(September 6) Dr. Abner Salgado: A Diffuse Interface Model For Electrowetting — Electrowetting refers to the phenomenon where the surface tension between two fluids can be locally modified by application of an electric field. Thus, wetting behavior can be controlled. This process has found applications in lab-on-a-chip systems, autofocus cell phone lenses, colored oil pixels and others. In this work we present a diffuse interface approach to model electrowetting. This model is fully three-dimensional and takes into account the most significant physical effects: differences between densities of the phases, other phase dependent quantities (conductivity, etc.), and contact line dynamics (including pinning). We show that the obtained model has an energy law and propose a fully discrete scheme that mimics this energy dissipation property. With the help of the proposed scheme we show that the semi-discrete problem (discrete in time, continuous in space) always has a solution. Moreover, the well-posedness of the scheme is not sensitive to the chosen dynamic contact line model. This is especially important since modeling of 3-phase contact line motion is still an open area of research. Preliminary numerical simulations will be presented that illustrate the capability of our model and method. Joint work with Ricardo H. Nochetto (UMD) and Shawn W. Walker (LSU).

(September 13) Prof. Sergio Preidikman: Developing Nonlinear Models for Aeroservoelastic Behavior of Large-scale Horizontal-axis Wind Turbines — Wind turbine technology has evolved rapidly over the last twenty years. The most obvious manifestation of this development is the exponential increase in machine size. The new large-scale horizontal-axis wind turbines (LHAWT) concepts are paradigm breakers; they have opened the doors to many exciting opportunities for modeling as well as for carrying out innovative and nontraditional designs. Despite the conservativeness in the structural designs of the conceived LHAWT systems, they are lightweight and they have high aspect-ratio blades. The aeroelastic behavior of these blades, which is not quite well understood yet, can have a significant influence on the considered LHAWTs performance. Hence, the ability to estimate reliable margins for aeroelastic instabilities is expected to be of major importance for an LHAWT designer. In this regard, there is an urgent need to develop a set of robust, accurate, and reliable prediction methods based on coupled aeroelasticity, structural dynamics, control systems, and nonlinear analysis. In particular, it is necessary to enhance and calibrate existing numerical tools and develop new numerical tools for predicting complex aeroservoelastic phenomena, including those due to aerodynamic and structural nonlinearities with a high level of accuracy.

The overall aim of this effort is to develop a fundamental understanding of the nonlinear aeroservoelastic behavior of LHAWT with high aspect-ratio blades and high flexibility. This understanding is to be realized by developing comprehensive computational tools, and the understanding gained through this study is to be used for predicting the uncontrolled and controlled responses of LHAWT. Novel aspects of this work are derived from the following: i) consideration of dynamic coupling between rigid-body modes and elastic modes of the flexible LHAWT structures, ii) combination of structural models and aerodynamic models to capture the coupled physics, iii) study of dynamic instabilities (for example, dynamic buckling, flutter) and post-instability motions such as limit-cycle oscillations (LCOs), and iv) use of nonlinear phenomenon such as modal saturation to design controllers and active (or smart/compliant) structures for attenuating oscillatory motions.

From a fundamental standpoint, this effort will help understand how to couple unsteady, nonlinear aerodynamic models with nonlinear structural dynamics models in studies of unsteady, nonlinear fluid-structure problems, the different possible nonlinear phenomena in such systems, and develop nonlinear phenomena based control strategies. The development of the proposed fluid-structure models and control strategies for highly flexible LHAWT is expected to provide an important foundation for the design of the next generation of LHAWT.

(September 20) Dr. William F. Mitchell: Comparison of hp-Adaptive Finite Element Strategies — Adaptive finite element methods have been studied for nearly 30 years now. Most of the work has focused on $h$-adaptive methods where the mesh size, $h$, is adapted locally by means of a local error estimator with the goal of placing the smallest elements in the areas where they will do the most good. $h$-adaptive methods for elliptic partial differential equations are quite well understood now, and widely used in practice.

Recently, the research community has begun to focus more attention on $hp$-adaptive methods where in addition to $h$-adaptivity one locally adapts the degree of the polynomials, $p$. One attraction of these methods is that they can achieve exponential rates of convergence. But the design of an optimal strategy to determine when to use $p$-refinement, when to use $h$-refinement, and what $p$'s to use in $h$-refined elements is an open area of research. Many such $hp$-adaptive strategies have been proposed over the past two decades. In this talk, we will briefly describe 13 $hp$-adaptive strategies and present the results of a numerical experiment to determine which strategies are most effective in terms of error vs. degrees of freedom in different situations.

