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131
On Nonreflecting Boundary Conditions
 J. COMPUT. PHYS
, 1995
"... Improvements are made in nonreflecting boundary conditions at artificial boundaries for use with the Helmholtz equation. First, it is shown how to remove the difficulties that arise when the exact DtN (DirichlettoNeumann) condition is truncated for use in computation, by modifying the truncated ..."
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Cited by 219 (4 self)
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Improvements are made in nonreflecting boundary conditions at artificial boundaries for use with the Helmholtz equation. First, it is shown how to remove the difficulties that arise when the exact DtN (DirichlettoNeumann) condition is truncated for use in computation, by modifying the truncated condition. Second, the exact DtN boundary condition is derived for elliptic and spheroidal coordinates. Third, approximate local boundary conditions are derived for these coordinates. Fourth, the truncated DtN condition in elliptic and spheroidal coordinates is modified to remove difficulties. Fifth, a sequence of new and more accurate local boundary conditions is derived for polar coordinates in two dimensions. Numerical results are presented to demonstrate the usefulness of these improvements.
Comparing Images Under Variable Illumination
, 1998
"... We consider the problem of determining whether two images come from different objects or the same object in the same pose, but under different illumination conditions. We show that this problem cannot be solved using hard constraints: even using a Lambertian reflectance model, there is always an obj ..."
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Cited by 63 (5 self)
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We consider the problem of determining whether two images come from different objects or the same object in the same pose, but under different illumination conditions. We show that this problem cannot be solved using hard constraints: even using a Lambertian reflectance model, there is always an object and a pair of lighting conditions consistent with any two images. Nevertheless, we show that for point sources and objects with Lambertian reflectance, the ratio of two images from the same object is simpler than the ratio of images from different objects. We also show that the ratio of the two images provides two of the three distinct values in the Hessian matrix of the object’s surface. Using these observations, we develop a simple measure for matching images under variable illumination, comparing its performance to other existing methods on a database of 450 images of 10 individuals.
The ADIFOR 2.0 System for the Automatic Differentiation of Fortran 77 Programs
 RICE UNIVERSITY
, 1994
"... Automatic Differentiation is a technique for augmenting computer programs with statements for the computation of derivatives based on the chain rule of differential calculus. The ADIFOR 2.0 system provides automatic differentiation of Fortran 77 programs for firstorder derivatives. The ADIFOR 2.0 s ..."
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Cited by 56 (16 self)
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Automatic Differentiation is a technique for augmenting computer programs with statements for the computation of derivatives based on the chain rule of differential calculus. The ADIFOR 2.0 system provides automatic differentiation of Fortran 77 programs for firstorder derivatives. The ADIFOR 2.0 system consists of three main components: The ADIFOR 2.0 preprocessor, the ADIntrinsics Fortran 77 exceptionhandling system, and the SparsLinC library. The combination of these tools provides the ability to deal with arbitrary Fortran 77 syntax, to handle codes containing single and doubleprecision real or complexvalued data, to fully support and easily customize the translation of Fortran 77 intrinsics, and to transparently exploit sparsity in derivative computations. ADIFOR 2.0 has been successfully applied to a 60,000line code, which we believe to be a new record in automatic differentiation.
SpaceTime Continuous Analysis Of Waveform Relaxation For The Heat Equation
 SIAM Journal on Scientific Computing
, 1997
"... . Waveform relaxation algorithms for partial differential equations (PDEs) are traditionally obtained by discretizing the PDE in space and then splitting the discrete operator using matrix splittings. For the semidiscrete heat equation one can show linear convergence on unbounded time intervals and ..."
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Cited by 49 (18 self)
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. Waveform relaxation algorithms for partial differential equations (PDEs) are traditionally obtained by discretizing the PDE in space and then splitting the discrete operator using matrix splittings. For the semidiscrete heat equation one can show linear convergence on unbounded time intervals and superlinear convergence on bounded time intervals by this approach. However the bounds depend in general on the mesh parameter and convergence rates deteriorate as one refines the mesh. Motivated by the original development of waveform relaxation in circuit simulation, where the circuits are split in the physical domain into subcircuits, we split the PDE by using overlapping domain decomposition. We prove linear convergence of the algorithm in the continuous case on an infinite time interval, at a rate depending on the size of the overlap. This result remains valid after discretization in space and the convergence rates are robust with respect to mesh refinement. The algorithm is in the clas...
