(September 7) Dr. Manuel Tiglio: Solving Einstein's equations for gravitational wave detection — Gravitational waves, for which there is indirect evidence but have not yet been directly detected, carry unique properties of Einstein's theory, which has not yet been tested or verified either in its highly non-linear regime, such as near black holes. Binary black hole collisions are some of the expected best sources to enter the range of upcoming updated gravitational wave detectors, and their detection relies on a bank of templates. I will discuss the status of Numerical Relativity -- the field of numerically solving Einstein's equations with supercomputers--, the challenges associated with generating a bank of templates (a parametrized problem with a rather high dimensional parameter space) and ongoing work in that direction.

(September 28) Dr. James F. Drake: Breaking field lines during magnetic reconnection — Magnetic reconnection is the driver of explosive releases of energy in the laboratory and nature, including solar and stellar flares and storms in the Earth's magnetosphere. The release of magnetic energy requires a topological change in magnetic geometry: field lines must break and reconnect to release energy. The dissipation mechanism that enables magnetic field lines to break during reconnection has remained a mystery since the first models of reconnection were proposed in the 1950s. Classical resistivity is too small to explain reconnection observations. We explore the dynamics of magnetic reconnection with 3-D particle-in-cell simulations and analytic analysis. The simulations reveal that strong currents and associated high electron-ion streaming velocities that develop near the x-line can drive instabilities. The electron scattering caused by this turbulence produces an enhanced drag, "anomalous resistivity", that has been widely invoked as the dissipation mechanism. We have demonstrated that these electron current layers become strongly turbulent. The surprise, however, is that the turbulence driven by an electron sheared-flow instability completely dominates traditional streaming instabilities and the associated turbulent driven "anomalous viscosity" balances the reconnection electric field and therefore breaks field lines. The talk will introduce the physics basis of magnetic reconnection to the general audience.

(October 5) Dr. Soeren Bartels: Approximation of harmonic maps and wave maps — Partial differential equations with a nonlinear pointwise constraint defined through a manifold occur in a variety of applications: The magnetization of a ferromagnet can be described by a unit length vector field and the orientation of the rod-like molecules that constitute a liquid crystal is often modeled by a vector field that attains its values in the real projective plane thus respecting the head-to-tail symmetry of the molecules. Other applications arise in geometric modeling, quantum mechanics, and general relativity. Simple examples reveal that it is impossible to satisfy pointwise constraints exactly by lowest order finite elements. For two model problems we discuss the practical realization of the constraint, the efficient solution of the resulting nonlinear systems of equations, and weak accumulation of approximations at exact solutions.

(November 2) Dr. Misha Kilmer: Factorization Strategies for Third Order Tensors with Imaging Applications — Operations with tensors, or multiway arrays, have become increasingly prevalent in recent years. The list of applications involving operations on tensors is lengthy and includes psychometrics, signal processing and datamining, to name but a few. A common feature in such applications is the need to derive a compressed representation of the data that take advantage of the multidimensional structure, since collapsing multiway data to matrices often has undesirable consequences.

In this talk, we give a new closed multiplication operation between third order tensors around which we build a framework that allows for extension of familiar concepts and algorithms in (numerical) linear algebra to be applied to third order tensors. In particular, we show that one way to represent a third order tensor is as a finite sum of matrix outer products. For example, we derive a new extension of the matrix SVD for third order tensors that is significantly different from other approaches in the tensor literature. More generally, using matrix outer products we can construct various compressed (i.e. approximate) representations for third order tensors, depending on what type of constraints we wish to impose. We illustrate the potential power of our framework in the context of two imaging applications: image deblurring and facial recognition.

This is joint work with Carla D. Martin (JMU).

(November 11) Dr. Chi-Wang Shu: Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin and finite volume schemes for conservation laws — We construct uniformly high order accurate discontinuous Galerkin (DG) and weighted essentially non-oscillatory (WENO) finite volume (FV) schemes satisfying a strict maximum principle for scalar conservation laws and passive convection in incompressible flows, and positivity preserving for density and pressure for compressible Euler equations. A general framework (for arbitrary order of accuracy) is established to construct a limiter for the DG or FV method with first order Euler forward time discretization solving one dimensional scalar conservation laws. Strong stability preserving (SSP) high order time discretizations will keep the maximum principle and make the scheme uniformly high order in space and time. One remarkable property of this approach is that it is straightforward to extend the method to two and higher dimensions. The same limiter can be shown to preserve the maximum principle for the DG or FV scheme solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible velocity field. A suitable generalization results in a high order DG or FV scheme satisfying positivity preserving property for density and pressure for compressible Euler equations. Numerical tests demonstrating the good performance of the scheme will be reported. This is a joint work with Xiangxiong Zhang.

(November 12) Prof. Chi-Wang Shu: Discontinuous Galerkin Finite Element Methods for High ORder Nonlinear PDEs — Discontinuous Galerkin (DG) finite element methods were first designed to solve hyperbolic conservation laws utilizing successful high resolution finite difference and finite volume schemes. More recently the DG methods have been generalized to solve convection dominated convection-diffusion equations, convection-dispersion and other high order nonlinear wave equations or diffusion equations. In this talk we will first give an introduction to the DG method and then we will move on to introduce a class of DG methods for solving high order PDEs, termed local DG (LDG) methods. We will highlight the important ingredient of the design of LDG schemes, namely the adequate choice of numerical fluxes, and emphasize the stability of the fully nonlinear DG approximations.

(November 16) Dr. Alexandre Ern: Implicit-explicit Runge-Kutta schemes with stabilized finite elements for advection-diffusion equations — We analyze a two-stage explicit-implicit Runge-Kutta scheme for time discretization of advection-diffusion equations. Space discretization uses continuous, piecewise affine finite elements with interelement gradient jump penalty; discontinuous Galerkin methods can be considered as well. The advective and stabilization operators are treated explicitly, whereas the diffusion operator is treated implicitly. Our analysis hinges on $L^2$-energy estimates on discrete functions in physical space. Our main results are stability and quasi-optimal error estimates for smooth solutions under a standard hyperbolic CFL restriction on the time step, both in the advection-dominated and in the diffusion-dominated regimes.

(November 23) Dr. Andy Wathen: Iterative Linear Solvers for PDE-Constrained Optimization Problems Involving Fluid Flow — The numerical approximation of Partial Differential Equation (PDE) problems leads typically to large dimensional linear or linearised systems of equations. For problems where such PDEs provide only a constraint on an Optimization problem (so-called PDE-constrained Optimization problems), the systems are many times larger in dimension.

We will discuss the solution of such problems by preconditioned iterative techniques in particular where the PDEs in question are the steady Stokes equations describing incompressible fluid flow and some very recent work on the time-dependent diffusion equation.

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