10:00 Daniel Szyld (Temple University):
Inexact and truncated parareal-in-time Krylov subspace methods for parabolic optimal control problems
We study the use of inexact and truncated Krylov subspace methods for the solution of the linear systems arising in the discretized solution of the optimal control of a parabolic partial differential equation. An all-at-once temporal discretization and a reduction approach are used to obtain a symmetric positive definite system for the control variables only, where a Conjugate Gradient (CG) method can be used at the cost of the solution of two very large linear systems in each iteration. We propose to use inexact Krylov subspace methods, in which the solution of the two large linear systems are not solved exactly, and their approximate solutions can be progressively less exact. The option we propose is the use of the parareal-in-time algorithm for approximating the solution of these two linear systems. The use of less parareal iterations makes it possible to reduce the time integration costs and to improve the time parallel scalability, and therefore, making it possible to really consider otimization in real time. We also show that truncated methods could be used without much delay in convergence, but with important savings in storage. Spectral bounds are provided and numerical experiments with inexact versions of CG, the full orthogonalization method (FOM), and of truncated FOM, are presented, illustrating the potential of the proposed methods.

Xiuhong Du, Marcus Sarkis, Christian E. Schaerer, and Daniel B. Szyld, Inexact and truncated Parareal-in-time Krylov subspace methods for parabolic optimal control problems, Research Report 12-02-06, Department of Mathematics, Temple University, February 2012.

Other references: (all with Valeria Simoncini)

10:20 Edward Phillips (U of Maryland, College Park):
Block preconditioners for an exact penalty viscoresistive MHD formulation
The magnetohydrodynamics equations are used to model the flow of electrically conducting fluids in such applications as liquid metals and plasmas. The equations are a system of non-self adjoint, non-linear PDEs which couple the Navier-Stokes equations for fluids and Maxwell’s equations for electromagnetics. They can span over a range of length- and time-scales, necessitating robust, accurate approximation techniques. There has been recent interest in fully coupled solvers for the MHD system because they allow for fast steady-state solutions that do not require pseudo-time stepping. When the fully coupled system is discretized, the strong coupling can make the resulting algebraic systems difficult to solve, requiring effective preconditioning of iterative methods for efficiency. In this work, we consider a finite element discretization of an exact penalty formulation for the stationary MHD equations (as proposed by Gerbeau, 2000). This formulation has the benefit of implicitly enforcing the divergence free condition on the magnetic field without requiring a Lagrange multiplier. We consider extending block preconditioning techniques developed for the Navier-Stokes equations (such as those of Elman et al., 2008) to the full MHD system. We analyze operators arising in block decompositions from a continuous perspective and apply approximate commutation arguments in order to develop new preconditioners that account for the multi-physics coupling. We demonstrate the quality of these preconditioners over a range of parameters on a variety of two-dimensional test problems.
10:40 Ana Maria Soane (U of Maryland, Baltimore County):
Multigrid solution of a distributed optimal control problem constrained by the Stokes equations
