(January 31) Dr. Weifeng (Frederick) Qiu: An analysis of the practical DPG method — In this work we give a complete error analysis of the Discontinuous Petrov Galerkin (DPG) method, accounting for all the approximations made in its practical implementation. Specifically, we consider the DPG method that uses a trial space consisting of polynomials of degree $p$ on each mesh element. Earlier works showed that there is a ``trial-to-test'' operator T, which when applied to the trial space, defines a test space that guarantees stability. In DPG formulations, this operator T is local: it can be applied element-by-element. However, an infinite dimensional problem on each mesh element needed to be solved to apply T. In practical computations, T is approximated using polynomials of some degree r > p on each mesh element. We show that this approximation maintains optimal convergence rates, provided that r> p+N, where N is the space dimension (two or more), for the Laplace equation. We also prove a similar result for the DPG method for linear elasticity. Remarks on the conditioning of the stiffness matrix in DPG methods are also included.

(February 7) Wujun Zhang: Convergence and quasi optimality of adaptive hybridizable discontinuous Galerkin methods — We establish the convergence and quasi optimality of adaptive hybridizable discontinuous Galerkin (AHDG) methods for the Poisson problem. We prove that the so-called quasi-error, that is, the sum of an energy-like error and a scaled error estimator, is a contraction between two consecutive loops. Moreover, we show that the AHDG methods achieve optimal rates of convergence.

(February 14) Prof. Soeren Bartels: Finite element approximation of large bending isometries — The mathematical description of the elastic deformation of thin plates can be derived by a dimension reduction from three-dimensional elasticity and leads to the minimization of an energy functional that involves the second fundamental form of the deformation and is subject to the constraint that the deformation is an isometry. We discuss two approaches to the discretization of the second order derivatives and the treatment of the isometry constraint. The first one relaxes the second order derivatives via a Reissner-Mindlin approximation and the second one employs discrete Kirchhoff triangles that define a nonconforming second order derivative. In both cases the deformation is decoupled from the deformation gradient and this enables us to employ techniques developed for the approximation of harmonic maps to impose the constraint on the deformation gradient at the nodes of a triangulation. The solution of the nonlinear discrete schemes is done by appropriate gradient flows and we demonstrate their energy decreasing behaviours under mild conditions on step sizes. Numerical experiments show that the proposed schemes provide accurate approximations for large vertical loads as well as compressive boundary conditions.

(February 21) Prof. Michael Hintermuller: Recent Advances in Fast Solvers for Variational Inequality Problems — In this talk a generalized Newton framework for solving variational inequalities (VIs) of the first and second kind is presented. Depending on regularity issues related to dual variables a Moreau-Yosida approximation of the feasible set of the problem is considered. In cases of VIs of the second kind, which exhibit a non-smooth part in the objective (which is not the indicator function of a convex set), an approach relying on the Fenchel dual is pursued. The talk also contains a report on the numerical behavior of the associated Newton-type solvers.

(February 22) Prof. Michael Hintermuller: Semismooth Newton Methods: Theory, Numerics and Applications — Non-smooth operator equations arise in many practical applications in biomedical or engineering sciences as well as mathematical imaging or finance. In this talk, for the numerical solution of such problems a generalized Newton framework in function space is discussed. Relying on the concept of semismoothness, locally superlinear convergence of the associated Newton iteration is established and its mesh independent convergence behavior upon discretization is shown. The efficiency and wide applicability of the method is highlighted by considering constrained optimal control problems for fluid flow, contact problems with or without adhesion forces, phase separation phenomena relying on non-smooth homogeneous free energy densities and restoration tasks in mathematical image processing.

(February 28) Prof. Thomas Seidman: FEM Approximation for Diffusion with Measure-Valued Source — Formally, the FEM with piecewise linear elements is defined for the Poisson equation or the heat equation with a measure-valued source term. Computational experiment suggests an order of convergence (for L^2 error of the heat equation) of h^{2-d/2) for dimensions d=2,3, consistent with earlier results by R. Scott for the Poisson equation with a single delta function as source. We obtain rigorous estimates for the problem to show that this does almost hold.

