## ACM Seminar

The Applied and Computational Mathematics (ACM) seminar series since 2020.

Friday, 05 November 2021 16:58

### Numerical methods for nonlocal models: asymptotically compatible schemes and multiscale modeling

#### Speaker: Xiaochuan Tian (University of California, San Diego)

Nonlocal continuum models are in general integro-differential equations in place of the conventional partial differential equations. While nonlocal models show their effectiveness in modeling a number of anomalous and singular processes in physics and material sciences, for example, the peridynamics model of fracture mechanics, they also come with increased difficulty in computation with nonlocality involved. In this talk, we will give a review of the asymptotically compatible schemes for nonlocal models with a parameter dependence. Such numerical schemes are robust under the change of the nonlocal length parameter and are suitable for multiscale simulations where nonlocal and local models are coupled. We will discuss finite difference, finite element and collocation methods for nonlocal models as well as the related open questions for each type of the numerical methods.

Time: November 12, 2021 3:30pm-4:30pm
Location: Virtually via Zoom
Host: Changhui Tan

Monday, 01 November 2021 09:37

### De Giorgi method for kinetic equations

#### Speaker: Weiran Sun (Simon Fraser University)

In this talk we explain how to generalize the De Giorgi level-set method for diffusion equations to a framework for kinetic equations with singular kernels. In particular, we use the non-cutoff Boltzmann and the Landau equations as examples to show how the De Giorgi method can be used to prove the existence of $$L^2\cap L^\infty$$ solutions in the near-equilibrium regime. The key idea is to make use of the strong averaging lemma to establish a nonlinear iteration for level-set energies which will give a local existence theory. We then extend the time interval to infinity by exploring the spectral structures of the linearized kinetic operators. This talk is based on recent works with Ricardo Alonso, Yoshinori Morimoto, and Tong Yang.

Time: November 5, 2021 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Changhui Tan

Wednesday, 27 October 2021 05:24

### Global-in-time domain decomposition methods for the coupled Stokes and Darcy flows

#### Speaker: Thi-Thao-Phuong Hoang (Auburn University)

In many engineering and biological applications (e.g., groundwater flow problems, flows in vuggy porous media, industrial filtrations, biofluid-organ interaction and cardiovascular flows), the Stokes-Darcy system is used to model the interaction of fluid flow with porous media flow, where the Stokes equations represent an incompressible fluid, and the Darcy equations represent a flow through a porous medium. The time scales in the Stokes and Darcy regions could be largely different, thus it is inefficient to use the same time step throughout the entire spatial domain.

In this talk, we present decoupling iterative algorithms based on domain decomposition for the time-dependent Stokes-Darcy model, in which different time step sizes can be used in the flow region and in the porous medium. The coupled system is formulated as a space-time interface problem based on either physical interface conditions or equivalent Robin-Robin interface conditions. Such an interface problem is solved iteratively by a Krylov subspace method (e.g., GMRES) which involves at each iteration parallel solution of time-dependent Stokes and Darcy problems. Consequently, local discretizations in both space and time can be used to efficiently handle multiphysics systems with discontinuous parameters. Numerical experiments with nonconforming time grids are considered to illustrate the performance of the proposed methods.

Time: November 19, 2021 2:30pm-3:30pm
Location: COL 2014 and Virtually via Zoom
Host: Lili Ju

Wednesday, 20 October 2021 15:53

### Applications of the shear-flow induced enhanced dissipation

#### Speaker: Siming He (Duke University)

In this talk, we consider the enhanced dissipation phenomena induced by shear flows. In the first part of the talk, I will introduce the idea of shear flow-induced enhanced dissipation and the recent developments on this topic. Then I will exhibit the applications of this phenomenon in various settings, ranging from suppression of chemotactic blow-ups to enhancement of chemical reactions.

Time: October 29, 2021 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Changhui Tan

Monday, 11 October 2021 20:24

### Energetic variational approaches for active/reactive fluids and applications

#### Speaker: Chun Liu (Illinois Institute of Technology)

We present a general framework for active fluids which convert chemical energy into various types of mechanical energies. This is the extension of the classical energetic variational approaches for isothermal mechanical systems. The methods will cover a wide range of both chemical reaction kenetics, thermal and mechanical processes. This is a joint project with many collaborators, in particular, Bob Eisenberg, Yiwei Wang and Tengfei Zhang.

