## Math/ICES Center of Numerical Analysis Seminar (Spring 2017)

Time and Location: Friday, 1:00-2:00PM, POB 6.304 Special time and locations are indicated in red.

 Dates Spekers and Hosts Title and Abstract 02/03/2017 Fernando Guevara-Vasquez University of Utah Wave manipulation of particles in a fluid Consider a compressible fluid that is subject to a standing acoustic wave. Particles within the fluid are subject to an acoustic radiation force and tend to move to minima of the associated potential. In many cases, these minima coincide with the nodal sets of the standing wave, which are solutions to the Helmholtz equation. We present two methods for solving the control problem of finding the settings for transducers lining a reservoir (i.e. boundary conditions), necessary to best approximate a desired particle pattern within the reservoir. In the first method, a discretized version of the control problem is reduced to finding the smallest eigenvalue of a matrix. In the second method, we use an approximation result of functions by entire solutions to the Helmholtz equation to give an efficient and explicit solution to the control problem. An application of this principle is to fabricate selectively reinforced composite materials, where the matrix is a photo-cured polymer and the inclusions are carbon nanotubes. 02/10/2017 02/17/2017 NO SEMINAR 02/24/2017 NO SEMINAR 03/03/2017 NO SEMINAR 03/07/2017 Jose Morales Escalante Technical University of Vienna, Austria DG Schemes for Collisional Electron Transport with Insulating Conditions on Rough Boundaries We consider the mathematical and numerical modeling of reflective boundary conditions, including diffusive reflection in addition to specularity, in the context of collisional electron transport in semiconductors, and their implementation in Discontinuous Galerkin (DG) schemes that solve the related Boltzmann kinetic model. We study the specular, diffusive and mixed (specular plus diffusive) reflection BC on physical boundaries of the problem. We develop a numerical approximation to model an insulating boundary condition, or equivalently, a zero flux mathematical condition for the electron transport equation. This condition balances the incident and reflective momentum flux at the microscopic level pointwise at the boundary, for the case of a more general mixed reflection with momentum dependant specularity probability. We compare the computational prediction of physical observables given by the numerical implementation of these different reflection conditions in our DG scheme for the Boltzmann model, and observe the influence of the diffusive condition in the kinetic moments over the position domain. 03/10/2017 Yunan Yang UT Austin Optimal Transport for Seismic Inversion Optimal transport has become a well developed topic in analysis since it was first proposed by Monge in 1781. Due to their ability to incorporate differences in both intensity and spatial information, the related Wasserstein metrics have been adopted in a variety of applications, including seismic inversion. Quadratic Wasserstein metric (W2) has ideal properties like convexity and insensitivity to noise, while conventional L2 norm is known to suffer from local minima. We propose two ways of using W2 in seismic inversion, a trace-by-trace comparison solved by sorting, and the global comparison which requires numerical solution to Monge-Ampere equation. 03/17/2017 SPRING BREAK 03/24/2017 Samuel Cole University of Illinois at Chicago A simple algorithm for spectral clustering of random graphs A basic problem in data science is to partition a data set into “clusters" of similar data. When the data are represented in a matrix, the spectrum of the matrix can be used to capture this similarity. This talk will consider how this spectral clustering performs on random matrices. Specifically, we consider the planted partition model, in which $n = ks$ vertices of a random graph are partitioned into $k$ clusters, each of size $s$. Edges between vertices in the same cluster and different clusters are included with constant probability $p$ and $q$, respectively (where $0 \le q < p \le 1$). We present a simple, efficient algorithm that, with high probability, recovers the clustering as long as the cluster sizes are are least $\Omega(\sqrt{n})$. 03/31/2017 04/07/2017 Peijun Li Purdue University Inverse Source Problems for Wave Propagation The inverse source problems, as an important research subject in inverse scattering theory, have significant applications in diverse scientific and industrial areas such as antenna design and synthesis, medical imaging, optical tomography, and fluorescence microscopy. Although they have been extensively studied by many researchers, some of the fundamental questions, such as uniqueness, stability, and uncertainty quantification, still remain to be answered. In this talk, our recent progress will be discussed on the inverse source problems for acoustic, elastic, and electromagnetic waves. I will present a new approach to solve the stochastic inverse source problem. The source is assumed to be a random function driven by the additive white noise. The inverse problem is to determine the statistical properties of the random source. The stability will be addressed for the deterministic counterparts of the inverse source problems. We show that the increasing stability can be achieved by using the Dirichlet boundary data at multiple frequencies. I will also highlight ongoing projects in random medium and time-domain inverse problems. 