# ACMS Abstracts: Fall 2015

### Li Wang (UCLA)

Singular shocks: From particle-laden flow to human crowd dynamics

In this talk, we will present two examples in which singular shock arises. The first example, gravity-driven thin film flow with a suspension of particles down an incline, is described by a system of conservation laws equipped with an equilibrium theory for particle settling and resuspension. Singular shock appears in the high particle concentration case that relates to the particle-rich ridge observed in the experiments. We analyze the formation of the singular shock as well as its local structure, and extend to the finite volume case, which leads to a linear relationship between the shock front with time to the one-third power. The second example, a panicking crowd with a spread of fear, is modeled via emotional contagion”. Singular shock happens in an extreme case whose continuum limit is a pressure less Euler equation. Such system is then modified with a nonlocal alignment to regularize the singularity. We will discuss the hierarchy of models and their mathematical properties. Novel numerical methods will be presented for both examples.

### Wai Tong (Louis) Fan

Reflected diffusions with partial annihilations on a membrane

Mathematicians and scientists use interacting particle models to gain understanding of the emergence of macroscopic phenomena from microscopic laws of nature. In this talk, I will introduce a class of interacting particle systems that can model the transport of positive and negative charges in solar cells. To connect the microscopic mechanisms with the macroscopic behaviors at two different scales, we obtain the hydrodynamic limits and the fluctuation limits for these systems. Proving these two types of limits represents establishing the law of large numbers and the central limit theorem, respectively, for the time-trajectory of the particle densities. We show that the hydrodynamic limit is a pair of deterministic measures whose densities solve a coupled nonlinear heat equations, while the fluctuation limit can be described by a Gaussian Markov process that solves a stochastic partial differential equation. This is joint work with Zhen-Qing Chen. This talk will focus on modeling methods and applications. A second talk on the probabilistic techniques involved in the proofs will be given in the Probability Seminar on Oct 15.

### Wenjia Jing (Chicago)

Limiting distributions of random fluctuations in stochastic homogenization

In this talk, I will present some results on the study of limiting distributions of the random fluctuations in stochastic homogenization. I will discuss first a framework of such studies for linear equations with random potential. The scaling factor and the scaling limit of the homogenization error turn out to depend on the singularity of the Green’s function and the correlation structure of the random potential. I will also present some results that extend the scope of the framework to the setting of oscillatory differential operators and to some nonlinear equations. Such results find applications, for example, in uncertainty quantification and Bayesian inverse problems.

### Arthur Evans (UW)

Ancient art and modern mechanics: using origami design to create new materials

The Japanese art of origami has been a purely aesthetic endeavor for hundreds of years, but recent decades have seen the application of cutting, creasing and folding to fields as diverse as architecture and nano-engineering. The key link between the artistry of paper-folding and the physics of cells and shells lies in the connection between geometry and mechanics. In this talk I will discuss the emergence of origami design for understanding the mechanics of thin structures, highlighting the physical and mathematical principles that drive the folding of a thin sheet. While much of origami-based engineering has relied on heuristic development, I will present here a method for generalizing material design in tessellated structures, and discuss some of the first steps in building a theory that adapts origami mechanics to non-Euclidean surfaces.

### Alfredo Wetzel (UW)

Direct scattering for the Benjamin-Ono equation with rational initial data

The Benjamin-Ono (BO) equation describes the weakly nonlinear evolution of one-dimensional interface waves in a dispersive medium. It is an integrable system with a known inverse scattering transform and can be viewed as a prototypical problem for the study of multi-dimensional integrable systems or Riemann-Hilbert problems with a nonlocal jump condition. In this talk, we propose a construction procedure for the scattering data of the BO equation for arbitrary rational initial conditions with simple poles, under mild restrictions. For this class of initial conditions, we are able to obtain explicit formulas for the Jost solutions and eigenfunctions of the associated spectral problem, yielding an Evans function for the eigenvalues and formulas for the phase constants and reflection coefficient. Lastly, we show that this procedure validates well-known formal results in the zero-dispersion limit.