We construct multigrid preconditioners to accelerate the solution process of a linear-quadratic optimal control problem constrained by the Stokes system. The first order optimality conditions of the control problem form a linear system (the KKT system) connecting the state, adjoint, and control variables. Our approach is to eliminate the state and adjoint variables by essentially solving two Stokes systems, and to construct efficient multigrid preconditioners for the Schur-complement of the block associated with the state and adjoint variables. These multigrid preconditioners are shown to be of optimal order with respect to the convergence properties of the discrete methods used to solve the Stokes system. In particular, the number of conjugate gradient iterations is shown to decrease as the resolution increases, a feature shared by similar multigrid preconditioners for elliptic constrained optimal control problems.
11:00 BREAK
11:30 Tobin Driscoll (U of Delaware):
Optimal interval splitting for spectral interpolation
The convergence rate of a spectrally accurate polynomial interpolant depends on the location of a determining singularity of the approximated function in the complex plane. If the approximation interval is split to create a piecewise interpolant, the original problem is replaced by two problems with different singularity locations. By balancing the convergence rates of the two subproblems, the optimal splitting location can be worked out as a function of the original singularity location. It turns out that a simple bisection heuristic in the Chebfun software system performs an effective binary search algorithm for the optimal location, but preliminary results suggest that stabilized Pade appproximations may be used to split more efficiently.
11:50 Qifeng Liao (U of Maryland, College Park):
Reduced basis collocation methods for partial differential equations with random coefficients
The sparse grid stochastic collocation method is a new method for solving partial differential equations with random coefficients. However, when the probability space has a high dimensionality, the number of points required for accurate collocation solutions can be large, and it may be costly to construct the solution. We show that the process can be made more efficient by combining collocation with reduced-basis methods, in which a greedy algorithm is used to identify a reduced problem to which the collocation method can be applied. Because the reduced model is much smaller, costs are reduced significantly. We demonstrate with numerical experiments that this is achieved with essentially no loss of accuracy.
12:10 Manuel Solano (U of Delaware):
Solving Dirichlet boundary-value problems on general domains
We present a technique for numerically solving Dirichlet boundary-value problems on a general domain. We do not assume the domain polygonal. This is achieved by using suitably defined extensions from polyhedral subdomains; the problem of dealing with curved boundaries is thus reduced to the evaluations of simple line integrals. The technique is independent of the representation of the boundary and of the space dimension. Moreover, it allows the use of only polyhedral elements and high order approximations. In the polyhedral subdomains, we use a hybridizable discontinuous Galerkin method and provide numerical experiments showing that the convergence properties of the resulting method are the same as those for the case when the domain is polygonal, whenever the distance between the boundary and the computational boundary is of order of the meshsize.
12:30 LUNCH (will be provided)
  1:30 Claudio Canuto (Turin, Italy):             INVITED SPEAKER
Adaptive Fourier-Galerkin methods
We study the performance of adaptive Fourier-Galerkin methods in a periodic box in Rd with dimension d≥1. These methods offer unlimited approximation power only restricted by solution and data regularity. They are of intrinsic interest but are also a first step towards understanding adaptivity for the hp-FEM.

