Fast Spectral-Galerkin Methods for High-Dimensional PDEs and Applications to the electronic Schrodinger equationMonday, October 31, 2011 - 14:00 , Location: Skiles 006 , Jie Shen , Purdue University, Department of Mathematics , Organizer: Haomin Zhou
Many scientific, engineering and financial applications require solving high-dimensional PDEs. However, traditional tensor product based algorithms suffer from the so called "curse of dimensionality".We shall construct a new sparse spectral method for high-dimensional problems, and present, in particular, rigorous error estimates as well as efficient numerical algorithms for elliptic equations in both bounded and unbounded domains.As an application, we shall use the proposed sparse spectral method to solve the N-particle electronic Schrodinger equation.
Monday, October 24, 2011 - 14:00 , Location: Skiles 006 , Jun Lu , GT Math , Organizer: Sung Ha Kang
We propose a new fast algorithm for finding the global shortest path connecting two points while avoiding obstacles in a region by solving an initial value problem of ordinary differential equations (ODE's). The idea is based on the factthat the global shortest path possesses a simple geometric structure. This enables us to restrict the search in a set of feasible paths that share the same structure. The resulting search space is reduced to a finite dimensional set. We use a gradient descent strategy based on the intermittent diffusion (ID) in conjunction with the level set framework to obtain the global shortest path by solving a randomly perturbed ODE's with initial conditions.Compared to the existing methods, such as the combinatorial methods or partial differential equation(PDE) methods, our algorithm is faster and easier to implement. We can also handle cases in which obstacles shape are arbitrary and/or the dimension of the base space is three or higher.
Monday, October 10, 2011 - 14:00 , Location: Skiles 006 , Bradley Lucier , Purdue University, Department of Mathematics , Organizer: Sung Ha Kang
We consider a variant of Rudin--Osher--Fatemi variational image smoothing that replaces the BV semi-norm in the penalty term with the B^1_\infty(L_1) Besov space semi-norm. The space B^1_\infty(L_1$ differs from BV in a number of ways: It is somewhat larger than BV, so functions inB^1_\infty(L_1) can exhibit more general singularities than exhibited by functions in BV, and, in contrast to BV, affine functions are assigned no penalty in B^1_\infty(L_1). We provide a discrete model that uses a result of Ditzian and Ivanov to compute reliably with moduli of smoothness; we also incorporate some ``geometrical'' considerations into this model. We then present a convergent iterative method for solving the discrete variational problem. The resulting algorithms are multiscale, in that as the amount of smoothing increases, the results are computed using differences over increasingly large pixel distances. Some computational results will be presented. This is joint work with Greg Buzzard, Antonin Chambolle, and Stacey Levine.
Monday, October 3, 2011 - 14:00 , Location: Skiles 006 , Zhimin Zhang , Wayne State University , Organizer: Yingjie Liu
Finite element approximations for the eigenvalue problem of the Laplace operator are discussed. A gradient recovery scheme is proposed to enhance the ﬁnite element solutions of the eigenvalues. By reconstructing the numerical solution and its gradient, it is possible to produce more accurate numerical eigenvalues. Furthermore, the recovered gradient can be used to form an a posteriori error estimator to guide an adaptive mesh reﬁnement. Therefore, this method works not only for structured meshes, but also for unstructured and adaptive meshes. Additional computational cost for this post-processing technique is only O(N) (N is the total degrees of freedom), comparing with O(N^2) cost for the original problem.
Efficient Numerical Algorithms for Image Reconstruction with Total Variation Regularization and Applications in clinical MRIMonday, September 26, 2011 - 14:00 , Location: Skiles 006 , Xiaojing Ye , School of Mathematics, Georgia Tech , Organizer: Haomin Zhou
We will discuss the recent developments of fast image reconstrcution with total variation (TV) regularization whose robustness has been justfied by the theory of compressed sensing. However, the solution of TV based reconstruction encounters two main difficulties on the computational aspect of many applications: the inversion matrix can be large, irregular, and severely ill-conditioned, and the objective is nonsmooth. We introduce two algorithms that tackle the problem using variable splitting and optimized step size selection. The algorithms also provide a general framework for solving large and ill-conditioned linear inversion problem with TV regularization. An important and successful application of TV based image reconstruction in magnetic resonance imaging (MRI) known as paratially parallel imaging (PPI) will be discussed. The numerical results demonstrate significantly improved efficiency and accuracy over the state-of-the-arts.