### Mohammed Lemou (Universite Rennes I)

A class of numerical schemes for multiscale parabolic problems

We consider a class of multiscale parabolic problems with diffusion coefficients oscillating in space at a possibly small scale. Numerical homogenization methods are popular for such problems, because they capture efficiently the asymptotic behavior as the oscillation parameter goes to 0, without using a dramatically fine spatial discretization at the scale of the fast oscillations. However, known such homogenization schemes are in general not accurate for both the highly oscillatory regime and the non oscillatory regime. In this paper, we introduce an Asymptotic Preserving method based on an exact micro-macro decomposition of the solution which remains consistent for both regimes.

### Victor Zavala (UW)

Large-scale nonlinear programming and applications to energy networks

We present advances in nonlinear programming that enable the solution of large-scale problems arising in the control of electrical and natural gas networks. Our advances involve new strategies to deal with negative curvature and Jacobian rank deficiencies in a matrix-free setting and the development of scalable numerical linear algebra strategies capable of exploiting embedded problem structures.

### Melvin Leok (UCSD)

Geometric numerical integration and computational geometric mechanics

Symmetry, and the study of invariant and equivariant objects, is a deep and unifying principle underlying a variety of mathematical fields. In particular, geometric mechanics is characterized by the application of symmetry and differential geometric techniques to Lagrangian and Hamiltonian mechanics, and geometric integration is concerned with the construction of numerical methods with geometric invariant and equivariant properties. Computational geometric mechanics blends these fields, and uses a self-consistent discretization of geometry and mechanics to systematically construct geometric structure-preserving numerical schemes.

In this talk, we will introduce a systematic method of constructing geometric integrators based on a discrete Hamilton's variational principle. This involves the construction of discrete Lagrangians that approximate Jacobi's solution to the Hamilton-Jacobi equation. Jacobi's solution can be characterized either in terms of a boundary-value problem or variationally, and these lead to shooting-based variational integrators and Galerkin variational integrators, respectively. We prove that the resulting variational integrator is order-optimal, and when spectral basis elements are used in the Galerkin formulation, one obtains geometrically convergent variational integrators.

We will also introduce the notion of a boundary Lagrangian, which is analogue of Jacobi's solution in the setting of Lagrangian PDEs. This provides the basis for developing a theory of variational error analysis for multisymplectic discretizations of Lagrangian PDEs. Equivariant approximation spaces will play an important role in the construction of geometric integrators that exhibit multimomentum conservation properties, and we will describe two approaches based on spacetime generalizations of Finite-Element Exterior Calculus, and Geodesic Finite Elements on the space of Lorentzian metrics.

Kinetic-fluid coupling: transition from the Boltzmann to the Euler

Kinetic equations (the Boltzmann, the neutron transport equation etc.) are known to converge to fluid equations (the Euler, the heat equation etc.) in certain regimes, but when kinetic and fluid regime co-exist, how to couple the two systems remains an open problem. The key is to understand the half-space problem that resembles the boundary layer at the interface. In this talk, I will present a unified proof for the well-posedness of a class of half-space equations with general incoming data, propose an efficient spectral solver, and utilize it to couple fluid with kinetics. Moreover, I will present complete error analysis for the proposed spectral solver. Numerical results will be shown to demonstrate the accuracy of the algorithm.

### Matthias Schlottbom (University of Münster)

On Galerkin schemes for time-dependent radiative transfer

The numerical solution of time dependent radiative transfer problems is challenging, both, due to the high dimension and the anisotropic structure of the underlying integro-partial differential equation. Starting from an appropriate variational formulations, we propose a general strategy for designing numerical methods based on a Galerkin discretization in space and angle combined with appropriate time stepping schemes. This allows us to systematically incorporate boundary conditions and to inherit basic properties like the conservation of mass and exponential stability from the continuous level. We also present the basic approximation error estimates. The starting point for our considerations is to rewrite the radiative transfer problem as a system of evolution equations which has a similar structure as more standard first order hyperbolic systems in acoustics or electrodynamics. This allows us to generalize the main arguments of the numerical analysis of such applications to the radiative transfer problems under investigation. We also discuss a particular discretization scheme based on a truncated spherical harmonic expansion in angle and a finite element discretization in space. This is a joint work with Herbert Egger (TU Darmstadt)