We examine two nonlinear approximation classes, one classical corresponding to algebraic decay of Fourier coefficients and another associated with exponential decay. We study the sparsity classes of the residual and show that they are the same as the solution for the algebraic class but not for the exponential one. This possible sparsity degradation for the exponential class can be compensated with coarsening, which we discuss in detail. We present several adaptive Fourier algorithms, and prove their contraction and optimal cardinality properties.

The extension of some of the previous results to the case of non-periodic (Legendre) expansions will also be considered.

This is a joint work with Ricardo H. Nochetto (University of Maryland) and Marco Verani (Politecnico di Milano).

  2:30 BREAK
  3:00 Lise-Marie Imbert (LJLL - UPMC, Paris, France):
A generalized plane wave numerical method for magnetic plasma
Maxwell's equations with hermitian dielectric tensor are used to model reflectometry in fusion plasma. Simplified models split them into two different propagation modes. Here we focus on the O-mode equation. We propose an original method based on generalized plane waves and approximated coeffcients for the numerical approximation. This is justified in dimension one by a high order convergence estimate rate. Some numerical results are presented in dimension one and two.
  3:20 Miao-Jung (Yvonne) Ou (U of Delaware):
Cartesian-grid finite volume modeling of wave propagation in anisotropic poroelastic media
Poroelasticity theory models the dynamics of porous, fluid-saturated media. It was pioneered by Maurice Biot in the 1930s through 1960s, and has applications in several fields, including geophysics and modeling of in vivo bone. A wide variety of methods have been used to model poroelasticity, including finite difference, finite element, pseudospectral, and discontinuous Galerkin methods. In this work we use a Cartesian-grid high-resolution finite volume method to numerically solve Biot's equations in the time domain for orthotropic materials, with the stiff relaxation source term in the equations incorporated using operator splitting. This class of finite volume method has several useful properties, including the ability to use wave limiters to reduce numerical artifacts in the solution,ease of incorporating material inhomogeneities, low memory overhead, and an explicit time-stepping approach. To the authors' knowledge, this is the first use of finite volume methods to model poroelasticity. The solution code uses the CLAWPACK finite volume method software, which also includes block-structured adaptive mesh refinement in its AMRCLAW variant. We present convergence results for known analytic plane wave solutions, and also compare against other numerical results from the literature. This is joint work with Grady I. Lemoine and Randall J. LeVeque.
  3:40 Harbir Antil (U of Maryland, College Park):
Application-specific fast, high accuracy reduced order quadratures and gravitational waves
Finding gravitational waveforms (GWs) is potentially going to open a new window to the universe. Einstein's equations are nonlinear parameter dependent (mass, spin etc.) PDEs that approximate the post-Newtonian equations (PN). PN without spin gives stationary phase approximations (SPA) to waveforms. Using matched filtering techniques, a LIGO search to find GWs, requires about 10,000 SPA templates and several months on supercomputers. Reduced basis and using discrete empirical interpolation points as quadrature nodes leads to significant savings. We will provide the a priori error estimates and numerical examples.
  4:00 BREAK
  4:30 Enrique Otarola (U of Maryland, College Park):
A FEM for the fractional Laplace operator
We consider fractional powers of the Laplace operator (-Δ)s, with s∈(0, 1), in a bounded domain. The fractional Laplace operator can be realized as the Dirichlet to Neumann operator of an extension problem posed on a semi-infinite cylinder. This extension problem involves a degenerate/singular elliptic equation, which we analyze in the framework of weighted Sobolev spaces. For numerical approximation we propose a suitable truncated problem, which can be justified by the rapid decay of the solution to the extension problem. A finite element approximation using Q1 elements is considered for the truncation. A priori error estimates are obtained for anisotropic elements in weighted Sobolev spaces and numerical experiments are presented illustrating the theory.
  4:50 Prince Chidyagwai (Temple University):
Discontinuous Galerkin method for 2D moment closures for radiative transfer
Radiative transfer plays an important role in many engineering and physics applications. We consider the radiative transport equation applied to electron radiotherapy. In this case the transport equation describes the distribution of electrons in time and space assuming that the electrons do not interact with each other. The full radiative transfer equation is computationally expensive to solve because it is a high dimensional equation. We present high order Discontinuous Galerkin (DG) schemes to solve systems that approximate the full radiative transport equation. The approximating systems are obtained from moment methods. These methods start with a system of infinitely many moments that are equivalent to the radiative transport equation. The truncation of this system is achieved using closure strategies results in both linear and non-linear hyperbolic systems. We show that the DG method is well suited to solve the approximating systems because it yields high order approximations and easily handles unstructured grids. We demonstrate that the method also scales very well on shared memory on a shared memory parallel implementation. We present numerical results verifying the high order of convergence of our numerical scheme on known smooth solutions as well as results form benchmark problems in radiative therapy test cases.
  5:10 Abner Salgado (U of Maryland, College Park):
Convergence and Optimality of a Goal Oriented Adaptive FEM
In many applications, one is not so much interested in the solution of a problem but rather in a so-called quantity of interest, that is, the value of a functional on the solution of this problem. At least heuristically, it is clear that if one approximates with sufficient accuracy the solution of the problem, the corresponding approximation of the quantity of interest will be accurate as well. However, the question that one can naturally ask is: Can we do better? Goal oriented methods aim at addressing this issue. In this talk we will present and analyze a goal oriented adaptive finite element method for a model diffusion problem. We will show that the method is convergent and optimal in the sense that the speed of convergence is as fast as the approximation classes of the involved problems allow us.
  5:30 END