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(February 5) Professor Christopher Beattie: Projection Methods for Structure-preserving Model Reduction - Dynamical systems are the basic framework for modeling and control of an enormous variety of complex systems. Direct numerical simulation of the associated models has been one of the few available means when goals include accurate prediction or control of complex physical phenomena. However, the ever increasing need for improved accuracy requires the inclusion of ever more detail in the modeling stage, leading inevitably to ever larger-scale, ever more complex dynamical systems.

Simulations in such large-scale settings can be overwhelming and make unmanageably large demands on computational resources; this is the main motivation for model reduction, which has as its goal production of a much lower dimensional system having the same input/output characteristics as the original system.

Rational Krylov subspaces are often capable of providing nearly optimal approximating subspaces for model reduction. A framework for model reduction is presented that includes rational Krylov-based methods as a special case. This broader framework allows retention of special structure in the reduced order models such as symmetry, second order structure, internal delays, and infinite dimensional subsystems.

The main cost to generate the required reduced order bases arises from solving large scale linear systems of equations having the same order as the dynamical system itself. Since the need for more detail and accuracy in the modeling stage causes the system dimension to reach levels on the order of millions, direct solvers for these linear systems are no longer feasible. In practice, iterative solvers become necessary and choice of termination criteria and effect of preconditioning influences the quality of final reduced order models. The effect on the underlying model reduction problem from the use of approximate subspaces generated from iterative solvers will be discussed as well.

(February 12) Professor Claudio Canuto: High-resolution methods for stochastic boundary-value problems with geometric uncertainties - In modern science and technology, there is an increasing need for numerical simulations supplemented by quantitative indications of their reliability. A posteriori error analysis and adaptivity provide an answer. We will rather focus on problems which are not fully deterministic, but depends on one or more random variables: in such cases, one is often interested in computing certain statistical quantities of the solution, which quantify the amount of uncertainty in the output of the model. For pde-based models, the use of Monte-Carlo techniques is often ruled out by the excessive cost of a single "call" to the numerical solver. The viable alternative is to transform the stochastic problem into a deterministic one, via suitable expansions of the random variables (Karhunen-Loeve or Polynomial Chaos expansions). High-resolution numerical methods, which "minimize" the number of unknowns per given target accuracy, may help in keeping small the complexity of the problem. We will illustrate these ideas in the case of boundary-value problems posed in random domains. We will present various approaches to handle the geometric uncertainty, such as mapping and fictitious domains. We will discuss the numerical analysis of the discretization methods, their implementation issues, as well as certain realistic applications to Wind Engineering (flow around a bridge deck).

(February 19) Professor Marco Verani: A Safeguarded Dual Weighted Residual Method - The dual weighted residual (DWR) method yields reliable a posteriori error bounds for linear output functionals provided that the error incurred by the numerical approximation of the dual solution is negligible. In that case its performance is generally superior than that of global energy norm error estimators which are unconditionally reliable. We present a simple numerical example for which neglecting the approximation error leads to severe underestimation of the functional error, thus showing that the DWR method may be unreliable. We propose a remedy that preserves the original performance, namely a DWR method safeguarded by additional asymptotically higher order a posteriori terms. In particular, the enhanced estimator is unconditionally reliable and asymptotically coincides with the original DWR method. These properties are illustrated via the aforementioned example. (joint work with R.H. Nochetto and A. Veeser)

(February 26) Professor Pedro Morin: Convergence of Adaptive Finite Elements for Eigenvalue Problems - We will discuss the convergence of adaptive finite element methods for second order elliptic eigenvalue problems using Lagrange elements of any degree. Convergence will be proved under a minimal refinement of marked elements, for all "reasonable" marking strategies, and starting from any initial mesh. This work is in collaboration with Eduardo Garau (Santa Fe, Argentina) and Carlos Zuppa (San Luis, Argentina)

(March 4) Professor Xiaobai Sun: Efficient Encoding of Irregular Geometric Structures in Matrix Computations - In this talk I describe the basic concepts and schemes of encoding geometric structures in matrices that arise from discretizing certain differential or integral equations over irregular domains and grids. The most noticeable examples include the Fast Multipole method and the Fourier transform, or the inverse Fourier transform, with non-Cartesian data. These methods are known for their low-order complexity in memory usage and arithmetic operations with respect to an arbitrarily specified accuracy tolerance and are typically employed in iterative solution methods. An efficient encoding scheme involves not only a compressive mathematical representation of the matrix defined on an irregular grid, but also a companion ordering for efficient traversal of the matrix and data in memory with respect to any particular memory hierarchy. During the talk I introduce a particular case of the inverse Fourier transform with three-dimensional radial data from a medical imaging application and describe the success and impact of the encoding approach.

(March 11) Professor Irene Livshits: Multiscale/Multigrid Method for Finding a Full Eigenbasis of the Schrodinger Operator - Many applications involving differential operators require a knowledge of their many eigenfunctions. In this talk, we will introduce a novel multigrid technique which, by employing a multiscale structure for eigenfunctions' approximation, allows calculating all eigenfunctions of the Schrodinger operator as well as performing many applications typical for eigenvalue problems in just O(N log N) operations, where N is the size of the discretized problem. Numerical results and a discussion of the further extension of the approach will conclude the talk.

(March 25) Professor Shari Moskow: Asymptotic and Numerical Approximation of the Resonances of Thin Photonic Structures - We study the computation of the resonances of thin dielectric structures by using both asymptotics and PML. Of great interest in the study of photonic band gap materials is when these structures are periodic with a defect. We derive a limiting resonance problem as a the thickness goes to zero, and for the case of a simple resonance find a first order correction. The limiting problem and correction have one dimension less, which can make the approach very efficient. Convergence estimates are proved for the asymptotics. We then proceed to compute these resonances by two very different methods: the asymptotic method by solving a dense, but small, nonlinear eigenvalue problem; and PML, which yields a large, but linear and sparse generalized eigenvalue problem. Both methods reproduce a photonic band gap type mode found previously by finite difference time domain methods.

