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396
The Variational Formulation of the FokkerPlanck Equation
 SIAM J. Math. Anal
, 1999
"... The FokkerPlanck equation, or forward Kolmogorov equation, describes the evolution of the probability density for a stochastic process associated with an Ito stochastic differential equation. It pertains to a wide variety of timedependent systems in which randomness plays a role. In this paper, ..."
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Cited by 285 (22 self)
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The FokkerPlanck equation, or forward Kolmogorov equation, describes the evolution of the probability density for a stochastic process associated with an Ito stochastic differential equation. It pertains to a wide variety of timedependent systems in which randomness plays a role. In this paper, we are concerned with FokkerPlanck equations for which the drift term is given by the gradient of a potential. For a broad class of potentials, we construct a timediscrete, iterative variational scheme whose solutions converge to the solution of the FokkerPlanck equation. The major novelty of this iterative scheme is that the time step is governed by the Wasserstein metric on probability measures. This formulation enables us to reveal an appealing, and previously unexplored, relationship between the FokkerPlanck equation and the associated free energy functional. Namely, we demonstrate that the dynamics may be regarded as a gradient flux, or a steepest descent, for the free energy wi...
ImplicitExplicit Methods For TimeDependent PDEs
 SIAM J. NUMER. ANAL
, 1997
"... Implicitexplicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods, for the time integration of spatially discretized PDEs of diffusionconvection type. Typically, an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection ..."
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Cited by 177 (6 self)
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Implicitexplicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods, for the time integration of spatially discretized PDEs of diffusionconvection type. Typically, an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term. Reactiondiffusion problems can also be approximated in this manner. In this work we systematically analyze the performance of such schemes, propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods. For the prototype linear advectiondiffusion equation, a stability analysis for first, second, third and fourth order multistep IMEX schemes is performed. Stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified. Numerical experiments demonstrate that weak decay of high freque...
Volumetric Transformation of Brain Anatomy
 IEEE TRANSACTIONS ON MEDICAL IMAGING
, 1997
"... This paper presents diffeomorphic transformations of threedimensional (3D) anatomical image data of the macaque occipital lobe and whole brain cryosection imagery and of deep brain structures in human brains as imaged via magnetic resonance imagery. These transformations are generated in a hierarc ..."
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Cited by 139 (11 self)
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This paper presents diffeomorphic transformations of threedimensional (3D) anatomical image data of the macaque occipital lobe and whole brain cryosection imagery and of deep brain structures in human brains as imaged via magnetic resonance imagery. These transformations are generated in a hierarchical manner, accommodating both global and local anatomical detail. The initial lowdimensional registration is accomplished by constraining the transformation to be in a lowdimensional basis. The basis is defined by the Green's function of the elasticity operator placed at predefined locations in the anatomy and the eigenfunctions of the elasticity operator. The highdimensional large deformations are vector fields generated via the mismatch between the template and targetimage volumes constrained to be the solution of a NavierStokes fluid model. As part of this procedure, the Jacobian of the transformation is tracked, insuring the generation of diffeomorphisms. It is shown that transformations constrained by quadratic regularization methods such as the Laplacian, biharmonic, and linear elasticity models, do not ensure that the transformation maintains topology and, therefore, must only be used for coarse global registration.
Swarming patterns in a twodimensional kinematic model for biological groups
 SIAM J. Appl. Math
, 2004
"... Abstract. We construct a continuum model for the motion of biological organisms experiencing social interactions and study its patternforming behavior. The model takes the form of a conservation law in two spatial dimensions. The social interactions are modeled in the velocity term, which is nonloc ..."
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Cited by 127 (19 self)
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Abstract. We construct a continuum model for the motion of biological organisms experiencing social interactions and study its patternforming behavior. The model takes the form of a conservation law in two spatial dimensions. The social interactions are modeled in the velocity term, which is nonlocal in the population density and includes a parameter that controls the interaction length scale. The dynamics of the resulting partial integrodifferential equation may be uniquely decomposed into incompressible motion and potential motion. For the purely incompressible case, the model resembles one for fluid dynamical vortex patches. There exist solutions which have constant population density and compact support for all time. Numerical simulations produce rotating structures which have circular cores and spiral arms and are reminiscent of naturally observed phenomena such as ant mills. The sign of the social interaction term determines the direction of the rotation, and the interaction length scale affects the degree of spiral formation. For the purely potential case, the model resembles a nonlocal (forwards or backwards) porous media equation. The sign of the social interaction term controls whether the population aggregates or disperses, and the interaction length scale controls the balance between transport and smoothing of the density profile. For the aggregative case, the population clumps into regions of high and low density. The characteristic length scale of the density pattern is predicted and confirmed by numerical simulations.