Time: December 3, 2021 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Qi Wang

Monday, 04 October 2021 14:14

### Structure preserving numerical methods for hyperbolic systems of conservation and balance laws

#### Speaker: Alina Chertock (North Carolina State University)

Many physical models, while quite different in nature, can be described by nonlinear hyperbolic systems of conservation and balance laws. The main source of difficulties one comes across when numerically solving these systems is lack of smoothness as solutions of hyperbolic conservation/balance laws may develop very complicated nonlinear wave structures including shocks, rarefaction waves and contact discontinuities. The level of complexity may increase even further when solutions of the hyperbolic system reveal a multiscale character and/or the system includes additional terms such as friction terms, geometrical terms, nonconservative products, etc., which are needed to be taken into account in order to achieve a proper description of the studied physical phenomena. In such cases, it is extremely important to design a numerical method that is not only consistent with the given PDEs, but also preserves certain structural and asymptotic properties of the underlying problem at the discrete level. While a variety of numerical methods for such models have been successfully developed, there are still many open problems, for which the derivation of reliable high-resolution numerical methods still remains to be an extremely challenging task.

In this talk, I will discuss recent advances in the development of two classes of structure preserving numerical methods for nonlinear hyperbolic systems of conservation and balance laws. In particular, I will present (i) well-balanced and positivity preserving numerical schemes, that is, the methods which are capable of exactly preserving some steady-state solutions as well as maintaining the positivity of the numerical quantities when it is required by the physical application, and (ii) asymptotic preserving schemes, which provide accurate and efficient numerical solutions in certain stiff and/or asymptotic regimes of physical interest.

Time: October 15, 2021 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Changhui Tan

Tuesday, 05 October 2021 14:14

### Neural nets and numerical PDEs

#### Speaker: Zhiqiang Cai (Purdue University)

In this talk, I will present our recent works on neural networks (NNs) and its application in numerical PDEs. The first part of the talk is to use NNs to numerically solve scalar linear and nonlinear hyperbolic conservation laws whose solutions are discontinuous. I will show that the NN-based method for this type of problems has an advantage over the mesh-based methods in terms of the number of degrees of freedom.

The second part of the talk is on our adaptive network enhancement (ANE) method. The ANE method is developed to address a fundamental, open question on how to automatically design an optimal NN architecture for approximating functions and solutions of PDEs within a prescribed accuracy. Moreover, to train the resulting non-convex optimization problem, the ANE method provides a natural process of obtaining a good initialization.

Time: October 22, 2021 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Wolfgang Dahmen

Thursday, 16 September 2021 19:55

### A new real-space method for the simulation of scanning transmission electron microscope images

#### Speaker: Christian Doberstein (University of South Carolina)

I will present a new method for the simulation of annular dark field (ADF) images in scanning transmission electron microscopy (STEM). While the simulation of a conventional transmission electron microscopy (TEM) image requires solving the Schrödinger equation only very few times, simulating an ADF STEM image requires solving the Schrödinger equation several times for every pixel in the output image. This makes it a computationally challenging task and it is therefore important to find algorithms that reduce the computation time to a reasonably short duration.

One of the classical approaches to simulating a STEM image is the Multislice algorithm. In this algorithm, the specimen is first divided into thin slices perpendicular to the beam direction. Afterwards, solutions to the Schrödinger equation are computed by transmitting the probe wave function (i.e. the initial condition) slice by slice through the specimen for every probe position. Recently, a new algorithm termed PRISM has been developed to speed up the Multislice computations. This algorithm makes use of the linearity of the Schrödinger equation and propagates a small set of certain elementary wave functions through the specimen instead of the probe wave functions themselves. The probe wave functions are then approximated by linear combinations of these elementary wave functions, where the number of elementary functions may be much smaller than the number of probe wave functions. Although PRISM is a mathematically elegant way to reduce the number of Multislice computations, it can introduce large errors and require prohibitive amounts of computer memory. This is due to the choice of the elementary wave functions as Dirac deltas in Fourier space and the fact that they are highly nonlocal in real space coordinates.

These problems give rise to the idea of approximating the probe wave functions by a different set of "elementary wave functions" that are localized in real space coordinates. I will present an example for such a set of elementary wave functions and show that this makes it possible to keep the speedup of PRISM while avoiding the precision and memory issues. Additionally, I will show how the Multislice computations can be performed entirely in real space coordinates using the GPU, which should further speed up the computations.

Time: September 24, 2021 3:30pm-4:30pm
Location: Virtually via Zoom

Wednesday, 14 April 2021 19:27

### 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

Monday, 05 April 2021 11:33

### 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

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