In this talk, we demonstrate one way of exploiting smooth structures hidden in convex functions to develop optimization algorithms. Our key idea is to generalize a powerful concept so-called "self-concordance" introduced by Y. Nesterov and A. Nemirovskii to a broader class of convex functions. We show that this structure covers many applications in statistics and machine learning. Then, we develop a…

he emergence of vascular networks is a long-standing problem which has been the subject of intense research in the past decades. One of the main reasons being the widespread applications that it has in tissue regeneration, wound healing, cancer treatment, etc. The mechanisms involved in the formation of vascular networks are complex and despite the vast amount of research devoted to it, there are still…

Much of the literature on the design and analysis of supersaturated designs (SSDs), in which the number of factors exceeds the number of runs, rests on design principles assuming a least-squares analysis.  More recently, researchers have discovered the potential of analyzing SSDs with penalized regression methods like the LASSO and Dantzig selector estimators.  There exists much theoretical work for these methods…

When running any iterative algorithm it is useful to know when to stop. Here we review LSQR and LSLQ, two iterative methods for solving \min_x \|Ax-b\|_2 based on the Golub-Kahan bidiagonalization process, as well as estimates for the 2-norm error \|x-x_*\|_2, where x_* is the minimum norm solution. We also review the closely related Craig's…

Mappings between domains are among the most basic and versatile tools used in the computational analysis and manipulation of shapes. Their applications range from animation in computer graphics to analysis of anatomical variation and anomaly detection in medicine and biology. My talk will start with a brief overview of discrete computational shape mapping, surface parameterization…

Tensors (aka multiway arrays) can be instrumental in revealing latent correlations residing in high dimensional spaces. Despite their applicability to a broad range of applications in machine learning, speech recognition, and imaging, inconsistencies between tensor and matrix algebra have been complicating their broader utility.  Researchers seeking to overcome those discrepancies have introduced several different candidate…

This event has been rescheduled for August 25. We present a fast and accurate numerical method for the simulation of time domain scattering problem. Both acoustic and electromagnetic scattering problems are discussed. Nonreflecting boundary conditions (NRBCs) are used to truncate the problem. We first derive analytic expressions for the underlying convolution kernels which allow for a rapid and accurate…

Large-scale optimization is ubiquitous in scientific and engineering applications. The end goal in most applications is the solution is a design, control, or inverse problem, constrained by complex high-fidelity models. Achieving this goal is challenging for many reasons, most notably, the computational complexity of the models and their numerous sources of uncertainty. This talk introduces…

Physics-informed neural networks (PINNs) have lately received great attention thanks to their flexibility in tackling a wide range of forward and inverse problems involving partial differential equations. However, despite their noticeable empirical success, little is known about how such constrained neural networks behave during their training via gradient descent. More importantly, even less is known…

Porous media and conduit coupled systems are heavily used in a variety of areas such as groundwater system, petroleum extraction, and biochemical transport. A coupled dual porosity Stokes model has been proposed to simulate the fluid flow in a dual-porosity media and conduits coupled system. Data assimilation is the discipline that studies the combination of mathematical models and observations. It…

We consider a so-called fractional gradient flow: an evolution equation aimed at the minimization of a convex and l.s.c. energy, but where the evolution has memory effects. This memory is characterized by the fact that the negative of the (sub)gradient of the energy equals the so- called Caputo derivative of the state. We introduce a…

We consider the problem of inferring the basal sliding coefficient field for an uncertain Stokes ice sheet forward model from surface velocity measurements. The uncertainty in the forward model stems from unknown (or uncertain) auxiliary parameters (e.g., rheology parameters). This inverse problem is posed within the Bayesian framework, which provides a systematic means of quantifying uncertainty in the solution. To account…

 Inverse problems are ubiquitous in many areas of Science and Engineering and, once discretized, they lead to ill-conditioned linear systems, often of huge dimensions: regularization consists in replacing the original system by a nearby problem with better numerical properties, in order to find a meaningful approximation of its solution. After briefly surveying some standard regularization…

The resurging interest in deep learning is commonly attributed to advances in hardware and growing data sizes and less so to new algorithmic improvements. However, cutting-edge numerical methods are needed to tackle ever larger and more complex learning problems. In this talk, I will illustrate numerical analysis tools for improving the effectiveness of deep learning…

Two-dimensional crystals have been intensely investigated both experimentally and theoretically since graphene was exfoliated from graphite. Physicists have recently developed the ability to stack one layer on another with a twist angle controlled to the scale of .1 degree with the goal of creating two dimensional materials with desired electronic, optical, and mechanical properties. Unusual…

Randomized methods are becoming increasingly popular in numerical linear algebra. However, few attempts have been made to use them in developing preconditioners. Our interest lies in solving large-scale sparse symmetric positive definite linear systems of equations where the system matrix is preordered to doubly bordered block diagonal form (for example, using a nested dissection ordering).…

The magnetohydrodynamics (MHD) equations are continuum models used in the study of a wide range of plasma physics systems. However, efficient numerical solution methods for MHD are extremely challenging due to disparate time and length scales, strong hyperbolic phenomena, and nonlinearity. Therefore the development of scalable, implicit MHD algorithms and high-resolution adaptive mesh refinement strategies…

 Nonlinear preconditioning refers to transforming a nonlinear algebraic system to a form for which Newton-type algorithms have improved success through quicker advance to the domain of quadratic convergence. We place these methods, which go back at least as far as the Additive Schwarz Preconditioned Inexact Newton (ASPIN, 2002) in the context of a proliferation distinguished…