regarding convex relaxations of the densest submatrix problem have established conditions under which this problem can be solved exactly in polynomial-time, despite the problem being NP-hard in general. However, the current state of the art algorithms for solving these relaxations are prohibitively slow (despite being polynomial-time) and are limited to solution of problem instances involving matrices with less than a few thousand rows and columns.
I will review several new relaxations of the densest submatrix problem based on non-convex quadratic programming, and will show that the earlier recovery guarantees extend to these new relaxations. Further, I will present a new family of first-order optimization algorithms for solving this quadratic program based on variants of the linearized alternating direction method of multipliers, establish their convergence properties, and provide empirical evidence of their improvement on the current state of the art.
- Sep 09, 2022
Denis Aslangil (Department of Aerospace Engineering and Mechanics, The University of Alabama)
Title: High-fidelity simulations of multi-physics turbulent flows
Abstract: In the real world, turbulence occurs in multi-material/phase flows, which in most cases involve materials with large density differences. Unlike incompressible single-fluid flows, the velocity field in multi-material flows with compositional variations is tightly coupled with the density field. These flows are called variable density (VD) flows. Such flows are of broad interest due to their occurrence in the form of Rayleigh-Taylor and Richtmyer-Meshkov instabilities, combustion applications in ramjet engines, high energy density processes like inertial confinement fusion (ICF), and convection regions in the atmosphere, oceans, and the Earth’s mantle. In this talk, the results of extremely large-resolution (domain size up to 2048^3) direct numerical simulations (DNS) of buoyancy-driven homogeneous VD turbulence will be presented to show the highly asymmetric VD flow development at large density ratios. In addition, Dr. Aslangil will discuss his research group’s current research directions at UA. His group aims to best use state-of-the-art computational tools to unravel novel physical insights of multi-physics turbulence, improve turbulence theory, and adapt the findings to develop efficient physics-informed engineering models.
- Sep 16, 2022
Shibin Dai (Department of Mathematics, The University of Alabama)
Title: Degenerate diffusion and interface motion of single layer and bilayer structures
Abstract: Degenerate diffusion plays an important role in the interface motion of complex structures. The degenerate Cahn-Hilliard equation is a widely used model for single layer structures. It has been commonly believed that degenerate diffusion eliminates diffusion in the bulk phases and results in surface diffusion only. We will show that due to the curvature effect there is porous medium diffusion in the bulk phases, and the geometric evolution of single layer structures is mediated by the porous medium diffusion process. We will also discuss the existence of weak solutions for the degenerate CH equation. For bilayer structures the Functionalized Cahn-Hilliard (FCH) equation is a new model that has been extensively studied in recent years. We will discuss the existence and nonnegativity of weak solutions for the degenerate FCH equation, and the corresponding interface motions.
- Sep 23, 2022
Ali Pakniyat (Department of Mechanical Engineering, The University of Alabama)
Title: Dualities in optimal control theory
Abstract: Duality is a mathematical principle which, when it emerges, signifies the intrinsic relations between two distinct concepts, theorems or structures. In this talk, I will present three dualities which emerge in optimal control theory: (i) the duality in the Minimum Principle (MP) between the finite dimensional spaces of state variations and that of co-state (adjoint) processes, (ii) the duality in Dynamic Programming (DP) between the infinite dimensional space of measures and that of continuous functions, and (iii) the duality in Feynman-Kac theory between partial differential equations (PDEs) and forward-backward stochastic differential equations (FBSDEs). I will present new versions of the MP and DP for deterministic and stochastic hybrid systems and illustrate their implementation and numerical computations for single-agent and multi-agent systems.
- Oct 7, 2022
Songting Luo (Department of Mathematics, Iowa State University)
Title: Fixed-point iteration methods for some high frequency wave propagation problems
Abstract: For wave propagation problems governed by high frequency Helmholtz equation or vector wave equations, we present a simple approach based on fixed-point iterations, where the problem is transferred into a fixed-point problem related to an exponential operator. The associated functional evaluations are achieved by unconditionally stable operator-splitting based spectral/pseudospectral schemes such that large step sizes are allowed to reach the approximated fixed point efficiently for prescribed termination requirement. The Anderson acceleration is further incorporated to accelerate the convergence of the fixed-point iterations. Numerical experiments are presented to demonstrate the method. This is a joint work with Qing Huo Liu (Duke University).
- Oct 28, 2022
Lorena Bociu (Department of Mathematics, North Carolina State University)
Title: Analysis and control in poroelastic systems with applications to biomedicine
Abstract: Fluid flows through deformable porous media are relevant for many applications in biology, medicine and bio-engineering, including tissue perfusion, fluid flow inside cartilages and bones, and design of bioartificial organs. Mathematically, they are described by quasi-static nonlinear poroelastic systems, which are implicit, degenerate, coupled systems of partial differential equations (PDE) of mixed parabolic-elliptic type. We answer questions related to tissue biomechanics via well-posedness theory, sensitivity analysis, and optimal control for the poroelastic PDE coupled systems mentioned above. One application of particular interest is perfusion inside the eye and its connection to the development of neurodegenerative diseases.