(October 4) Prof.Andrei Draganescu: Multigrid preconditioners for linear systems arising in PDE constrained optimization — We will discuss the problem of finding optimal order multigrid preconditioners for linear systems involved in the solution process of large-scale, distributed optimal control problems constrained by partial differential equations. Multigrid methods have long been associated with large-scale linear systems, the paradigm being that the solution process can be significantly accelerated by using multiple resolutions of the same problem. However, the exact embodiment of the multigrid paradigm depends strongly on the class of problems considered, with multigrid methods for differential equations (elliptic, parabolic, flow problems) being significantly different from methods for PDE constrained optimization problems, where the linear systems often resemble integral equations. In this talk we will present a number of model problems for which we were able to construct optimal order multigrid preconditioners, as well as problems where we have been less successful. The test-problems include (a) linear and semi-linear elliptic constrained problems, (b) optimal control problems constrained by Stokes flow (both (a) and (b) without control-constraints), and (c) control-constrained problems with linear-elliptic PDE constraints.

(October 11) Dr. Qifeng Liao: Effective a posteriori error estimators for low order finite elements — This talk focuses on a posteriori error estimation for (bi-)linear and (bi-)quadratic elements. At first, the simple diffusion problem is tested for introducing the methodology we adopted for doing error estimation, which is based on solving local Poisson problems. Next, this methodology is applied to dealing with classical mixed approximations of incompressible flow problems. Computational results suggest that our error estimators are cost-effective, both from the perspective of accurate estimation of the global error and for the purpose of selecting elements for refinement within a contemporary self-adaptive refinement algorithm.

(October 18) Dr. Gunay Dogan: Fast reconstruction methods for diffuse imaging with many sources — In this work, we address a 2d tomography problem, where we try to reconstruct the absorption coefficient of an elliptic partial differential equation (PDE) from boundary measurements induced by a large number of sources. Our motivation for this problem is diffuse optical tomography, a new imaging modality that can be used for breast tumor detection and functional brain imaging. We pose our model problem on a square geometry where the light sources and measurements are located regularly on opposite sides of the domain. This corresponds to a popular data acquisition pattern. The problem in this form requires solving a large nonlinear inverse problem, where the forward problem is given by multiple elliptic PDEs, and is thus computationally intensive. To address this, we propose to solve a linearized version of the problem based on the Born approximation and show that substantial gains can be made in computation. By revealing the special structure of the problem, we design fast methods to assemble the coefficient matrix for the linearized problem. We also propose fast matrix-vector product routines that can be used to solve the linear system with iterative methods or sparse singular value decomposition. Finally we introduce a fast inversion algorithm that produces the solution of the inverse problem by solving a sequence of small systems. We demonstrate the effectiveness of our method with several examples. This is joint work with George Biros.

(October 25) Prof. Annalisa Quaini: A Fluid-Structure Interaction Model to Simulate Mitral Valve Regurgitant Flow — We discuss the numerical approximation of a fluid-structure interaction (FSI) problem involving an incompressible fluid (blood). This fluid surrounds an elastic wall containing a geometric orifice which mimics a leaky mitral valve. Our goal is to simulate hemodynamics conditions encountered in patients with mitral regurgitation in order to show strengths and limitations of echocardiographic methods to assess the severity of the pathology. To solve the coupled problem, we propose a semi-implicit monolithic approach which consists in preconditioning the linear system obtained after the space-time discretization and linearization of the FSI problem with a suitable diagonal scaling combined with an ILU preconditioner. Mitral regurgitant jets can be of two kinds: central (i.e. flowing in center of the atrium) and eccentric (i.e. flowing close to the atrial wall). The wall-hugging behavior of the latter kind has been associated with the Coanda effect. We will present numerical results for central jets and some preliminary work on jets undergoing the Coanda effect.

(November 08) Prof. Dongbin Xiu: Flexible Stochastic Collocation Algorithm on Arbitrary Nodes — Stochastic collocation method have become the dominating methods for uncertainty quantification and stochastic computing of large and complex systems. Though the idea has been explored in the past, its popularity is largely due to the recent advance of employing high-order nodes such as sparse grids. These nodes allow one to conduct UQ simulations with high accuracy and efficiency.

The critical issue is, without any doubt, the standing challenge of ``curse-of-dimensionality''. For practical systems with large number of random inputs, the number of nodes for stochastic collocation method can grow fast and render the method computationally prohibitive. Such kind of growth is especially severe when the nodal construction is structured, e.g., tensor grids, sparse grids, etc. One way to alleviate the difficulty is to employ adaptive approach, where the nodes are added only in the region that is needed. To this end, it is highly desirable to design stochastic collocation methods that work with arbitrary number of nodes on arbitrary locations. Another strong motivation is the practical restriction one may face. In many cases one can not conduct simulations at the desired nodes.

In this work we present a fundamental algorithm development that allows one to construct high-order polynomial responses based on stochastic collocation on arbitrary nodes. We present its rigorous mathematical framework, its practical implementation details, and its applications in high dimensions.

