Dates

Spekers and Hosts

Title and Abstract

01/26/2018

Li Wang

SUNY Buffalo

Front capturing schemes for nonlinear PDEs with a free boundary limit

Evolution in physical or biological systems often involves interplay between nonlinear interaction among the constituent “particles”, and convective or diffusive transport, which is driven by density dependent pressure. When pressure-density relationship becomes highly nonlinear,  the evolution equation converges to a free boundary problem as a stiff limit. In terms of numerics, the nonlinearity and degeneracy bring great challenges, and there is lack of standard mechanism to capture the propagation of the front in the limit. In this talk, I will introduce a numerical scheme for tumor growth models based on a prediction-correction reformulation, which naturally connects to the free boundary problem in the discrete sense. As an alternative, I will present a variational method for a class of continuity equations (such as Keller-Segel model) using the gradient flow structure, which has built-in stability, positivity preservation and energy decreasing property, and serves as a good candidate in capturing the stiff pressure limit.

02/02/2018

Christian Klingenberg

Wuerzburg University

Multi-species kinetic and fluid models and applications

We consider a multi component gas mixture. This mixture is modelled by a system of kinetic BGK equations. Consistency of the model is proven, also existence, uniqueness and the positivity of solutions. We can extend our model to an ES-BGK model and to polyatomic mixtures. By taking moments, this allows us to derive macroscopic two-species conservation laws. We present numerical simulations using an adaptive kinetic-fluid models for plasma simulations. This is joint work with Marlies Pirner (Wuerzburg University, Germany) and Gabriella Puppo (Universita Insubria, Italy).

02/05/2018

4:00-5:00PM

RLM 6.104

(Mathematics Colloqium)

Liliana Borcea

University of Michigan

Untangling the nonlinearity in inverse scattering using data-driven reduced order models

We discuss an inverse problem for the wave equation, where an array of sensors probes an unknown, heterogeneous medium with pulses and measures the scattered waves. The goal in inversion is to determine from these measurements scattering structures in the medium, modeled mathematically by a reflectivity function. Most imaging methods assume a linear mapping between the unknown reflectivity and the array data. The linearization, known as the Born (single scattering) approximation is not accurate in strongly scattering media, so the reconstruction of the reflectivity may be poor. We show that  it is possible to remove the multiple scattering (nonlinear) effects from the data using a reduced order model (ROM). The ROM is defined by an orthogonal projection of the wave propagator operator on the subspace spanned by the time snapshots of the solution of the wave equation. The snapshots are known only at the sensor locations, which is enough information to construct the ROM. The main result discussed in the talk is a novel, linear-algebraic algorithm that uses the ROM to map the data to its Born approximation. 

02/06/2018

1:00-2:00PM

RLM 10.176

Liliana Borcea

University of Michigan

Pulse Reflection in a Random Waveguide with a Turning Point

Guided waves arise in a variety of applications like underwater acoustics, optics, the design of musical instruments, and so on. We present an analysis of wave propagation and reflection in an acoustic waveguide with random sound soft boundary and a turning point. The waveguide has slowly bending axis and variable cross section. The variation consists of a slow and monotone change of the width of the waveguide and small and rapid fluctuations of the boundary, on the scale of the wavelength. These fluctuations are modeled as random. The turning point is many wavelengths away from the source, which emits a pulse that propagates toward the turning point, where it is reflected.  We consider a regime where scattering at the random boundary has a significant effect on the reflected pulse. We determine from first principles when this effects amounts to a deterministic pulse deformation. This is known as a pulse stabilization result. The reflected pulse shape is not the same as the emitted one. It is damped, due to scattering at the boundary, and is deformed by dispersion in the waveguide. An example of an application of this result is in inverse problems, where the travel time of reflected pulses at the turning points can be used  to determine the geometry of the waveguide.

02/23/2018

Alexander Mamonov

University of Houston

Acoustic Imaging and Multiple Removal via Model Order Reduction

We introduce a novel framework for imaging and removal of multiples from waveform data based on model order reduction. The reduced order model (ROM) is an orthogonal projection of the wave equation propagator (Green's function) on the subspace of discretely sampled time domain wavefield snapshots. The projection can be computed just from the knowledge of the boundary waveform data using the block Cholesky factorization. Once the ROM is found, its use is twofold.

First, given a rough knowledge of kinematics, the projected propagator can be backprojected to obtain an image of reflectors in the medium. ROM computation implicitly orthogonalizes the wavefield snapshots. This highly nonlinear procedure differentiates our approach from the conventional linear migration methods (Kirchhoff, RTM). It allows to resolve the reflectors independently of the knowledge of the kinematics and to untangle the nonlinear interactions between the reflectors. As a consequence, the resulting images are almost completely free from the multiple reflection artifacts.