04/14/2017 Luis Chacon LANL A Multiscale, Conservative, Implicit 1D-2V Multispecies Vlasov-Fokker-Planck Solver for ICF Capsule Implosion Simulations Plasma collisionality conditions during the implosion of an ICF capsule vary widely. Early in the implosion process, the plasma is cold and very collisional. Later in the implosion, however, the plasma becomes very hot, and the collisional mean free path becomes a large fraction of the system size. In this regime, kinetic phenomena may become important, and a fully kinetic treatment is needed to assess their impact on compression and yield in ICF capsules. Modeling such kinetic behaviors at any level of fidelity, however, demands a quantum leap in algorithmic complexity from standard radiation-hydrodynamics models currently in use. Kinetic physics in semi-collisional plasmas is governed by the multispecies Vlasov-Fokker-Planck equation, which is a high-dimensional (3D+3V+time), multiscale set of equations supporting very disparate time and length scales. The Fokker-Planck collision operator is nonlinear and nonlocal, and features strict conservation properties in the continuum that must be numerically enforced for long-term accuracy. Even in reduced dimensionality (1D-2V, spherically symmetric), a naive numerical treatment of this set of equations for ICF simulation is impractical, demanding circa 10^10 time steps and 10^12 degrees of freedom, which are way beyond exascale computing. In this talk, we present a fully conservative (mass, momentum, and energy), fully implicit multispecies Vlasov-Rosenbluth-Fokker-Planck solver in 1D-2V. The approach achieves exact numerical conservation by nonlinearly enforcing the collision operator symmetries, and by enslaving numerical truncation errors [1]. Positivity is enforced by taking advantage of the advection-diffusion structure of the Fokker-Planck collision operator [1,2]. The approach features an adaptive scheme in velocity space that optimally resolves the distribution function locally, thus substantially decreasing the velocity space resolution requirements regardless of temperature disparity and variations [2]. Solver-wise, the code relies on optimal multigrid-preconditioned Jacobian-free Newton-Krylov strategies [3], which we generalize here to deal with multiple ion species. Our proposed algorithm has been specifically designed to deal with spatio-temporal temperature disparities such as those present in ICF capsules, and as a result it is able to simulate ICF implosions at a fraction of the cost of the naive estimates (10^5 time steps and 10^6 degrees of freedom), well within current computational capabilities. The resulting code, iFP, has been thoroughly verified in planar and spherical geometries, and we have begun exercising it for the simulation of kinetic interfaces [4] and spherical ICF implosions [5]. In this talk, we will provide a number of numerical examples demonstrating the accuracy and efficiency of the scheme, and we will provide first insights into the importance of kinetic ion-species segregation effects in the reactivity of ICF capsules. [1] W. T. Taitano, L. Chacón, A. N. Simakov, K. Molvig, J. Comput. Phys., 297, 257-380 (2015) [2] W. T. Taitano, L. Chacón, A. N. Simakov, J. Comput. Phys., 318, 391–420 (2016) [3] L. Chacón, D. C. Barnes, D. A. Knoll, G. H. Miley, J. Comput. Phys., 157, 654-682 (2000). [4] L. Yin, B. J. Albright, W. Taitano, E. L. Vold, L. Chacón, and A. Simakov, Phys. Plasmas, 23, 112302 (2016) (2016) [5] W. Taitano, L. Chacón, A. N. Simakov, “An adaptive, mass, momentum, and energy conserving, 1D-2V mulit-ion Vlasov-Rosenbluth-Fokker-Planck solver with fluid electrons,” In preparation. 04/21/2017 Mike O'Neil Courant Institute Integral equation methods for the Laplace-Beltrami problem The reformulation of many of the classical constant-coefficient PDEs of mathematical physics (e.g. Laplace, Helmholtz, Maxwell, etc.) as boundary integral equations is a standard mathematical tool, which, when coupled with iterative solvers and fast algorithms such as fast multipole methods (FMMs), allows for the nearly optimal-time solution of these PDEs. Extending these methods to variable coefficient PDEs, especially those defined along surfaces, is not straightforward and ongoing research. In this talk, we will address the problem of solving the Laplace-Beltrami problem along surfaces in three dimensions. The Laplace-Beltrami problem is a variable coefficient PDE, with applications in electromagnetics, fluid-structure interactions, and surface diffusions. Our resulting integral equation is ready for acceleration using standard FMMs for Laplace potentials; several numerical examples will be provided. 04/28/2017 Jean Ragusa Texas A&M TBA 05/05/2017 Carlos Borges ICES, UT Austin High Resolution Solution of Inverse Scattering Problems I describe a fast, stable framework for the solution of the inverse acoustic scattering problem. Given full aperture far field measurements of the scattered field for multiple angles of incidence, the recursive linearization is used to obtain high resolution reconstructions of properties of the scatterer. Despite the fact that the underlying optimization problem is formally ill-posed and non-convex, recursive linearization requires only the solution of a sequence of linear least squares problems at successively higher frequencies. By seeking a suitably band-limited approximation of the sound speed prole, each least squares calculation is well-conditioned and involves the solution of a large number of forward scattering problems. For two dimension problems we employ spectrally accurate, fast direct solvers. For the largest problems considered, approximately one million partial differential equations were solved, requiring approximately two days to compute using a parallel MATLAB implementation on a multi-core workstation.