| Thomas Hudson, CERMICS – École des Ponts ParisTech |
| Title: Stability and motion of screw dislocations in a lattice model |
| Abstract:Dislocations are defects found in crystalline solids, and their motion provides a mechanism through which such materials may plastically deform. In this talk, I will present a series of results concerning a lattice model in which configurations containing screw dislocations may be identified as local minima of a certain energy. Based upon this description, a stochastic model is then proposed for thermally-driven dislocation motion. In the limit of low temperature, a Large Deviations Principle for the evolution can be found, showing that the most probable trajectory of the system is along solutions of a generalized gradient flow. These results provide a first rigorous identification of a regime in which Discrete Dislocation Dynamics models, commonly simulated by Materials Scientists in practice, is valid. |
AUTHOR
Duan Chen
Wednesday May 4th, at 2:00PM Math Conference room
Categories: Spring 2022
Friday April 29th, at 11:00AM Math Conference room
Categories: Spring 2022
| Kelin Xia, Michigan State University |
| Title: Topological modeling and analysis of complex data in biomolecules |
| Abstract: Proteins are the most important biomolecules for living organisms. The understanding of protein structure, function, dynamics, and transport is one of the most challenging tasks in biological science. We have introduced persistent homology for extracting molecular topological fingerprints (MTFs) based on the persistence of molecular topological invariants. MTFs are utilized for protein characterization, identification, and classification. The multidimensional persistent homology is proposed and further used to quantitatively predict the stability of protein folding configurations generated by steered molecular dynamics. An excellent consistence between my persistent homology prediction and molecular dynamics simulation is found. Further, we introduce multiresolution persistent homology to handle complex biomolecular data. The essential idea is to match the resolution with the scale of interest so as to represent large scale datasets with appropriate resolution. By appropriately tuning the resolution of a density function, we are able to focus the topological lens on the scale of interest. The proposed multiresolution topological method has potential applications in arbitrary data sets, such as social networks, biological networks and graphs. |
Friday April 22nd, at 11:00AM Math Conference room
Categories: Spring 2022
| Xiu Yang, |
| Title: Enhancing Sparsity in Uncertainty Quantification Problems by Iterative Rotations |
| Abstract: Compressive sensing has become a powerful addition to uncertainty quantification in recent years. It helps to extract information from limited data. We propose a new method to enhance the sparsity of the representation of the uncertainty of the quantity of interest. Specifically, we consider rotation-based linear mappings which are determined iteratively for generalized polynomial chaos expansions.This procedure increases the accuracy of the compressive sensing-based uncertainty quantification method. We demonstrate the efficiency of this method with several examples. |
Friday April 1st, at 11:00AM Math Conference room
Categories: Spring 2022
| Christoph Ortner, Mathematics Institute, University of Warwick |
| Title: The Dimer Method for Saddle Point Computations |
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Abstract:The dimer method is a simple hessian-free algorithm for computing index-1 saddles. I will review this algorithm and describe some improvements to its efficiency, in particular adding preconditioning capabilities and line-search based on a local merit function. I will demonstrate the efficiency of the new variant on a range of applications from academic toy problems, an atomistic problem and a PDE problem. Despite these new improvements, we can currently give no global convergence guarantee. Indeed, we can construct counterexamples to global convergence. I will conclude my talk by explaining some of the difficulties we encountered and posing a challenge for the optimisation community. |
Friday March 18th, at 11:00AM Math Conference room
Categories: Spring 2022
| Yue Yu, Department of Mathematics, Lehigh University |
| Title: Stabilized numerical methods for fluid-structure interactions with application in cerebral aneurysm simulations |
| Abstract: Cerebral aneurysm is a diseased dilatation of the arterial walls in brain, and its rupture can lead to intracranial hemorrage and subsequent death. While the current clinical technology could not provide detailed in vivo measurements for intra-aneurysm flow patterns, fluid (blood) structure (arterial wall) interaction simulations then appear as an effective alternative approach for un- derstanding the mechanisms behind aneurysm growth and rupture. There are two approaches in formulating the discrete systems in simulating fluid-structure interaction (FSI) problems: the monolithic approach, and the partitioned approach. The former is efficient for small problems but does not scale up to realistic sizes, whereas the latter suffers from numerical stability issues. Here we consider the partitioned approach and we develop new stabilized algorithms. In par- ticular, in cerebral aneurysm simulations where the mass ratio between the structure and the fluid is relatively small, the partitioned approach gives rise to the so-called added-mass effect which renders the simulation unstable. I will present two new numerical methods to handle this added-mass effect: (1) by introducing fictitious pressure (acceleration) terms in the fluid (structure) equations to balance the added-mass effect, which stabilizes the coupled formulation and reduces drastically the number of subiterations in each time step; (2) by relaxing the exact no-slip boundary condition and introducing proper penalty terms on the fluid-structure inter- face, which enables the possibility of stable explicit coupling procedure. For both methods we obtained the optimal parameters via theoretical analysis, and numerically verified that stability can be achieved irrespective of the fluid-structure mass ratio. Moreover, I will also discuss an application in three-dimensional large scale simulations which were obtained for patient- specific cerebral aneurysms. With stabilized FSI method applied, the 3D fractional-order PDEs (FPDEs) were investigated which better describe the viscoelastic behavior of cerebral arterial walls. |