Second, the ROM computed from the original, multiply scattered waveform data can be used to generate the Born data set, i.e. the data that the measurements would produce if the propagation of waves in the unknown medium obeyed Born approximation instead of the full wave equation. Obviously, such data only contains primary reflections and the multiples are removed. Moreover, the multiply scattered energy is mapped back to primaries. Consecutively, existing linear imaging and inversion techniques can be applied to Born data to obtain reconstructions in a direct, non-iterative manner.

03/06/2018

Alexei Novikov

Penn State University 

Data Structures for Robust Multifrequency Imaging

We consider certain imaging problems that can be cast in the form of an underdetermined linear system of equations. When a single measurement vector is available, a sparsity promoting ℓ1-minimization based algorithm may be used to solve the imaging problem efficiently. A suitable algorithm in the case of multiple measurement vectors would be the MUltiple SIgnal Classification (MUSIC) which is a subspace projection method. In this talk I will look at both algorithms from a basic linear algebra point of view. It allows to explain conditions when the ℓ1-minimization problem and the MUSIC method admit an exact solution. We also examine the performance of these two approaches when the data are noisy. Several imaging configurations that fall under the assumptions of the theory are discussed such as active imaging with single or multiple frequency data and phaseless imaging. This is a joint work with M.Moscoso, G.Papanicolaou and C.Tsogka.

03/16/2018

SPRING BREAK: NO SEMINAR

04/02/2018

Monday

11:00-12:00

 Nat Trask

Sandia National Laboratories

Spatially compatible meshfree discretization

While meshfree methods have long promised a natural means of handling problems with large deformation with little numerical dissipation, they have struggled to obtain properties that are often taken for granted in mesh-based methods. From the perspective of compatible discretization, the lack of a mesh means that there is no chain complex upon which to develop a discrete exterior calculus. For this reason, particle methods that are simultaneously able to achieve high-order accuracy and discrete conservation principles have remained elusive. In this talk, we will present recent work from the Compadre (compatible particle discretization) project where we seek to develop meshfree discretizations that achieve these properties. In the first part of the talk, we present a computationally efficient meshfree Gauss divergence theorem which assigns virtual notions of volume and area to particles. With a consistent summation-by-parts theorem in hand, we then develop a meshfree analogue to the finite volume method and demonstrate its robustness when considering Darcy flows with jumps in material properties. In the second part of the talk, we present a meshfree approach to remedy well-known issues with numerical discretizations of peridynamics. While the non-local continuum theory of peridynamics provides an attractive framework for studying fracture with reduced regularity restrictions, particle discretizations of peridynamics fail to obtain a notion of asymptotic compatibility in which the discrete non-local solution recovers the exact local solution as the non-local interaction is reduced. We present a new optimization-based strong form method constructed to enforce reproduction of a given class of nonlocal operators, for which we prove asymptotic compatibility and demonstrate its implementation in a standard engineering workflow.

04/06/2018

Braxton Osting

University of Utah 

A generalized MBO diffusion generated method for constrained harmonic maps

A variety of tasks in inverse problems and data analysis can be formulated as the variational problem of minimizing the Dirichlet energy of a function that takes values in a certain submanifold and possibly satisfies additional constraints. These additional constraints may be used to enforce fidelity to data or other structural constraints arising in the particular problem considered. I'll present a generalization of the Merriman-Bence-Osher (MBO) method for minimizing such a functional. I’ll give examples of how this method can be used for the geometry processing task of generating quadrilateral meshes, finding Dirichlet partitions, and constructing smooth orthogonal matrix valued functions. For this last problem, I'll prove the stability of the method by introducing an appropriate Lyapunov function, generalizing a result of Esedoglu and Otto to matrix-valued functions. I'll also state a convergence result for the method. I’ll conclude with some applications in inverse problems for manifold-valued data. This is joint work with Dong Wang, Ryan Viertel, and Todd Reeb.

04/13/2018

 Lars Ruthotto

Emory University

An Optimal Control Framework for Efficient Training of Deep Neural Networks

One of the most promising areas in artificial intelligence is deep learning, a form of machine learning that uses neural networks containing many hidden layers. Recent success has led to breakthroughs in applications such as speech and image recognition. However, more theoretical insight is needed to create a rigorous scientific basis for designing and training deep neural networks, increasing their scalability, and providing insight into their reasoning.

In this talk, we present a new mathematical framework that simplifies designing, training, and analyzing deep neural networks. It is based on the interpretation of deep learning as a dynamic optimal control problem similar to path-planning problems.  We will exemplify how this understanding helps design, analyze, and train deep neural networks. First, we will focus on ways to ensure the stability of the dynamics in both the continuous and discrete setting and on ways to exploit discretization to obtain adaptive neural networks.  Second, we will present new multilevel and multiscale approaches, derived from he continuous formulation. Finally, we will discuss adaptive higher-order discretization methods and illustrate their impact on the optimization problem.

04/20/2018

04/27/2018

05/04/2018