### Chris Rycroft (Harvard)

Interfacial dynamics of dissolving objects in fluid flow

An advection--diffusion-limited dissolution model of an object being eroded by a two-dimensional potential flow will be presented. By taking advantage of conformal invariance of the model, a numerical method will be introduced that tracks the evolution of the object boundary in terms of a time-dependent Laurent series. Simulations of several dissolving objects will be shown, all of which show collapse to a single point in finite time. The simulations reveal a surprising connection between the position of the collapse point and the initial Laurent coefficients, which was subsequently derived analytically using residue calculus.

### Theodoros Katasounis (KAUST)

A posteriori error control and adaptivity for Schrodinger equations

We derive optimal order a posteriori error estimates for fully discrete approximations of linear Schrodinger-type equations. For the discretization in time we use the Crank-Nicolson method, while for the space discretization we use finite element spaces that are allowed to change in time. The derivation of the estimators is based on a novel elliptic reconstruction that leads to estimates which reflect the physical properties of Schrodinger equations. The final estimates are obtained using energy techniques and residual-type estimators. Various numerical experiments for the one-dimensional linear Schrodinger equation in the semiclassical regime, verify and complement our theoretical results. The numerical implementations are performed with both uniform partitions and adaptivity in time and space. For adaptivity, we further develop and analyze an existing time-space adaptive algorithm to the cases of Schrodinger equations. The adaptive algorithm reduces the computational cost substantially and provides efficient error control for the solution and the observables of the problem, especially for small values of the Planck constant. The analysis is extended also for the nonlinear Schrodinger eq.

### Roummel Marcia (UC Merced)

This is SPIRAL-TAP: Sparse Poisson Intensity Reconstruction ALgorithms - Theory and Practice

In many medical imaging applications (e.g., SPECT, PET), the data are counts of the number of photons incident on a detector array. When the number of photons is small, the measurement process is best modeled with a Poisson distribution. The problem addressed in this talk is the estimation of an underlying intensity from photon-limited projections where the intensity admits a sparse or low-complexity representation. This approach is based on recent inroads in sparse reconstruction methods inspired by compressed sensing. However, unlike most recent advances in this area, the optimization formulation we explore uses a penalized negative Poisson log-likelihood objective function with nonnegativity constraints (since Poisson intensities are naturally nonnegative). This talk describes computational methods for solving the nonnegatively constrained sparse Poisson inverse problem. In particular, the proposed approach incorporates key ideas of using separable quadratic approximations to the objective function at each iteration and penalization terms related to l1-norms of coefficient vectors, total variation seminorms, and non-convex p-norms.

Joint work with Rebecca Willett (UW-Madison), Zachary Harmany (UC Davis) and Lasith Adhikari (UC Merced)

### Christof Sparber (UIC)

Dimension reduction for rotating Bose-Einstein condensates with anisotropic confinement

We consider the three-dimensional time-dependent Gross-Pitaevskii equation arising in the description of rotating Bose-Einstein condensates and study the corresponding scaling limit of strongly anisotropic confinement potentials. The resulting effective equations in one or two spatial dimensions, respec- tively, are rigorously obtained as special cases of an averaged three dimensional limit model. In the particular case where the rotation axis is not parallel to the strongly confining direction the resulting limiting model(s) include a negative, and thus, purely repulsive quadratic potential, which is not present in the original equation and which can be seen as an effective centrifugal force counteracting the confinement. This is joint work with Florian Mehats.

Scalable fully-coupled Newton-Krylov-AMG solution methods for implicit Finite Element Resistive MHD

The computational solution of the governing balance equations for mass, momentum, heat transfer and magnetic induction for resistive magnetohydrodynamics (MHD) systems can be extremely challenging. These difficulties arise from both the strong nonlinear coupling of fluid and electromagnetic phenomena, as well as the significant range of time- and length-scales that the interactions of these physical mechanisms produce. To enable accurate and stable approximation of these systems a range of spatial and temporal discretization methods are commonly employed. In the context of finite element spatial discretization methods these include mixed integration, stabilized methods and structure-preserving (physics compatible) approaches. For effective time integration for these systems some form of implicitness is required.

