Accurate estimation of microseismic events (small earthquakes) generated during hydraulic fracturing of low permeability rocks such as shale enables important characterization of hydraulic fracture networks. Determining the orientation of the fracture is important in characterizing the effectiveness of the stimulation process. We consider the source as separable in time and space and invert for a…
Geophysical seismic exploration, as well as radar and sonar imaging, require the solution of large scale forward and inverse problems for hyperbolic systems of equations. In this talk, I will show how model order reduction can be used to address some intrinsic difficulties of these problems. In model order reduction, one approximates the response (transfer…
The leading eigenvalue problems arise in many applications. When the dimension of the matrix is super huge, such as for applications in quantum many-body problems, conventional algorithms become impractical due to computational and memory complexity. In this talk, we will describe some recent works on new algorithms for the leading eigenvalue problems based on randomized and coordinate-wise methods (joint work with…
In nonlinear conservation laws the flux function f(u) is usually single valued, but in many important applications it is hysteretic, i.e., it assigns different values depending on whether the input u(t) is increasing or decreasing in t. We present our recent results on a hysteresis model built with a collection of auxiliary ODEs under constraints. The model shares some similarities…
When a hurricane threatens North Carolina, researchers use computational models to predict how the ocean waters will rise, and what areas will be flooded. Emergency managers rely on fast and accurate storm surge predictions from these models to make decisions and estimate damages during storm events. These models use unstructured, finite-element meshes to describe the…
(Brief Note: In this talk, we utilize Jupyter notebooks to re-create some of our published results in real-time and also build a "computational intuition" for the ideas presented. In this way, we are (mostly) transparent about all the computations involved in our work. I will email these materials to anyone interested after the presentation.) Models are useful for…
In this talk, I will review some mathematical challenges posed by the modeling of collective dynamics and self-organization. Then, I will focus on two specific problems, first, the derivation of fluid equations from particle dynamics of collective motion and second, the study of phase transitions and the stability of the associated equilibria.
We discuss the efficiency of recent advances on the vector penalty-projection methods including the kinematic version which uses fast discrete Helmholtz-Hodge decompositions on edge-based generalized MAC-type unstructured meshes. These methods are especially designed for the computation of incompressible or low-Mach multiphase flows under strong constraints (large density or viscosity ratios, large surface tension) and with…
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…