From signal transduction to spatial pattern formation in E. coli: A paradigm for multiscale modeling in biology, Multiscale Model
 Simul
, 2005
"... Abstract. The collective behavior of bacterial populations provides an example of how celllevel decision making translates into populationlevel behavior and illustrates clearly the difficult multiscale mathematical problem of incorporating individuallevel behavior into populationlevel models. He ..."
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Cited by 38 (16 self)
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Abstract. The collective behavior of bacterial populations provides an example of how celllevel decision making translates into populationlevel behavior and illustrates clearly the difficult multiscale mathematical problem of incorporating individuallevel behavior into populationlevel models. Here we focus on the flagellated bacterium E. coli, for which a great deal is known about signal detection, transduction, and celllevel swimming behavior. We review the biological background on individual and populationlevel processes and discuss the velocityjump approach used for describing populationlevel behavior based on individuallevel intracellular processes. In particular, we generalize the momentbased approach to macroscopic equations used earlier [R. Erban and H. G. Othmer, SIAM J. Appl. Math., 65 (2004), pp. 361–391] to higher dimensions and show how aspects of the signal transduction and response enter into the macroscopic equations. We also discuss computational issues surrounding the bacterial pattern formation problem and technical issues involved in the derivation of macroscopic equations.
Stochastic modelling of reactiondiffusion processes: algorithms for bimolecular reactions
 Phys. Biol
"... Abstract. Several stochastic simulation algorithms (SSAs) have been recently proposed for modelling reactiondiffusion processes in cellular and molecular biology. In this paper, two commonly used SSAs are studied. The first SSA is an onlattice model described by the reactiondiffusion master equat ..."
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Cited by 37 (11 self)
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Abstract. Several stochastic simulation algorithms (SSAs) have been recently proposed for modelling reactiondiffusion processes in cellular and molecular biology. In this paper, two commonly used SSAs are studied. The first SSA is an onlattice model described by the reactiondiffusion master equation. The second SSA is an offlattice model based on the simulation of Brownian motion of individual molecules and their reactive collisions. In both cases, it is shown that the commonly used implementation of bimolecular reactions (i.e. the reactions of the form A+B → C, or A+A → C) might lead to incorrect results. Improvements of both SSAs are suggested which overcome the difficulties highlighted. In particular, a formula is presented for the smallest possible compartment size (lattice spacing) which can be correctly implemented in the first model. This implementation uses a new formula for the rate of bimolecular reactions per compartment (lattice site).
A practical guide to stochastic simulations of reactiondiffusion processes, 35 pages, available as http://arxiv.org/abs/0704.1908
, 2007
"... Abstract. A practical introduction to stochastic modelling of reactiondiffusion processes is presented. No prior knowledge of stochastic simulations is assumed. The methods are explained using illustrative examples. The article starts with the classical Gillespie algorithm for the stochastic modell ..."
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Cited by 29 (13 self)
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Abstract. A practical introduction to stochastic modelling of reactiondiffusion processes is presented. No prior knowledge of stochastic simulations is assumed. The methods are explained using illustrative examples. The article starts with the classical Gillespie algorithm for the stochastic modelling of chemical reactions. Then stochastic algorithms for modelling molecular diffusion are given. Finally, basic stochastic reactiondiffusion methods are presented. The connections between stochastic simulations and deterministic models are explained and basic mathematical tools (e.g. chemical master equation) are presented. The article concludes with an overview of more advanced methods and problems. Key words. stochastic simulations, reactiondiffusion processes AMS subject classifications. 60G05, 92C40, 60J60, 92C15
Statistical symmetric shape from shading for 3d structure recovery of faces
 In European Conference on Computer Vision
, 2004
"... Abstract. In this paper, we aim to recover the 3D shape of a human face using a single image. We use a combination of symmetric shape from shading by Zhao and Chellappa and statistical approach for facial shape reconstruction by Atick, Griffin and Redlich. Given a single frontal image of a human fac ..."
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Cited by 25 (3 self)
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Abstract. In this paper, we aim to recover the 3D shape of a human face using a single image. We use a combination of symmetric shape from shading by Zhao and Chellappa and statistical approach for facial shape reconstruction by Atick, Griffin and Redlich. Given a single frontal image of a human face under a known directional illumination from a side, we represent the solution as a linear combination of basis shapes and recover the coefficients using a symmetry constraint on a facial shape and albedo. By solving a single leastsquares system of equations, our algorithm provides a closedform solution which satisfies both symmetry and statistical constraints in the best possible way. Our procedure takes only a few seconds, accounts for varying facial albedo, and is simpler than the previous methods. In the special case of horizontal illuminant direction, our algorithm runs even as fast as matrixvector multiplication. 1