To enable robust, scalable and efficient solution of the large-scale sparse linear systems generated by the Newton linearization, fully-coupled multilevel preconditioners are developed. The multilevel preconditioners are based on two differing approaches. The first technique employs a graph-based aggregation method applied to the nonzero block structure of the Jacobian matrix. The second approach utilizes approximate block decomposition methods and physics-based preconditioning approaches that reduce the coupled systems into a set of simplified systems to which multilevel methods are applied. A critical aspect of these methods is the development of approximate Schur complement operators that encode the critical cross-coupling physics of the system. To demonstrate the flexibility and performance of these methods we consider application of these techniques to resistive MHD models for challenging prototype systems. In this context robustness, efficiency, and the parallel and algorithmic scaling of the preconditioning methods are discussed. These results include weak-scaling studies on up to 256K cores.

(This is joint work with Roger Pawlowski, Eric Cyr, Edward Phillips, Ray Tuminaro, Paul Lin, and Luis Chacon.)

### Ken Kamrin (MIT)

Continuum modeling of flowing grains

Granular matter is very common --- sands, soils, raw materials, food stuffs, pills, powders --- but the challenge of predicting the motion of a collection of flowing grains has proven to be a difficult one, from both computational and theoretical perspectives. Grain-by-grain discrete element methods can be used, but these approaches become computationally unrealistic for large bodies of material and long times. A broadly accurate continuum model would be ideal if it could be found, as it would provide a much more rapid means of calculating flows in real-world problems, such as those encountered in industry and geo-engineering.

With this challenge in mind, in this talk we present a new constitutive relation for granular matter, which produces quantitatively accurate predictions. The model is constructed in a step-by-step fashion. First we compose a local relation based on existing granular rheological approaches (i.e. the principle of "inertial" rheology) and point out where the model succeeds and where it does not. The clearest missing ingredient is shown to be the lack of an intrinsic length scale. To tie flow features more carefully to the characteristic grain size, we justify a nonlocal modification which takes the form of a size-dependent term in the rheology (with one new material parameter). The nonlocal model is then numerically implemented with a custom-written User-Element in the Abaqus package, where it is shown to greatly improve flow predictions compared to the local model. In fact, it is the first model to accurately predict all features of flows in `split-bottom cell' geometries, a decade-long open question in the field. In total, we will show that this new model, using three material parameters, quantitatively matches the flow and stress data from over 160 experiments in several different families of geometries. We then show how the same model can be used to reconcile many of the "strange" features of granular media that have been documented in the literature, such as the observation that thinner granular layers behave as if they are stronger, and the motion-induced "quicksand" effect wherein flow at one location effectively removes the yield stress everywhere.

The talk will close with a discussion of our recent mesh-free numerical algorithm that can implement the above model alongside a gas-like disconnected response, allowing us to solve all phases of granular motion in one setting.

### Lei Zhang (SJTU)

Numerical Homogenization with Localized Basis

Numerical homogenization concerns the finite dimensional approximation of the solution space of, for example, divergence form elliptic equation with $L^\infty$ coefficients which allows for nonseparable scales. Standard methods such as finite-element method with piecewise polynomial elements can perform arbitrarily badly for such problems. In this talk, I will introduce an approach for numerical homogenization which precomputes H^{-d} localized bases on patches of size H log(1/H). The localization is due to the exponential decay of the corresponding fine scale solutions with Lagrange type constraints. Interestingly, this approach can be reformulated as a Bayesian inference or decision theory problem. Furthermore, the numerical homogenization method can be used to construct efficient and robust fine scale multigrid solver or domain decomposition preconditioner, and generalized to time dependent problems such as wave propagation in heterogeneous media. This is joint work with Houman Owhadi.