Friday March 25th, at 2:00PM Math Conference room
Categories: Spring 2022
| Jennifer L Sinclair, Department of Mathematics, Georgia Gwinnett College |
| Title: On applications of generalized tempered stable processes |
| Abstract: This seminar introduces the class of generalized tempered stable (GTS) processes and describes the formulation of GTS processes as a natural extension of tempered stable processes, which have applications in physics and finance. GTS processes encompass variations of tempered stable processes that have been introduced in the field, including modified tempered stable processes, layered stable processes, and Lamperti stable processes. Recent variations of tempered stable processes in the literature include new applications in financial mathematics and “p-tempered distributions.” In this talk, short and long time behavior of GTS Lévy processes is characterized and the absolute continuity of GTS processes with respect to the underlying stable processes is established. Series representations of GTS Lévy processes are derived. |
Thursday March 10th, at 2:00PM Math Conference room
Categories: Spring 2022
| Wayne Lawton, Department of Mathematics, National University of Singapore |
| Title: Prediction Theory and Spectral Factorization |
| Abstract: Prediction theory studies stationary random functions on ordered groups. Time series are functions on the integer group characterized by classical harmonic analysis results such as the Fejer and Szego spectral factorization theorems. Images and wavelet filters are functions on higher rank groups. We apply results in entire functions, ergodic theory and harmonic analysis to characterize multivariable spectral factors. |
Friday February 19th, at 10:00AM Room: Fret 315
Categories: Spring 2022
| Derek Olson, Department of Mathematics, University of Minnesota |
| Title: Blended-force based quasicontinuum method for multilattices |
| Abstract: Atomistic-to-continuum (AtC) methods are a vast class of multiphysics models which couple atomistic models of materials, such as molecular statics or molecular dynamics, with continuum mechanics models, such as nonlinear elasticity. The last thirty years has seen a surge of interest in the development of these methods, especially for crystalline materials, and even a thorough mathematical understanding of these methods has begun to emerge in the last five years. However, the applicability of these methods to crystalline materials has often been restricted only to materials comprised of a Bravais lattice. In this talk, we will present the blended force-based quasicontinuum method for modeling defects in multilattices, which allows for technologically important materials such as alloys and graphene to be modeled. Error estimates in terms of the computational cost of implementing the method will also be discussed. This is joint work with Xingjie Li, Christoph Ortner, and Mitchell Luskin. |
Thursday February 11th, at 3:00PM Room: Fret 223
Categories: Spring 2022
| Harbir Antil, Department of Mathematical Sciences, George Mason University |
| Title: Optimal control of fractional order PDEs |
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Abstract:Diffusion is the tendency of a substance to evenly spread into an available space, and is one of the most common physical processes. The classical models of diffusion lead to well-known equations. However, in recent times, it has become evident that many of the assumptions involved in these models are not always satisfactory or not even realistic at all. Consequently, different models of diffusion have been proposed, fractional diffusion being one of them. The latter has received a great deal of attention recently, mostly fueled by applications in diverse areas such as finance, turbulence and quasi-geostrophic flow models, image processing, peridynamics, biophysics, and many others. This talk will serve as an introduction to fractional diffusion equation – fractional derivative in both space and time. A novel PDE result by Caffarelli and Silvestre ’07 has led to innovative schemes to realize the fractional order operators. We will discuss these numerical methods and their application to PDE constrained optimization problems. |
Friday January 15th, at 9:45AM in Fretwell 128
Categories: Spring 2022
| Takumi Saegusa, Department of Mathematics, University of Maryland, College Park Hosted by Yanqing Sun |
| Title: Statistical Methods for Two-phase Stratified Sampling |
| Abstract: Two-phase stratified sampling is a sampling technique for cost reduction and improving efficiency. Examples includes stratified case control study and exposure-stratified case cohort study. A main theoretical challenge is a dependent and biased sample due to sampling without replacement in each stratum. A biostatistical approach to this issue is to approximate design by stratified Bernoulli sampling and focus on few specific models including the Cox model, resulting in paucity of research in general methodology such as bootstrap. An approach from sampling theory is to impose general conditions regardless of designs, leading to implicit asymptotic distributions and inefficient statistical methods applicable for any design. Our approach, which extends empirical process theory to two-phase stratified sampling, explicitly obtains asymptotic distributions, and yields general methodology tailored to two-phase stratified sampling. In this talk, we consider three statistical problems, model selection, improving efficiency and variance estimation, arising from the RV144 case control study. Our approach illustrates inadequacy of existing methods in these problems, and naturally introduces our proposed methods. Finite sample properties are investigated in simulation studies using the logistic regression model and the Cox model. |