Monday, September 19, 2011 - 14:05 , Location: Skiles 006 , Jeff Geronimo , School of Mathematics, Georgia Tech , Organizer:
Using the technique of intertwining multiresolution analysis piecewise linear, continuous, orthogonal, wavelets on a regular hexagon are constructed. We will review the technique of intertwining multiresolution analysis in the one variable case then indicate the modifications necessary for the two variable construction. This is work with George Donovan and Doug Hardin.
Monday, September 12, 2011 - 14:00 , Location: Skiles 006 , Rick Chartrand , Los Alamos National Laboratory, Theoretical Division , Organizer: Haomin Zhou
There has been much recent work applying splitting algorithms to optimization problems designed to produce sparse solutions. In this talk, we'll look at extensions of these methods to the nonconvex case, motivated by results in compressive sensing showing that nonconvex optimization can recover signals from many fewer measurements than convex optimization. Our examples of the application of these methods will include image reconstruction from few measurements, and the decomposition of high-dimensional datasets, most notably video, into low-dimensional and sparse components.
Monday, April 18, 2011 - 14:00 , Location: Skiles 005 , JungHa An , California State University, Stanislaus , Organizer: Sung Ha Kang
Medical imaging is the application of mathematical and engineering models to create images of the human body for clinical purposes or medical science by using a medical device. One of the main objectives of medical imaging research is to find the boundary of the region of the interest. The procedure to find the boundary of the region of the interest is called a segmentation. The purpose of this talk is to present a variational region based algorithm that is able to deal with spatial perturbations of the image intensity directly. Image segmentation is obtained by using a Gamma-Convergence approximation for a multi-scale piecewise smooth model. This model overcomes the limitations of global region models while avoiding the high sensitivity of local approaches. The proposed model is implemented efficiently using recursive Gaussian convolutions. The model is applied to magnetic resonance (MR) images where image quality depends highly on the acquisition protocol. Numerical experiments on 2-dimensional human liver MR images show that our model compares favorably to existing methods.This work is done in collaborated with Mikael Rousson and Chenyang Xu.
Friday, April 15, 2011 - 14:00 , Location: Skiles 005 , Xun Jia , University of California, San Diego, Department of Radiation Oncology , Organizer: Sung Ha Kang
Cone Beam Computer tomography (CBCT) has been broadly applied incancer radiation therapy, mainly for positioning patients to align withtreatment radiation beams. As opposed to tomography reconstruction problemsfor diagnostic purposes, CBCT reconstruction in radiotherapy requires a highcomputational efficiency, since it is performed while patient is lying on acouch, waiting for the treatment. Moreover, the excessive radiation dosefrom frequent scans has become a clinical concern. It is therefore desirableto develop new techniques to reconstruct CBCT images from low dose scans. Inthis talk, I will present our recent work on an iterative low dose CBCTreconstruction technique via total variation regularization and tight frameregularization. It is found that 40~60 x-ray projections are sufficient toreconstruct a volumetric image with satisfactory quality in about 2min. Wehave also studied 4 dimensional CBCT (4DCBCT) reconstruction problem viatemporal non-local means (TNLM) and high quality 4DCBCT images can beobtained. Our algorithms have been fully implemented on a graphicsprocessing unit. Detailed implementation techniques will also be addressed.
Monday, April 11, 2011 - 14:00 , Location: Skiles 005 , Alex Alexeev , Georgia Tech Mechanical Engineering , Organizer:
Biomimetic synthetic cilia can be effectively utilized for regulating microscale transport processes at interfaces. Using computer simulations, we examine how polymeric cilia can be harnessed to control the motion of microscopic particles suspended in a viscous fluid. The cilia are modeled as deformable, elastic filaments and our simulations capture the complex fluid-structure interactions among these filaments, channel walls and surrounding solution. We show that non-motile cilia that are tilted with respect to the surface can hydrodynamically direct solid particles towards channel walls, thereby, inducing their rapid deposition. When synthetic cilia are actuated by a sinusoidal force that is applied at the free ends, the beating cilia can either drive particles downwards toward the substrate or expelled particles into the fluid above the actuated cilial layer. This dynamic behavior can be regulated by changing the driving frequency. The findings uncover new routes for controlling the deposition of microscopic particles in microfluidic devices.