Seminar on numerical methods

Organizers: Marcus Sarkis and Zhongqiang (Handy) Zhang

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Schedule Spring 2019

Wednesday, May 01, 2019
11:00-11:50 am, Room SH203
Hengguang Li (Wayne State University)

New 3D anisotropic meshes: regularity and a priori analysis.

Abstract. New mesh algorithms were developed for numerical approximations of elliptic equations with singularities. These algorithms are simple, intuitive, and impose fewer geometric constraints on the domain. The resulting mesh is generally anisotropic and may not possess the maximum angle condition. In this talk, we present new regularity results on polyhedral domains and propose optimal anisotropic finite element algorithms approximating the 3D singular solution.

Thursday, Feb 21, 2019
11:00-11:50 am, Room SH 203
Amanda Diegel (Mississippi State University)

Finite Element Methods for Phase Field Models

Abstract. In this talk, we investigate numerical methods relating to phase field models with a particular attention to two-phase flow models. Due to the vastness in applicability, two-phase flow models have drawn the attention of a number of researchers in recent years. One of the biggest challenges to modeling these systems lies in the difficulties regarding the moving interfaces (or boundaries) between the various phases. The traditional sharp interface models usually lead to almost unsolvable theoretical problems, not to mention the hardships found while attempting to derive stable and convergent computational schemes for these problems. To overcome these hardships, a phase field approach is taken such that the Cahn-Hilliard equation is coupled with fluid flow equations, thus creating a diffuse interface and eliminating the need to explicitly track a sharp interface. The numerical methods we discuss in this talk mimic the energy dissipation laws inherent to models using a phase field approach. Creating numerical methods in this way makes it possible to rigorously prove three key properties: unconditional stability, unconditional unique solvability, and optimal convergence. Convergence results provide valuable feedback concerning the approximation properties of a numerical scheme and unconditional stability leads to enhanced convergence estimations which leads to high confidence that the numerical schemes accurately estimate solutions to the equations upon which they are designed. If time allows, we will conclude the talk with an extension of the ideas presented to the development of a novel finite element method for a phase field model of nematic liquid crystal droplets.

Schedule Fall 2018

Thursday, November 29, 2018
11:00-11:50 am, Room SH 203
Calina Copos (Courant Institute)

Title. Modeling the cell cytoplasm rheology in confined environments

Abstract. Microfluidic devices have found numerous applications in biology and medicine because of their ability to efficiently control and replicate microenvironments. Cell migration through microfluidic channels has gained interest as an experimental method for one-dimensional, directed migration and has been applied to study red blood cell flow, differentiation of cancer cells, and the role of interstitial flow in tumor cell migration. In such confined microenvironments, the rheology of the cytoplasm becomes an important factor in determining the escape time across the channel. With this goal in mind, we consider a poroelastic immersed boundary method in which a fluid permeates a porous, elastic structure of negligible volume fraction, and extend this method to include stress relaxation of a moving, deforming material. Finally, we use this modeling framework to study the passage of a cell through a microfluidic channel. In this confined experimental setup, we demonstrate that the rheology of the cell cytoplasm is important for capturing the transit time through a narrow channel in the presence of a pressure drop in the extracellular fluid.

Faculty Host. Professor Sarah Olson

Thursday, November 01, 2018
11:00-11:50 am, Room SH 203
Jiayu Zhai (UMASS Amherst)

Title: Convergence Analysis of a Finite Element Approximation of Minimum Action Methods

Abstract. The Freidlin-Wentzell (F-W) theory of large deviations is a rigorous mathematical tool to study small-noise-induced transitions in a dynamical system as rare events. In this work, we address the convergence of a finite element approximation of the minimizer of the F-W action functional for non-gradient dynamical systems perturbed by small noise.

Thursday, October 04, 2018
11:00-11:50 am, Room SH 203
Mikhail Zaslavsky (Schlumberger-Doll Research)

Joint with Statistics Seminar

Title: Clustering of graph vertex subset via Krylov subspace model reduction

Abstract. Clustering via graph-Laplacian spectral imbedding is ubiquitous in data science and machine learning. It provides a low dimensional parametrization of the data manifold which makes the subsequent clustering (with, say, k-means or any of its approximations) much easier. However, it becomes less efficient for large data sets due to two factors. First, computing the partial eigendecomposition of the graph-Laplacian typically requires a large Krylov subspace. Second, after the spectral imbedding is complete, the clustering is typically performed with various relaxations of k-means, which may lose robustness with respect to the initial guess, become prone to getting stuck in local minima and scale poorly in terms of computational cost for large data sets.