(March 13) Prof. Dianne O'Leary: Divide and Conquer: Using Spectral Methods on Partitioned Images — Spectral filtering methods such as Tikhonov regularization are powerful tools in deblurring images, but their usefulness is limited by two factors: the expense of the computation and the lack of flexibility in the filter function.

This talk focuses on overcoming these limitations by partitioning the data in order to better process each segment. The data division can take several forms: for example, partitioning by spectral frequency, by input channel, or by subimages.

Benefits of a divide and conquer approach to imaging can include smaller error, better conditioning of subproblems, and faster computations. We give examples using various data partitionings.

Joint with Julianne Chung and Glenn Easley.

(March 30) Prof. Fabio Nobile: Collocation methods for PDEs with random input data — We consider the problem of numerically approximating the solution of a partial differential equation (PDE), whose input data (coefficients, forcing terms, boundary conditions, geometry, etc.) are uncertain and described by a finite or countably infinite number of random variables. This situation includes the case of infinite dimensional random fields suitably expanded in e.g. Karhunen-Loeve or Fourier expansions.

We focus on collocation strategies where the PDE is solved for a well chosen set of points in the probability space (realizations of the random variables) and the solutions thus obtained are used to build a global polynomial approximation with respect to the underlying random variables.

In particular we discuss the case of sparse tensor grids of Gauss points with respect to the underlying probability density function and present recent results on optimized sparse grids in which the selection of points is based on a knapsack approach and relies on sharp estimates of the decay of the coefficients of the polynomial chaos expansion of the solution on the random variables. Some theoretical and numerical results will be presented for an elliptic equation with random coefficients.

We will briefly comment also a regression strategy where the collocation points are chosen randomly and a discrete L2 projection on a polynomial space is performed to recover an approximated polynomial chaos expansion of the solution.

References:

I. Babuska, F. Nobile, and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM Review, vol. 52, pp. 317-355, June 2010.

J. Back, F. Nobile, L. Tamellini, and R. Tempone, On the optimal polynomial approximation of stochastic PDEs by Galerkin and collocation methods, MOX-Report 23-2011, Department of Mathematics, Politecnico di Milano, Italy, 2011. accepted for publication on M3AS.

G. Migliorati, F. Nobile, E. von Schwerin, and R. Tempone, Analysis of the discrete L2 projection on polynomial spaces with random evaluations, MOX-Report 46-2011, Department of Mathematics, Politecnico di Milano, Italy, 2011. Submitted

L. Tamellini, "Polynomial approximation of PDEs with stochastic coefficients", PhD thesis, Politecnico di Milano, 2012.

(April 10) Minghao Wu: Lyapunov Inverse Iteration for Computing a Few Rightmost Eigenvalues of Large Generalized Eigenvalue Problems — In linear stability analysis of a large-scale dynamical system, we need to compute the rightmost eigenvalue(s) for a series of large generalized eigenvalue problems. Existing iterative eigenvalue solvers are not robust when no estimate of the rightmost eigenvalue(s) is available. In this study, we show that such an estimate can be obtained from Lyapunov inverse iteration applied to a special eigenvalue problem of Lyapunov structure. Furthermore, we generalize the analysis to a deflated version of this problem, and propose an algorithm that computes a few rightmost eigenvalues for the eigenvalue problems arising from linear stability analysis. Numerical experiments demonstrate the robustness of the algorithm.

(April 17) Prof. Andre Tits: Constraint Reduction for Linear and Convex Optimization — Constraint reduction is a technique by which, in the context of interior-point methods, at each iteration, the search direction computation involves only a {\em subset} of the inequality constraints. For problems with many more constraints than variables, this can yield a dramatic speedup. We provide some background, outline schemes, discuss convergence properties, and report numerical results, both on randomly generated problems and on problems arising in specific applications.

