Computationally feasible methods based on Krylov subspaces to solve large-scale, constrained, and time dependent inverse problems
Speaker: Mirjeta Pasha (Arizona State University)
Ill-posed inverse problems arise in many fields of science and engineering. Their solution, if it exists, is very sensitive to perturbations in the data. The challenge of working with linear discrete ill-posed problems comes from the ill-conditioning and the possible large dimension of the problems. Regularization methods aim to reduce the sensitivity by replacing the given problem with a nearby one, whose solution is less affected by perturbations. The methods in this talk are concerned with solving large scale problems by projecting them into a Krylov or generalized Krylov subspace of fairly small dimension. The first type of methods discussed are based on Bregman-type iterative methods that even though the high quality reconstruction that they deliver, they may require a large number of iterations and this reduces their attractiveness. We develop a computationally attractive linearized Bregman algorithm by projecting the problem to be solved into an appropriately chosen low-dimensional Krylov subspace. Recently, the use of a \(p\)-norm to measure the fidelity term and a \(q\)-norm to measure the regularization term has received considerable attention. For applications such as image reconstruction, where the pixel values are non-negative, we impose a non-negativity constraint to make sure the reconstructed solution lies in the non-negative orthant. We propose techniques to select the regularization parameter without any significant computational cost. This makes the proposed method more efficient and useful especially for large-scale problems. In addition, we explore how to estimate maximum a posteriori when the available data are perturbed with non-Gaussian noise. Near the end of the talk will be discussed current work in progress on solving time-dependent inverse problems with the goal to preserve edges and model small changes in time, solving separable nonlinear inverse problems as well as learning strategies based on optimal experimental design and Krylov subspaces. Numerical examples illustrate the performances of the approaches proposed in terms of both accuracy and efficiency. We consider two-dimensional problems, with a particular attention to the restoration of blurred and noisy images.
Time: April 23, 2021 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Zhu Wang
First-order image restoration models for staircase reduction and contrast preservation
Speaker: Wei Zhu (University of Alabama)
In this talk, we will discuss two novel first-order variational models for image restoration. In the literature, lots of higher-order models were proposed to fix the staircase effect. In our first model, we consider a first-order variational model that imposes stronger regularity than total variation on regions with small image gradients in order to achieve staircase reduction. In our second model, we further propose a novel regularizer that presents a lower growth rate than any power function with a positive exponent for regions with large image gradients. Besides removing noise and keeping edges effectively, this regularizer also helps preserve image contrasts during the image restoration process. We employ augmented Lagrangian method (ALM) to minimize both models and provide the convergence analysis. Numerical experiments will be then presented to demonstrate the features of the proposed models.
Time: April 16, 2021 3:30pm-4:30pm
Location: Virtually via Zoom
Host: Yi Sun
Access to the video of the talkA parallel cut-cell algorithm for the free-boundary Grad-Shafranov problem
Speaker: Shuang Liu (University of California, San Diego)
A parallel cut-cell algorithm is described to solve the free boundary problem of the Grad-Shafranov equation. The algorithm reformulates the free-boundary problem in an irregular bounded domain and its important aspects include a searching algorithm for the magnetic axis and separatrix, a surface integral along the irregular boundary to determine the boundary values, an approach to optimize the coil current based on a targeting plasma shape, Picard iterations with Aitken's acceleration for the resulting nonlinear problem and a Cartesian grid embedded boundary method to handle the complex geometry. The algorithm is implemented in parallel using a standard domain-decomposition approach and a good parallel scaling is observed. Numerical results verify the accuracy and efficiency of the free-boundary Grad-Shafranov solver.
Time: March 26, 2021 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Xinfeng Liu
Symmetry and uniqueness via a variational approach
Speaker: Yao Yao (Georgia Institute of Technology)
For some nonlocal PDEs, its steady states can be seen as critical points of an associated energy functional. Therefore, if one can construct perturbations around a function such that the energy decreases to first order along the perturbation, this function cannot be a steady state. In this talk, I will discuss how this simple variational approach has led to some recent progresses in the following equations, where the key is to carefully construct a suitable perturbation.