Normalized graph-Laplacian is intimately related to the random walk on the graph, and we will exploit this connection in our algorithms. In particular, we propose two novel algorithms for spectral clustering of a subset of the graph vertices (target subset) based on the theory of model order reduction. They rely on realizations of a reduced order model (ROM), that accurately approximates the transfer function of the random walk on the original graph for inputs and outputs restricted to the target subset. While our focus is limited to this subset, our algorithms produce its clusterization that is consistent with the overall structure of the graph and thus with the full graph clustering if one would perform such. In particular, it preserves such parameters of the random walk on the full graph as diffusion and commute-time distances between subset nodes. Moreover, working with a small target subset reduces greatly the required dimension of Krylov subspace and allows to exploit the approximations of k-means in the regimes when they are most robust and efficient.

There are several uses for our algorithms. First, they can be employed on their own to clusterize a representative subset in cases when the full graph clustering is either infeasible of simply not required. Second, they may be used for quality control and filtering of noisy data, i.e., outliers. Third, as they drastically reduce the clustering problem size, they enable the application of more sophisticated and powerful approximations of k-means like those based on semi-definite programming (SDP) instead of the conventional Lloyd's algorithm. Finally, they can be used as building blocks of a divide-and conquer type algorithm for the full graph clustering (in progress).

I'll provide the results of numerical experiments with synthetic data as well as real-world statistical data for companies email connections and for citations in ArXiv repository. Time permitting, I'll discuss preliminary results for ongoing project with financial statistics of the stock market data.

This is joint work with Vladimir Druskin (Worcester Polytechnic Institute) and Alexander Mamonov (University of Houston)

Thursday, September 27, 2018
11:00-11:50 am, Room SH 203
Zhicheng Wang (MIT)

Title: A phase-field method for large-eddy simulation of two-phase slug flow in a horizontal pipe

Abstract. We present a phase-field method that are capable of large-eddy simulation (LES) of the initial- ization and development of the two-phase slugs from a stratified flow in a long horizontal pipe using realistic parameters of density ratio, viscosity ratio, Weber number and Reynolds number. The method is implemented both in a three-dimensional spectral-elment and a spectral-element/Fourier code that make the simulation be fast with a high accuracy. The free interface is solved by the Cahn-Hilliard equation that leads to excellent mass conservation. The Entropy Viscosity Method (EVM) is employed as LES subgrid eddy viscosity model. The visualization of the simulation re- sult exhibits the richness of the evolving topology of the slugs, while the predicted slug length and frequency are in good agreement with experimental measurements.

Thursday, September 20, 2018
11:00-11:50 am, Room SH 203
Cheng Wang (UMASS Dartmouth)

Title: Epitaxial thin film growth model and its numerical simulation

Abstract. A nonlinear PDE model of thin film growth model, with or without slope selection, are presented in the talk. A global in time solution with Gevrey regularity is established for the one with slope selection. For the numerical simulation, an idea of convex-concave splitting of the corresponding physical energy is applied, which gives to an implicit treatment for the convex part and an explicit treatment for the concave part. That in turn leads to a numerical scheme with a non-increasing energy. Both the first and second order splittings in time will be considered in the work. Some numerical simulation results are also presented in the talk.

Thursday, August 30, 2018
11:00-11:50 am, Room SH 203
Xiaoning Zheng (Division of Applied Mathematics, Brown University)

Title: A 3D phase-field model for multiscale simulation of the interaction between blood and thrombus in arterial vessels

Abstract: We developed a three dimensional multi-phase computational model that is calibrated by existing in vivo and in vitro experimental data to study the interactions between the blood clot and the blood. The model is validated with data from experiments and literature. Simulations provide new insights into mechanisms underlying clot deformation and permeability that cannot be studied experimentally at this time. In particular, model simulations show that flow-induced changes in size, shape and internal structure of the clot are largely determined by shear-dependent mechanisms. These results can be used in future to predict risk of thrombotic event based on the data about permeability and deformability of a clot under different blood flow conditions.

Past talks

2017-2018 list of talks

2016-2017 list of talks

List of Seminars at Department of mathematical Sciences