Numerical simulation of large-scale dynamical systems plays a crucial role and may be the only possibility in studying a great variety of complex physical phenomena with applications ranging from heat transfer to fluid dynamics, to signal propagation and interference in electronic circuits, and many more. However, these large-scale dynamical systems present significant computational difficulties when…

I will discuss formulations and algorithms for computing sparse optimal controls in systems governed by PDEs. These sparse solutions can guide the placement of control devices in applications.  After reviewing results for elliptic and parabolic PDEs, I will focus on recent work on sparse optimal control governed by linear PDEs with uncertain coefficients. Here, we aim at finding stochastic controls that…

An increasing number of scientific and enterprise data sets are multidimensional, where data is gathered for every configuration of three or more parameters. For example, physical simulations often track a set of variables in two or three spatial dimensions over time, yielding 4D or 5D data sets. Tensor decompositions are structured representations of multidimensional data…

The numerical solution of large-scale risk-averse PDE-constrained optimization problems requires substantial computational effort due to the discretization in physical and stochastic dimensions. Managing the cost is essential to tackle such problems with high dimensional uncertainties. In this work, we combine an inexact trust-region (TR) algorithm from with a local, reduced basis (RB) approximation to efficiently solve risk-averse optimization problems…

Semidefinite programs (SDPs) -- optimization problems with linear constraints, linear objective, and semidefinite matrix variables --  are some of the most useful, versatile, and pervasive optimization problems to emerge in the last 30 years. They find applications in combinatorial optimization, machine learning, and statistics, to name just a few areas. Unfortunately, SDPs often behave pathologically: the optimal values of the primal…

In general, solutions to the Laplacian equation enjoy relatively high smoothness. However, they can exhibit singular behaviors at domain corners or points where boundary conditions change type. In this talk, I will focus on the mixed Dirichlet-Neumann boundary conditions for Laplacian equation, and discuss how singularities in this case adversely affect the accuracy and convergence…

Poroelastic models have been widely used in Biomechanics. For example, modeling brain edema and cancellous bones. We aim at solving the Biot model under the MAC Finite Difference discretization and the stabilized finite element discretizations. To solve the resulting saddle point linear systems, some iterative methods are proposed and compared. In these methods, the outer iteration solver can be the GMRES…

Asynchronous methods refer to parallel iterative procedures where each process performs its task without waiting for other processes to be completed, i.e., with whatever information it has locally available and with no synchronizations with other processes. In this talk, an asynchronous version of the optimized Schwarz method is presented for the solution of differential equations on a parallel computational environment. Convergence is…

The study of the propagation of undular bores in channels is relevant to many applications which go from the propagation of tsunami waves, to that of tidal bores/waves, to the propagation of strong waves in manmade channels due to hazards (e.g. dam breaking). In absence of banks, the flow exhibits a transition across which undulating waves  start breaking and transform…

What to do when the size and complexity of your model essentially  prevent you from using it? Well, get a smaller and simpler model... At the heart of this dimension reduction process is the notion of parameter importance which, ultimately, is part of the modeling process itself. Global Sensitivity Analysis (GSA) aims at efficiently  identifying important…

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…

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…