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Current contacts: Benjamin Seibold or Daniel B. Szyld
The Seminar usually takes place on Wednesdays at 4:00 PM in Room 617 on the sixth floor of Wachman Hall. Click on title for abstract.
Jingmei Qiu, University of Delaware
I will provide overview of low rank time integrators for time dependent PDEs.These include an explicit scheme that involves a time stepping followed by a SVD truncation procedure with application to the Vlasov equations; two implicit schemes: Reduced Augmentation Implicit Low rank (RAIL) scheme and a Krylov subspace low rank scheme with applications to the heat equation and the Fokker-Planck equation; as well as implicit-explicit low rank integrators for advection-diffusion equations.
Rodolfo Ruben Rosales, Massachusetts Institute of Technology
A small liquid drop placed above the vibrating surface of a liquid, will not (under appropriate conditions) fall and merge. In fact it will bounce from the surface, and can be made to do so for very many bounces (hundreds of thousands). If the liquid below is just under the Faraday threshold, the drop excites waves with each bounce, and via these waves it can extract momentum from the fluid underneath it, and starts moving "walking" at some preferred speed. The drop-wave system then becomes a peculiar active system, where the active elements interact with each other (if there are many drops) via waves, as well as with their own past via the waves generated earlier in their history. This system have many special properties, some reminiscent of quantum mechanics. In this talk I will focus on one such property:
If the drop is constrained to move in a bounded region by some external force (e.g.: Coriolis), then its path exhibits radial quantization: the statistics for the radius of curvature along the drop path is concentrated on a discrete set of values. The question is why? There are various models that predict this, but the question is not about the model(s), but about what is the mechanism behind the behavior. An obvious answer is that it is because the drop motion is caused via waves. This is, basically, correct; but too vague, even misleading. First of all, the drop does not move on some "external" wave field, but on a self-generated one. Second, the waves decay, hence the wave field is dominated by the waves produced in the recent path. Yet, if one discards all but the recent past, the quantization disappears --- the recent past selects the preferred speed, but does not quantize. It turns out that the effect is caused by (exponentially suppressed) waves emitted in the past at "special" regions where constructive interference magnifies their effect. As I hope to show, this gives a simple and intuitive explanation of how the radii selection occurs.
Sean McQuade, Rutgers University Camden
Chemical networks, such as metabolism, can be simulated to assist in an array of research including new drug discovery, personalized medicine, and testing high-risk treatment before applying it to humans. Improved biochemical simulations can also reduce our dependence on animal testing before clinical trials. This talk demonstrates a mathematical framework for biochemical systems that was designed with two goals in mind: 1. improved early phase drug discovery and 2. personalized medicine. The talk also addresses a particular contribution that can be made by deep learning models.
Chun Liu, Illinois Institute of Technology
I will present the dynamic boundary conditions in the general
energetic variational approaches. The focus is on the coupling between
the bulk effects with the active boundary conditions.
In particular, we will study applications in the evolution of grain
boundary networks, in particular, the drag of trip junctions. This is a
joint work with Yekaterina Epshteyn (University of Utah) and Masashi
Mizuno (Nihon University).
Erik Boman, Sandia National Laboratory
Randomization has become a popular technique in numerical linear algebra in recent years, with applications in several areas from scientific computing to machine learning. We review some problems where it works well. Sketching is a powerful way to reduce a high-dimensional problem to a lower-dimensional problem. Sketch-and-solve and sketch-and-precondition are the two main approaches for linear systems and least squares problems. Finally, we describe two recent applications in more detail: Fast and stable orthogonalization (QR on tall, skinny matrices), and spectral graph partitioning.
Chen Greif, University of British Columbia
Double saddle-point systems are drawing increasing attention in the past few years, due to the importance of multiphysics and other relevant applications and the challenge in developing efficient iterative numerical solvers. In this talk we describe some of the numerical properties of the matrices arising from these problems. We derive eigenvalue bounds and analyze the spectrum of preconditioned matrices, and it is shown that if Schur complements are effectively approximated, the eigenvalue structure gives rise to rapid convergence of Krylov subspace solvers. A few numerical experiments illustrate our findings.
Hongchang Gao, Temple University
Stochastic Bilevel Optimization (SBO) has widespread applications in machine learning, such as meta learning, hyperparameter optimization, and network architecture search. To train those machine learning models on large-scale distributed data, it is necessary to develop distributed SBO algorithms. Therefore, Decentralized Stochastic Bilevel Optimization (DSBO) has been actively studied in recent years due to the efficiency and robustness of decentralized communication. However, it is challenging to estimate the stochastic hypergradient on each worker due to the loss function's bilevelstructure and decentralized communication.
In this talk, I will present our recent work on decentralized stochastic bilevel gradient descent algorithms. On the algorithmic design side, I will talk about how to estimate the hypergradient without incurring large communication overhead under both homogeneous and heterogeneous settings. On the theoretical analysis side, I will describe the convergence rate of our algorithms, showing how the communication topology, the number of workers, and heterogeneity affect the theoretical convergence rate. Finally, I will show the empirical performance of our algorithms.
Yekaterina Epshteyn, University of Utah
Many technologically useful materials are polycrystals composed of small monocrystalline grains that are separated by grain boundaries of crystallites with different lattice orientations. One of the central problems in materials science is to design technologies capable of producing an arrangement of grains that delivers a desired set of material properties.
A method by which the grain structure can be engineered in polycrystalline materials is through grain growth (coarsening) of a starting structure. Grain growth in polycrystals is a very complex multiscale multiphysics process. It can be regarded as the anisotropic evolution of a large cellular network and can be described by a set of deterministic local evolution laws for the growth of individual grains combined with stochastic models for the interaction between them. In this talk, we will present new perspectives on mathematical modeling, numerical simulation, and analysis of the evolution of the grain boundary network in polycrystalline materials. Relevant recent experiments will be discussed as well.
Francois-Henry Rouet, Ansys
Element-by-element preconditioners were an active area of research in the 80s and 90s, and they found some success for problems arising from Finite Element discretizations, in particular in structural mechanics and fluid dynamics (e.g., the "EBE" preconditioner of Hughes, Levit, and Winget). Here we consider problems arising from Boundary Element Methods, in particular the discretization of Maxwell's equations in electromagnetism. The matrix comes from a collection of elemental matrices defined over all pairs of elements in the problem and is therefore dense. Inspired by the EBE idea, we select subsets of elemental matrices to define different sparse preconditioners that we can factor with a direct method. Furthermore, the input matrix is rank-structured ("data sparse") and is compressed to accelerate the matrix-vector products. We use the Block Low-Rank approach (BLR). In the BLR approach, a given dense matrix (or submatrix, in the sparse case) is partitioned into blocks following a simple, flat tiling; off-diagonal blocks are compressed into low-rank form using a rank-revealing factorization, which reduces storage and the cost of operating with the matrix. We demonstrate results for industrial problems coming from the LS-DYNA multiphysics software.
Joint work with Cleve Ashcraft and Pierre L'Eplattenier
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