I will start with the aggregation-diffusion equation, which is a nonlocal PDE driven by two competing effects: nonlinear diffusion and long-range attraction. We show that all steady states are radially symmetric up to a translation (joint with Carrillo, Hittmeir and Volzone), and give some criteria on the uniqueness/non-uniqueness of steady states within the radial class (joint with Delgadino and Yan).
I will also discuss the 2D Euler equation, where we aim to understand under what condition must a stationary/uniformly-rotating solution be radially symmetric. Using a variational approach, we settle some open questions on the radial symmetry of rotating patches, and also show that any smooth stationary solution with compactly supported and nonnegative vorticity must be radial (joint with Gómez-Serrano, Park and Shi).
Time: April 2, 2021 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Changhui Tan
Discontinuous Galerkin methods with local time stepping for the nonlinear shallow water equations
Speaker: Yulong Xing (Ohio State University)
Shallow water equations (SWEs) with a non-flat bottom topography have been widely used to model flows in rivers and coastal areas. In this presentation, we will talk about the applications of high-order well-balanced and positivity-preserving discontinuous Galerkin methods to this system. With carefully chosen numerical fluxes, we will show that the proposed methods preserve the still water steady state exactly, and at the same time maintain the non-negativity of the water height. For the temporal discretization, we propose the high order ADER-differential transform approach. Local time stepping strategy will also be studied to allow elements of different sizes to use different time steps. One- and two-dimensional numerical tests are performed to verify the well-balanced property, high-order accuracy, and good resolution for general solutions.
Time: March 19, 2021 3:30pm-4:30pm
Location: Virtually via Zoom
Host: Lili Ju
Analysis and numerics for nematic liquid crystals
Speaker: Shawn Walker (Louisiana State University)
I start with an overview of nematic liquid crystals (LCs) and their applications, including how they are modeled, such as Oseen-Frank, Landau-de Gennes, and the Ericksen model. For the rest of the talk, I will focus on Landau-de Gennes (LdG) and Ericksen.
Next, I describe some of the analytical difficulties of these models. For example, the Ericksen model exhibits a non-smooth constraint (for the PDE solution), and the LdG model with uniaxiality enforced as a hard constraint is also non-smooth. I will then discuss related numerical analysis issues that arise and how we handle these difficulties with a structure-preserving finite element method (FEM) for computing energy minimizers. We prove stability and consistency of the method without regularization, and \(\Gamma\)-convergence of the discrete energies towards the continuous one as the mesh size goes to zero. Numerical simulations will be presented in two and three dimensions, some of which include non-orientable line fields, using a provably robust minimization scheme. Finally, I will conclude with some current problems and an outlook to future directions.
Time: March 5, 2021 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Qi Wang
From signal representations to musical creation: a geometric approach
Speaker: Carmine Emanuele Cella (University of California, Berkeley)
Carmine Emanuele Cella, assistant professor in music and technology at CNMAT, will present work done in the last years in searching good signal representations that permit high-level manipulation of musical concepts. After the definition of a geometric approach to signal representation, the theory of sound-types and its application to music will be presented. Finally, recent research on assisted orchestration will be shown and some possible musical applications will be proposed, with connections to deep learning methods.
This is a joint seminar with the Composition Seminar in the School of Music.
Time: February 26, 2021 2:30pm-4:00pm
Location: Virtually via Zoom
Host: Qi Wang and Yi Sun
Numerical methods for solving nonlinear differential equations from homotopy methods to machine learning
Speaker: Wenrui Hao (Pennsylvania State University)
Many systems of nonlinear differential equations are arising from engineering and biology and have attracted research scientists to study the multiple solution structure such as pattern formation. In this talk, I will present several methods to compute the multiple solutions of nonlinear differential equations. First, I will introduce the homotopy continuation technique to compute the multiple steady states of nonlinear differential equations and also to explore the relationship between the number of steady-states and parameters. Then I will use the machine learning techniques to solve nonlinear differential equations and learn the multiple solutions by developing a randomized Newton's method for the neural network discretization. Several benchmark problems will be used to illustrate these ideas.