This is joint work with Jay Gopalakrishnan and Fadil Santosa

(April 1) Professor Mili Shah: Calculating a Symmetric Approximation via the Symmetry Preserving Singular Value Decomposition - Determining symmetry within a collection of spatially oriented points is a problem that occurs in many fields. In these applications, large amounts of data are generally collected, and it is desirable to approximate this data with a compressed representation. In some situations, the data is known to obey certain symmetry conditions, and it is profitable to preserve such symmetry in the compressed approximation. I accomplish this task by providing a symmetry preserving singular value decomposition. I will present the formulation of this factorization along with an application of this method to molecular dynamics.

(April 8) Professor Ricardo Nochetto: Adaptive FEM for the Laplace-Beltrami Operator and Applications to Geometric Flows - The Laplace-Beltrami operator, or surface Laplacian, is ubiquitous in geometric problems and has a natural variational structure. This allows for finite element discretizations of surface and PDE of arbitrary polynomial degree, and the use of refinement/coarsening techniques that lead to adaptivity. We first discuss key properties and present applications such as surface diffusion, optimal shape control and image segmentation, along with several simulations exhibiting large deformations as well as pinching and topological changes in finite time. This part is joint with G. Dogan, P. Morin and M. Verani. We next discuss a posteriori error analysis for the Laplace-Beltrami operator and prove a conditional contraction property of the ensuing AFEM; this is joint with K. Mekchay and P. Morin. We finally discuss the geometrically consistent refinement of polyhedral surfaces, which is critical for large domain deformations and new paradigm in adaptivity. This is joint with A. Bonito and M.S. Pauletti.

(April 15) Professor Jennifer Erway: Recent Developments in Iterative Methods for Large-Scale Unconstrained Optimization - Numerical optimization involves the minimization or maximization of an "objective function," possibly subject to functional constraints on the variables. Over the past decade, advances in methods for optimization problems with partial differential equation (PDE) constraints have highlighted the need for optimization algorithms that can handle problems with huge numbers of variables and constraints. This has lead to a resurgence of interest in methods that solve a constrained problem as a sequence of unconstrained problems. The algorithm for each unconstrained problem must be iterative in nature, i.e., it must not rely on matrix factorizations. Two of the biggest challenges for this class of algorithms are reliability and efficiency.

Trust-region methods are one of the most popular approaches to solving unconstrained optimization problems. This talk will present recent developments in trust-region methods for large-scale unconstrained optimization.

(April 21) Professor Vidar Thomee: On Maximum-Principles in Elliptic and Parabolic Finite Element Problems - We consider finite element discretizations of Dirichlet's problem for Laplace's equation and the corresponding initial-boundary value problem for the homogeneous heat equation. We survey work on discrete versions of the associated maximum-principles, using piecewise linear finite elements on triangulations of the underlying spatial domain. In the elliptic case the maximum-principle holds if the triangulation is of Delaunay type, but that this is is not a necessary condition. For the spatially semidiscrete parabolic problem, the maxiumum-principle does not hold in general, but, as was shown by Fujii in 1973, it is valid for the lumped mass variant when the triangulation is of Delaunay type. We show that these conditions are essentially sharp. We also study conditions for the solution operator acting on the discrete initial data, with homogeneous lateral boundary conditions, to be a contraction in the maximum-norm, or a positive operator.

(April 22) Professor Rick Falk: A New Approach to Finite Element Methods for the Equations of Linear Elasticity - We exploit the connection between the de Rham sequence and a corresponding exact sequence for elasticity to derive new finite element methods for the approximation of the equations of linear elasticity. These methods use a Lagrange multiplier to weakly impose the symmetry of the stress, thus allowing the use of standard mixed finite elements for scalar second order elliptic problems.

(April 29) Professor Houman Owhadi: From Stochastic Variational Integrators to Ballistic Diffusion at Uniform Temperature - We present a continuous and discrete Lagrangian theory for stochastic Hamiltonian systems on manifolds. Analogously to discrete mechanics and variational integrators, this theory leads to structure preserving numerical integrators for noisy mechanical systems by extremizing a discrete stochastic action. In this talk, we will focus on a particular one, designed to approximate the solutions of Langevin equations with uniform friction and noise. This new integrator is characterized by the following two properties:

* It exactly preserves the exponential rate of decay of the symplectic form.

* It inherits a near Boltzmann-Gibbs measure preservation property from the near energy preservation property of its associated noise/friction-free symplectic Euler integrator.

By applying our integrator to a fluctuation-driven magnetic motor we exhibit a paradigm for isothermal, mechanical rectification of stochastic fluctuations.

The central idea is to transform energy injected by (random/thermal) fluctuations into rigid-body rotational kinetic energy. We show that although directed motion is not possible at uniform temperature, thermal noise can be used to obtain ballistic diffusion. This is a joint work with Nawaf Bou-Rabee.

(May 13) Professor Daniele Boffi: Mimetic Finite Differences and Quadrilateral Finite Elements - It is well known that quadrilateral finite elements may produce unsatisfactory results when used on distorted meshes. Many commonly used finite elements do not achieve optimal convergence properties: the list of suboptimal elements include serendipity elements and basically all elements for the approximation of H(div) (like Raviart-Thomas or Brezzi-Douglas-Marini spaces). On the other hand, mimetic finite differences have become popular for the approximation of problems involving H(div) on very general geometries. We show how to use the ideas of mimetic finite differences for the stabilization of Raviart-Thomas element on general quadrilateral meshes.