Imaging and Computing Seminar — Fall 2010
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Th 09/30, at 2:00pm, in room 54-517 (special time, special room!)
Maarten de Hoop, Mathematics, Purdue University.
Construction and properties of solutions of the acoustic and elastic wave equations with coefficients of limited smoothness
Abstract: We discuss the construction of weak solutions of the acoustic and elastic wave equations with $C^{1,1}$ coefficients using curvelets. We obtain a concentration of curvelets result and analyze polarization decoupling. We introduce a procedure to compute the approximate solutions appearing in the construction making use of prolate spheroidal wave functions and discuss the formation of caustics. We then establish a regularity estimate which implies the proper choice of quadrature to generate the curvelet-to-curvelet scattering. We briefly mention the possibility to incorporate random fluctuations in the coefficients and discuss applications in imaging based on reverse time migration.
Joint research with V. Brytik, S. Holman, H. Smith, G. Uhlmann, R.D. van der Hilst and H. Wendt.
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Th 10/07, at 4:15pm, in room 2-131
Julius Kusuma, Schlumberger-Doll Research.
A parametric approach to acquisition of signals with a finite rate of innovation: results and challenges
Abstract: The science and engineering of sparse signal acquisition have gathered significant interest in recent years. Both non-parametric and parametric approaches have been studied, promising novel ways to sample certain signals at rates well below the Nyquist rate. In this talk we focus on the work of Vetterli et al. that showed it is possible to reconstruct parametrically described signals from samples taken at or near their innovation rate. A majority of these approaches rely on the solution to a form of exponential fitting, for both real-valued and complex-valued cases. Therefore, analysis and understanding of how algorithms for solving exponential fitting behave in the presence of noise is very important.
In this talk we will review some basic results, discuss the prospects in various application areas including novel signal acquisition technologies, and highlight what insights and features might be desirable to bring this theoretical idea to further practice. Questions such as how to analyze and deal with undersampling/aliasing and how to build efficient devices are explored.
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Th 10/14, at 4:15, in room 54-517
Thomas Gallot, Universite Joseph Fourier, Grenoble.
Passive elastography : a correlation-based shear wave tomography in soft solids
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Th 10/21, no seminar.
See instead the talk of Charles Fefferman at 4:30pm in room 4-237 on Interpolation of data in R^n.
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Th 10/28, no seminar.
See the talk of Thomas Hoft in the Schlumberger-Tufts computational mathematics seminar.
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Th 11/04, at 4:30pm, in room 2-131
Dan Kushnir, Dept of Mathematics, Yale University
Anisotropic diffusion on sub-manifolds with application to Earth structure classification
Abstract: We show a method for computing extendable independent components of stochastic data sets generated by nonlinear mixing. In particular, we describe the empirical solution of an inverse problem in electro-magnetic measurements of geological formations. This method suggests a general tool for solving empirically inverse problems.
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Th 11/11, no seminar.
Veteran's day holiday.
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Th 11/18, at 4:30pm, in room 2-131
Jalal Fadili, ENSI Caen
Dequantizing Compressed Sensing: from recovery guarantees to nonsmooth optimization
Abstract: This talk would be divided in two parts. The first one will focus on several optimization problems involved in linear inverse problems where the solution is assumed to be sparsely represented in an overcomplete dictionary of waveforms. These problems are formalized within a unified framework of convex optimization theory, and invoke tools from convex analysis (e.g. duality, proximity operators) and maximal monotone operator splitting. Fast iterative splitting algorithms are proposed to solve them. This framework includes some previously proposed algorithms as a special case. In the second part, we study the problem of recovering sparse or compressible signals from uniformly quantized measurements. Toward this goal, a new class of convex optimization decoders is introduced, coined Basis Pursuit DeQuantizer of moment $p$ (BPDQ$_p$), that model the quantization distortion more faithfully than the commonly used BPDN or Lasso. We show that in oversampled situations, the performance of the BPDQ$_p$ decoders are significantly better than that of BPDN, with reconstruction error due to quantization divided by $\sqrt{p+1}$. This reduction relies on a modified Restricted Isometry Property of the sensing matrix expressed in the $\ell_p$-norm (RIP$_p$); a property satisfied by Gaussian random matrices with high probability. The challenging BPDQ$_p$ optimization program are solved by monotone operator splitting methods presented in the first part.
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Th 11/25, no seminar.
Thanksgiving holiday.
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Th 12/02, no seminar.
See the 12/08 talk of Laurent Demanet in the Schlumberger-Tufts computational mathematics seminar.
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Th 12/09, at 3:00pm, in room 54-517
Andreas Kloeckner, Courant Institute, New York University
Machine-adapted Methods: High-order Discontinuous Galerkin Wave Propagation on GPUs
Abstract: Having recently shown that high-order unstructured discontinuous Galerkin (DG) methods are a spatial discretization that is well-matched to execution on GPUs, in this talk I will explore both core and supporting components of high-order DG wave solvers for their suitability for and performance on modern, massively parallel architectures. Components examined range from linear forward solvers and the software components needed to make them work well to shock capturing schemes and time steppers, for which I will present a selection of design considerations, algorithms, and performance data.
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Mo 12/13, at 3:00pm, in room 2-147
Yaniv Plan, Applied and Computational Mathematics, Caltech
A probabilistic and RIPless theory of compressed sensing
Abstract: We introduce a novel, simple, and very general theory of compressive sensing. In this theory, the sensing mechanism simply selects sensing vectors independently at random from a probability distribution F; it includes all models — e.g. Gaussian, frequency measurements — commonly discussed in the literature, but also provides a framework for new measurement strategies as well. We prove that if the probability distribution F obeys a simple incoherence property and an isotropy property, one can faithfully recover approximately sparse signals from a minimal number of noisy measurements. The novelty is that our recovery results do not require the restricted isometry property (RIP) — they make use of a much weaker notion — or a random model for the signal. As an example, the theory demonstrates that a signal with s nonzero entries can be faithfully recovered from about s log n Fourier coefficients that are contaminated with noise.