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Monday, September 21, 2009 - 13:00 ,
Location: Skiles 255 ,
Yuliya Babenko ,
Department of Mathematics and Statistics, Sam Houston State University ,
Organizer: Doron Lubinsky

In this talk we first present the exact asymptotics of the optimal

error in the weighted L_p-norm, 1\leq p \leq \infty, of linear spline

interpolation of an arbitrary bivariate function f \in C^2([0,1]^2). We

further discuss the applications to numerical integration and adaptive

mesh generation for finite element methods, and explore connections

with the problem of approximating the convex bodies by polytopes. In

addition, we provide the generalization to asymmetric norms.

We give a brief review of known results and introduce a series of new

ones. The proofs of these results lead to algorithms for the

construction of asymptotically optimal sequences of triangulations for

linear interpolation.

Moreover, we derive similar results for other classes of splines and

interpolation schemes, in particular for splines over rectangular

partitions.

Last but not least, we also discuss several multivariate

generalizations.

error in the weighted L_p-norm, 1\leq p \leq \infty, of linear spline

interpolation of an arbitrary bivariate function f \in C^2([0,1]^2). We

further discuss the applications to numerical integration and adaptive

mesh generation for finite element methods, and explore connections

with the problem of approximating the convex bodies by polytopes. In

addition, we provide the generalization to asymmetric norms.

We give a brief review of known results and introduce a series of new

ones. The proofs of these results lead to algorithms for the

construction of asymptotically optimal sequences of triangulations for

linear interpolation.

Moreover, we derive similar results for other classes of splines and

interpolation schemes, in particular for splines over rectangular

partitions.

Last but not least, we also discuss several multivariate

generalizations.

Wednesday, September 9, 2009 - 13:00 ,
Location: Skiles 114 ,
Amy Novick-Cohen ,
Technion ,
Organizer: John McCuan

Grain boundaries within polycrystalline materials are known to be governed by motion by mean curvature. However, when the polycrystalline specimen is thin, such as in thin films, then the effects of the exterior surfaces start to play an important role. We consider two particularly simple geometries, an axi-symmetric geometry, and a half loop geometry which is often employed in making measurements of the kinetic coefficient in the motion by mean curvature equation, obtaining corrective terms which arise due to the coupling of grain boundaries to the exterior surface. Joint work with Anna Rotman, Arkady Vilenkin & Olga Zelekman-Smirin

Monday, August 31, 2009 - 13:00 ,
Location: Skiles 255 ,
Nicola Guglielmi ,
Università di L&#039;Aquila ,
guglielm@univaq.it ,
Organizer: Sung Ha Kang

In this talk I will address the problem of the computation of the jointspectral radius (j.s.r.) of a set of matrices.This tool is useful to determine uniform stability properties of non-autonomous discrete linear systems. After explaining how to extend the spectral radius from a single matrixto a set of matrices and illustrate some applications where such conceptplays an important role I will consider the problem of the computation ofthe j.s.r. and illustrate some possible strategies. A basic tool I willuse to this purpose consists of polytope norms, both real and complex.I will illustrate a possible algorithm for the computation of the j.s.r. ofa family of matrices which is based on the use of these classes of norms.Some examples will be shown to illustrate the behaviour of the algorithm.Finally I will address the problem of the finite computability of the j.s.r.and state some recent results, open problems and conjectures connected withthis issue.

Tuesday, August 18, 2009 - 14:00 ,
Location: Skiles 255 ,
Justin W. L. Wan ,
Computer Science, University of Waterloo ,
Organizer: Sung Ha Kang

In image guided procedures such as radiation therapies and computer-assisted surgeries, physicians often need to align images that are taken at different times and by different modalities. Typically, a rigid registration is performed first, followed by a nonrigid registration. We are interested in efficient registrations methods which are robust (numerical solution procedure will not get stuck at local minima) and fast (ideally real time). We will present a robust continuous mutual information model for multimodality regisration and explore the new emerging parallel hardware for fast computation. Nonrigid registration is then applied afterwards to further enhance the results. Elastic and fluid models were usually used but edges and small details often appear smeared in the transformed templates. We will propose a new inviscid model formulated in a particle framework, and derive the corresponding nonlinear partial differential equations for computing the spatial transformation. The idea is to simulate the template image as a set of free particles moving toward the target positions under applied forces. Our model can accommodate both small and large deformations, with sharper edges and clear texture achieved at less computational cost. We demonstrate the performance of our model on a variety of images including 2D and 3D, mono-modal and multi-modal, synthetic and clinical data.

Thursday, April 23, 2009 - 13:00 ,
Location: Skiles 255 ,
Per-Gunnar Martinsson ,
Dept of Applied Mathematics, University of Colorado ,
Organizer: Haomin Zhou

Note special day

Linear boundary value problems occur ubiquitously in many areas of

science and engineering, and the cost of computing approximate

solutions to such equations is often what determines which problems

can, and which cannot, be modelled computationally. Due to advances in

the last few decades (multigrid, FFT, fast multipole methods, etc), we

today have at our disposal numerical methods for most linear boundary

value problems that are "fast" in the sense that their computational

cost grows almost linearly with problem size. Most existing "fast"

schemes are based on iterative techniques in which a sequence of

incrementally more accurate solutions is constructed. In contrast, we

propose the use of recently developed methods that are capable of

directly inverting large systems of linear equations in almost linear

time. Such "fast direct methods" have several advantages over

existing iterative methods:

(1) Dramatic speed-ups in applications involving the repeated solution

of similar problems (e.g. optimal design, molecular dynamics).