Time: February 19, 2021 3:30pm-4:30pm
Location: Virtually via Zoom
Host: Qi Wang
Aggregation with intrinsic interactions on Riemannian manifolds
Speaker: Razvan Fetecau (Simon Fraser University)
We consider a model for collective behaviour with intrinsic interactions on Riemannian manifolds. We establish the well-posedness of measure solutions, defined via optimal mass transport, on several specific manifolds (sphere, hypercylinder, rotation group \(SO(3)\)), and investigate the mean-field particle approximation. We study the long-time behaviour of solutions, where the primary goal is to establish sufficient conditions for a consensus state to form asymptotically. The analytical results are illustrated with numerical experiments that exhibit various asymptotic patterns.
Time: February 12, 2021 3:30pm-4:30pm
Location: Virtually via Zoom
Host: Changhui Tan
Transport information dynamics with applications
Speaker: Wuchen Li (University of South Carolina)
In this talk, I briefly reviewed several dynamical equations, raised in optimal transport, information geometry and mean field game modeling. I will discuss the applications of these dynamics in AI and Bayesian sampling problems.
Time: January 22, 2021 2:30pm-3:30pm
Location: Virtually via Zoom
More...
Linear interpolation convexity/concavity in the minimization of attractive-repulsive energy
Speaker: Ruiwen Shu (University of Maryland)
Energy minimization problems of attractive-repulsive pairwise interactions are very important in the study of pattern formation in biological and social sciences. In this talk, I will discuss some recent progress (joint work with Jose Carrillo) on the study of Wasserstein-\(\infty\) local energy minimizers by using the method of linear interpolation convexity/concavity. In the first part, we prove the radial symmetry and uniqueness of local minimizers for interaction potentials satisfying the 'linear interpolation convexity', which generalizes the result of O. Lopes 17' for global minimizers. In the second part, we show that the failure of linear interpolation convexity could lead to the formation of small scales in the support of local minimizers, and construct interaction potentials whose local minimizers are supported on fractal sets. To our best knowledge, this is the first time people observe fractal sets as the support of local minimizers.
Time: Febrary 5, 2021 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Changhui Tan
Self-similar Solutions of Active Scalars with Critical Dissipation
Speaker: Dallas Albritton (New York University)
In PDE analyses of fluid models, often we may identify a so-called critical space that lives precisely at the borderline between well-posedness and ill-posedness. What happens at this borderline? We explore this question in two active scalar equations with critical dissipation. In the critical surface quasi-geostrophic equations, we investigate the connection between non-uniqueness and large self-similar solutions that was established by Jia, Sverak, and Guillod in the Navier-Stokes equations. This is joint work with Zachary Bradshaw. In the critical Burgers equation, and more generally in critical scalar conservation laws, the analogous self-similar solutions are unique, and we show that all front-like solutions converge to a self-similar solution at the diffusive rates. This is joint work with Raj Beekie.
Time: November 13, 2020 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Changhui Tan
Some Analytic Properties of a Singular Potential in the Laudau-de Gennes Theory for Liquid Crystals
Speaker: Xiang Xu (Old Dominion University)
The Landau-de Gennes theory is a type of continuum theory that describes nematic liquid crystal configurations in the framework of the Q-tensor order parameter. In the free energy, there is a singular bulk potential which is considered as a natural enforcement of a physical constraint on the eigenvalues of symmetric, traceless Q-tensors. In this talk we shall discuss some analytic properties related to this singular potential.
Time: October 30, 2020 3:30pm-4:30pm
Location: Virtually via Zoom
Host: Changhui Tan
A Proximal-gradient Algorithm for Crystal Surface Evolution
Speaker: Li Wang (University of Minnesota)
We develop a novel semi-implicit scheme for the crystal surface evolution equation, which suffers from significant stiffness that prevents simulation on fine spatial grids. Our method leverages the formal structure of the equation as the gradient flow of the total variation energy, with respect to a weighted \(H^{-1}\) norm. Inspired by the classical minimizing movement scheme, we reformulate the semi-implicit time discretization into an optimization problem, and then use a primal-dual hybrid gradient (PDHG) method to compute the minimizer. In one dimension, we prove the convergence of PDHG to the semi-implicit scheme, and show that, at the discrete level, our PDHG method converges at a rate independent of the grid size. This is joint work with Katy Craig, Jian-Guo Liu, Jianfeng Lu and Jeremy Marzuola.
Time: October 23, 2020 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Changhui Tan