- Oct 28, 2022
Teresa Portone (Optimization and Uncertainty Quantification Department, Sandia National Laboratory)
Title: Quantifying model-form uncertainty with an application to subsurface transport
Abstract: Computational models are increasingly used to make predictions affecting high-consequence engineering design and policy decisions. However, incomplete information about the represented phenomena and limitations in computational resources require approximations and simplifications that can lead to uncertainties in the computational models’ forms and errors in predicted quantities of interest. Addressing these uncertainties are essential for understanding the reliability of such predictions. This talk will provide a survey of current approaches to address model-form uncertainty and detail its characterization for an upscaled/homogenized model of transport through a heterogeneous porous medium.
- Oct 28, 2022
Abhishek Halder (Department of Applied Mathematics, University of California at Santa Cruz)
Title: A distributed algorithm for Wasserstein proximal operator splitting: theory & applications
Abstract: Many problems in control, statistics, machine learning and solving partial differential equations can be cast as infinite dimensional gradient flows with respect to the Wasserstein metric that appears in the theory of optimal mass transport. The main idea is to construct the solution of interest as the gradient descent of a suitable measure valued functional that is convex along the generalized geodesics with respect to the Wasserstein metric. These connections have unfolded a rapid development across several disciplines in the past two decades. Many time stepping algorithms are now available in the literature to numerically realize the Wasserstein proximal updates, which generalize the concept of gradient steps in the manifold of probability measures with finite second moments. Motivated by the observation that most practical problems of interest have additive objectives, this talk will present a distributed algorithm to perform the Wasserstein proximal updates. The proposed algorithm generalizes the finite dimensional Euclidean consensus ADMM algorithm to the measure valued Wasserstein, and its entropy regularized Sinkhorn variants. We will point out how the proposed algorithm differs compared to the standard Euclidean case, and present numerical case studies. This is joint work with Ph.D. student Iman Nodozi at UC Santa Cruz.
- Nov 11, 2022
Chi-Wang Shu (Division of Applied Mathematics, Brown University)
Title: Explicit-implicit-null (EIN) time-marching for high order PDEs
Abstract: Time discretization is an important issue for time-dependent partial differential equations (PDEs). For the k-th (k is at least 2) order PDEs, the explicit time-marching method may suffer from a severe time step restriction τ = O(hk) (where τ and h are the time step size and spatial mesh size respectively) for stability. The implicit and implicit-explicit (IMEX) time-marching methods can overcome this constraint. However, for the equations with nonlinear high derivative terms, the IMEX methods are not good choices either, since a nonlinear algebraic system must be solved (e.g. by Newton iteration) at each time step. The explicit-implicit-null (EIN) time-marching method is designed to cope with the above mentioned shortcomings. The basic idea of the EIN method is to add and subtract a sufficiently large linear highest derivative term on one side of the considered equation, and then apply the IMEX time-marching method to the equivalent equation. The EIN method so designed does not need any nonlinear iterative solver, and the severe time step restriction for explicit methods can be removed. Coupled with the EIN time-marching method, we will discuss high order finite difference and local discontinuous Galerkin schemes for solving high order dissipative and dispersive equations. For simplified equations with constant coefficients, we perform analysis to guide the choice of the coefficient for the added and subtracted highest order derivative terms in order to guarantee stability for large time steps. Numerical experiments show that the proposed schemes are stable and can achieve optimal orders of accuracy for both one-dimensional and two-dimensional linear and nonlinear equations. This talk is based on joint work with Haijin Wang, Qiang Zhang and Shiping Wang, and with Meiqi Tan and Juan Cheng.
- Nov 18, 2022
Qiwei Feng (Mathematical and Statistical Sciences, University of Alberta)
Title: Compact finite difference schemes for interface problems
Abstract: Interface problems arise in many applications such as modeling of underground waste disposal, oil reservoirs, composite materials, and many others. The coefficient a, the source term f, the solution u and the flux a∇ u∙ n are possibly discontinuous across the interface curve Γ in such problems. In order to obtain reasonable numerical solutions, higher order numerical schemes are desirable. Firstly, we propose a sixth order compact 9-point finite difference method (FDM) on uniform Cartesian grids, for Poisson interface problems with singular sources in a rectangular domain. The matrix A in the resulting linear system Ax=b, following from the proposed compact 9-point scheme, is independent of any source terms f, jump conditions, and interface curves Γ. We prove the sixth order convergence rate for the proposed compact 9-point scheme using the discrete maximum principle. For elliptic interface problems with discontinuous and high-contrast piecewise smooth coefficients in a rectangle, we propose a high order compact 9-point FDM and a high order local calculation for approximation of the solution u and the gradient ∇ u respectively. The scheme is developed on uniform Cartesian grids, avoiding the transformation into local coordinates. This is joint work with Bin Han, and Peter Minev.