(November 15) Hanan Samet: Sorting in Space — The representation of spatial data is an important issue in computer graphics, computer vision, geographic information systems, and robotics. A wide number of representations is currently in use. Recently, there has been much interest in hierarchical data structures such as quadtrees, octrees, R-trees, etc. The key advantage of these representations is that they provide a way to index into space. In fact, they are little more than multidimensional sorts. They are compact and depending on the nature of the spatial data they save space as well as time and also facilitate operations such as search. In this talk we give a brief overview of hierarchical spatial data structures and related research results. In addition we demonstrate the SAND Browser (found at http://www.cs.umd.edu/~brabec/sandjava) and the VASCO JAVA applet which illustrate these methods (found at http://www.cs.umd.edu/~hjs/quadtree/index.html).

(November 29) Prof. Jorg Frauendiener: Exploring the corner: numerical relativity near space-like infinity — Numerical Relativity has made tremendous advances over the last decade. However, there are still some issues which need clarification. One of these is the question of the outer boundary. It is well known that modeling an infinite domain using a finite outer boundary creates problems, not only numerically but even more so conceptually. A partial solution of this issue is obtained by focussing on the hyperboloidal initial value problem using the conformal field equations or by regularising the equations at infinity. This still leaves the problem of having to specify hyperboloidal initial data in contrast to asymptotically Euclidean data based on a Cauchy hypersurface (that we might be more used to) and one needs to find a way of connecting data on a Cauchy surface with those on a hyperboloidal hypersurface. In this talk I will describe a particular attempt to tackle this problem based on methods developed by H. Friedrich.

(December 01) Prof. Jinchao Xu: Using Divergent Elements to Precondition Convergent Elements — For biharmonic equations (on concave domains), some "natural" mixed finite element methods may be non-optimal or simply divergent. But this type of elements, as demonstrated in this talk based on a joint work with Shuo Zhang, can be used for constructing (nearly) optimal preconditioner for both conforming (such as Agyris) and nonconforming (such as Morley) finite elements for biharmonic equations discretized on unstructured grids. The resulting preconditioner reduces the solution of a discrete biharmonic equation to the solution of several discrete Laplacian equations together with some local relaxation methods (such as Gauss-Seidel).

(December 02) Prof. Jinchao Xu: Optimal and Practical Algebraic Solvers for Discretized PDEs — An overview of fast solution techniques (such as multi-grid, two-grid, one-grid and nil-grid methods) will be given in this talk on solving large scale systems of equations that arise from the discretization of partial differential equations (such as Poisson, elasticity, Stokes, Navier-Stokes, Maxwell, MHD, and black-oil models). Mathematical optimality, practical applicability and parallel (CPU/GPU) scalability will be addressed for these algorithms and applications.

(December 06) Alfred S. Carasso: Unexplored Territory in some Nonstandard Parabolic Equations and their Application — This talk will highlight some interesting questions arising in following areas of application:

1. Fractional diffusion and image denoising: Helium Ion Microscopes (HIM) are capable of acquiring images with better than one nanometer resolution, and HIM images are particularly rich in morphological surface detail. However, such images are generally quite noisy. A major challenge is to denoise these images while preserving delicate surface information. A highly effective slow motion denoising technique, based on solving fractional diffusion equations, will be discussed.

2. Plausible, but false, backward parabolic reconstructions: Identifying sources of groundwater pollution, and deblurring galaxy images, are two important applications requiring numerical computation of time-reversed parabolic equations. However, while backward uniqueness generally holds in parabolic equations, the precise data needed for the existence of a particular backward solution is seldom available. This talk will exhibit previously unsuspected examples of physically meaningful, well-behaved, yet completely false reconstructions from slightly imprecise data.

3. Logarithmic diffusion and deblurring of Hubble imagery. Generalized Linnik processes and associated logarithmic diffusion equations can be constructed by appropriate Bochner randomization of the time variable in Brownian motion and the related heat conduction equation. Remarkably, over a large frequency range, behavior in Linnik characteristic functions matches Fourier domain behavior in a large class of Hubble telescope images. A powerful blind deconvolution procedure, based on postulating system optical transfer functions in the form of Linnik characteristic functions, will be discussed.

(December 13) Dr. Claude J. Gittelson: Multilevel, Sparse Tensor and Adaptive Discretizations of Random Elliptic PDE — We consider an elliptic PDE with a random diffusion coefficient, which we assume to be expanded in a series. The solution to such an equation depends on the coefficients in this series in addition to the spatial variables. Spectral discretizations typically use tensorized polynomials to represent the parameter-dependence of the solution. Each coefficient of the solution with respect to such a polynomial basis is a function of the spatial variable, and can be approximated by finite elements.

Since the importance of the polynomial basis functions can vary greatly, one would expect a gain in efficiency if the coefficients are approximated in different finite element spaces. We show under what conditions such a multilevel construction reaches a higher convergence rate than optimal approximations employing just a single spatial discretization. Sparse tensor products, which impose additional structure on the multilevel approximation, can always essentially attain the optimal convergence rate.

In practice, discretizations can be constructed using estimates of some norm of the coefficients. We consider adaptive strategies based on adaptive wavelet algorithms that can construct efficient discretizations if suitable estimates are not available. A simple computational example demonstrates advantages and shortcomings of all of the proposed methods.

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