(April 24) Prof. Claudio Canuto: Multilevel Preconditioning of Discontinuous-Galerkin Spectral-Element Methods — We propose and discuss preconditioners for symmetric interior-penalty Discontinuous-Galerkin Spectral-Element (also known as h-p FEM) discretizations of second order elliptic boundary value problems. The discretizations use locally refined quadrilateral or hexahedral partitions, with possible hanging nodes and variable polynomial degrees that can be arbitrarily large provided some weak grading constraint is respected. The conceptual foundation of the envisaged preconditioners is the Auxiliary Space Method (ASM), or in fact, an iterated variant of it. It will be shown that in this framework one obtains optimal preconditioners that are fully robust with respect to spatial refinements as well as with respect to polynomial degrees. We also explore a second strategy, in which the ASM defines an outer precodnitioner only, whereas the inner preconditioner is accomplished by the BDDC method. The resulting preconditioner is quasi-optimal, since the condition number exhibits a polylogarithmic dependence on the polynomial degree distribution and the spatial refinement.

The talk is based on joint works with Kolja Brix and Wolfgang Dahmen, and with Luca Pavarino and Alexandre Pieri.

(May 1) Dr. Marco Verani: A Mimetic Discretization of Elliptic Obstacle Problems — We present a Finite Element method (FEM) which can adopt very general meshes with polygonal elements for the numerical approximation of elliptic obstacle problems. This kind of methods are also known as mimetic discretization schemes, which stem from the Mimetic Finite Difference (MFD) method. The first-order convergence estimate in a suitable (mesh-dependent) energy norm is established. Numerical experiments confirming the theoretical results are also presented.

(May 8) Prof. Blaise Bourdin: The Variational Approach to Fracture: Implementation, Validation and Verification — The variational approach to fracture recasts Griffith's celebrated criterion as a sequence of unilateral global minimization problem of a Free Discontinuity Energy. In the last 15 years, its analysis and numerical implementation have been the subject of an intense scrutiny.

In this talk, I will briefly introduce the model and describe its numerical implementation. I will then describe its potential applications to reservoir stimulation. I will present two series of experiments that can account for specific situations in a controlled environment. The first one deals with the drying of colloidal suspensions and the second with crack propagation in ceramics subject to thermal shocks. in both cases, were able to perform perform quantitative comparison between numerical simulations, experiments, and exact solutions, or simulations.

(May 9) Prof. Howard Elman: Efficient Solution Algorithms for Partial Differential Equations with Random Coefficients — We consider new computational methods for solving partial differential equations (PDEs) when components of the problem such as diffusion coefficients are random fields. In recent years, several computational techniques have been developed for such models that offer potential for improved efficiencies compared with traditional Monte-Carlo methods. These include stochastic Galerkin methods, which use an augmented weak formulation of the PDE derived from averaging with respect to expected value, and stochastic collocation methods, which use a set of samples relatively small in cardinality that captures the character of the solution space. We discuss the relative advantages of these two methods and explore their performance. For problems in which the dependence on uncertain parameters is linear, the Galerkin systems can be solved efficiently by multigrid methods so that the overall cost of solution is significantly lower than for collocation. We also discuss refinements of these methodologies to handle nonlinearities. A commonly used model takes the diffusion coefficient to be of log-normal form, which ensures well-posedness but causes the dependence on uncertain parameters to be nonlinear. We show that this difficulty can be alleviated by reformulating the problem in convection-diffusion form. Finally, computation of statistical properties of solutions such as moments requires explicit knowledge of the joint density function associated with the (finite number of) random variables that determine the uncertain coefficients. This is often not available, and instead, computation is enabled under a simplifying and possibly false assumption that the joint density is simply the product of the marginal densities. We discuss the use of kernel density estimation to circumvent this difficulty.

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