(2) The ability to solve inherently ill-conditioned problems (such as

scattering problems) without the use of custom designed preconditioners.

(3) The ability to construct spectral decompositions of differential

and integral operators.

(4) Improved robustness and stability.

In the talk, we will also describe how randomized sampling can be used

to rapidly and accurately construct low rank approximations to matrices.

The cost of constructing a rank k approximation to an m x n matrix A

for which an O(m+n) matrix-vector multiplication scheme is available

is O((m+n)*k). This cost is the same as that of the well-established

Lanczos scheme, but the randomized scheme is significantly more robust.

For a general matrix A, the cost of the randomized scheme is O(m*n*log(k)),

which should be compared to the O(m*n*k) cost of existing deterministic

methods.

science and engineering, and the cost of computing approximate

solutions to such equations is often what determines which problems

can, and which cannot, be modelled computationally. Due to advances in

the last few decades (multigrid, FFT, fast multipole methods, etc), we

today have at our disposal numerical methods for most linear boundary

value problems that are "fast" in the sense that their computational

cost grows almost linearly with problem size. Most existing "fast"

schemes are based on iterative techniques in which a sequence of

incrementally more accurate solutions is constructed. In contrast, we

propose the use of recently developed methods that are capable of

directly inverting large systems of linear equations in almost linear

time. Such "fast direct methods" have several advantages over

existing iterative methods:

(1) Dramatic speed-ups in applications involving the repeated solution

of similar problems (e.g. optimal design, molecular dynamics).

(2) The ability to solve inherently ill-conditioned problems (such as

scattering problems) without the use of custom designed preconditioners.

(3) The ability to construct spectral decompositions of differential

and integral operators.

(4) Improved robustness and stability.

In the talk, we will also describe how randomized sampling can be used

to rapidly and accurately construct low rank approximations to matrices.

The cost of constructing a rank k approximation to an m x n matrix A

for which an O(m+n) matrix-vector multiplication scheme is available

is O((m+n)*k). This cost is the same as that of the well-established

Lanczos scheme, but the randomized scheme is significantly more robust.

For a general matrix A, the cost of the randomized scheme is O(m*n*log(k)),

which should be compared to the O(m*n*k) cost of existing deterministic

methods.

Monday, April 20, 2009 - 13:00 ,
Location: Skiles 255 ,
Tiancheng Ouyang ,
Brigham Young ,
Organizer: Chongchun Zeng

In this talk, I will show many interesting orbits in 2D and 3D of the N-body problem. Some of them do not have symmetrical property nor with equal masses. Some of them with collision singularity. The methods of our numerical optimization lead to search the initial conditions and properties of preassigned orbits. The variational methods will be used for the prove of the existence.

Friday, April 17, 2009 - 13:00 ,
Location: Skiles 255 ,
Gilad Lerman ,
University of Minnesota ,
Organizer: Sung Ha Kang

Note special day.

We propose a fast multi-way spectral clustering algorithm for multi-manifold data modeling, i.e., modeling data by mixtures of manifolds (possibly intersecting). We describe the supporting theory as well as the practical choices guided by it. We first develop the case of hybrid linear modeling, i.e., when the underlying manifolds are affine subspaces in a Euclidean space, and then we extend this setting to more general manifolds. We exemplify the practical use of the algorithm by demonstrating its successful application to problems of motion segmentation.

Monday, April 13, 2009 - 13:00 ,
Location: Skiles 255 ,
Stacey Levine ,
Duquesne University ,
Organizer: Sung Ha Kang

We present new finite difference approximations for solving

variational problems using the TV and Besov smoothness penalty

functionals. The first approach reduces oversmoothing and anisotropy

found in common discrete approximations of the TV functional. The

second approach reduces the staircasing effect that arises from TV

type smoothing. The algorithms converge and can be sped up using a

multiscale algorithm. Numerical examples demonstrate both the

qualitative and quantitative behavior of the solutions.

variational problems using the TV and Besov smoothness penalty

functionals. The first approach reduces oversmoothing and anisotropy

found in common discrete approximations of the TV functional. The

second approach reduces the staircasing effect that arises from TV

type smoothing. The algorithms converge and can be sped up using a

multiscale algorithm. Numerical examples demonstrate both the

qualitative and quantitative behavior of the solutions.

Monday, March 30, 2009 - 13:00 ,
Location: Skiles 255 ,
Richardo March ,
Istituto per le Applicazioni del Calcolo &quot;Mauro Picone&quot; of C.N.R. ,
Organizer: Haomin Zhou

We consider ordered sequences of digital images. At a given pixel a time course is observed which is related to the time courses at neighbour pixels. Useful information can be extracted from a set of such observations by classifying pixels in groups, according to some features of interest. We assume to observe a noisy version of a positive function depending on space and time, which is parameterized by a vector of unknown functions (depending on space) with discontinuities which separate regions with different features in the image domain. We propose a variational method which allows to estimate the parameter functions, to segment the image domain in regions, and to assign to each region a label according to the values that the parameters assume on the region. Approximation by \Gamma-convergence is used to design a numerical scheme. Numerical results are reported for a dynamic Magnetic Resonance imaging problem.

Wednesday, March 25, 2009 - 13:00 ,
Location: Skiles 255 ,
Junping Wang ,
NSF ,
Organizer: Haomin Zhou

This talk will first review domain decomposition methods for second order elliptic equations, which should be accessible to graduate students. The second part of the talk will deal with possible extensions to the Stokes equation when discretized by finite element methods. In particular, we shall point out the difficulties in such a generalization, and then discuss ways to overcome the difficulties.