Modeling and Rendering of Weathered Stone
 SIGGRAPH'99
, 1999
"... Stone is widespread in its use as a building material and artistic medium. One of its most remarkable qualities is that it changes appearance as it interacts with the environment. These changes are mainly confined to the surface but involve complex volumetric effects such as erosion and mineral diss ..."
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Cited by 112 (12 self)
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Stone is widespread in its use as a building material and artistic medium. One of its most remarkable qualities is that it changes appearance as it interacts with the environment. These changes are mainly confined to the surface but involve complex volumetric effects such as erosion and mineral dissolution. This paper presents an approach for the modeling and rendering of changes in the shape and appearance of stone. To represent stone, we introduce a slab data structure, which is a surfacealigned volume confined to a narrow region around the boundary of the stone. Our weathering model employs a simulation of the flow of moisture and the transport, dissolution, and recrystallization of minerals within the porous stone volume. In addition, this model governs the erosion of material from the surface. To render the optical effects of translucency and coloration due to the composition of minerals near the surface, we simulate the scattering of light inside the stone using a general subsurface Monte Carlo ray tracer. These techniques can capture many aspects of the timedependent appearance of stone. We demonstrate the approach with models of granite and marble statues, as well as a sandstone column.
Deformable Shape Models For Anatomy
, 1994
"... Medical imaging modalities, such as magnetic resonance (MR), histological images, and positron emission tomography (PET), enable study of anatomy and function in animals and humans. The technology to collect such data greatly exceeds tools to analyze it. This research seeks to address this issue by ..."
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Cited by 76 (0 self)
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Medical imaging modalities, such as magnetic resonance (MR), histological images, and positron emission tomography (PET), enable study of anatomy and function in animals and humans. The technology to collect such data greatly exceeds tools to analyze it. This research seeks to address this issue by developing methods that automatically synthesize labeled electronic atlases tailored to individuals. The approach
Physical Modeling with the 2D Digital Waveguide Mesh
, 1993
"... An extremely efficient method for modeling wave propagation in a membrane is provided by the multidimensional extension of the digital waveguide. The 2D digital waveguide mesh is constructed out of bidirectional delay units and scattering junctions. We show that it coincides with the standard finit ..."
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Cited by 71 (7 self)
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An extremely efficient method for modeling wave propagation in a membrane is provided by the multidimensional extension of the digital waveguide. The 2D digital waveguide mesh is constructed out of bidirectional delay units and scattering junctions. We show that it coincides with the standard finite difference approximation scheme for the 2D wave equation, and we derive the dispersion error. Applications may be found in physical models of drums, soundboards, cymbals, gongs, smallbox reverberators, and other acoustic constructs where a onedimensional model is less desirable. 1 Background Theory There are many musical applications of the onedimensional digital waveguide ranging from the generation of wind and string instrument tones, to flanging effects [Van Duyne and Smith, 1992], to reverberation [Smith, 1987]. We review the theoretical derivation of onedimensional traveling waves as a basis for development of the twodimensional digital waveguide mesh. 1.1 The 1D Wave Equation...
Stability of largeamplitude shock waves of compressible NavierStokes equations
, 2003
"... We summarize recent progress on one and multidimensional stability of viscous shock wave solutions of compressible Navier–Stokes equations and related symmetrizable hyperbolic–parabolic systems, with an emphasis on the largeamplitude regime where transition from stability to instability may be ..."
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Cited by 66 (38 self)
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We summarize recent progress on one and multidimensional stability of viscous shock wave solutions of compressible Navier–Stokes equations and related symmetrizable hyperbolic–parabolic systems, with an emphasis on the largeamplitude regime where transition from stability to instability may be expected to occur. The main result is the establishment of rigorous necessary and sufficient conditions for linearized and nonlinear planar viscous stability, agreeing in one dimension and separated in multidimensions by a codimension one set, that both extend and sharpen the formal conditions of structural and dynamical stability found in classical physical literature. The sufficient condition in multidimensions is new, and represents the main mathematical contribution of this article. The sufficient condition for stability is always satisfied for sufficiently smallamplitude shocks, while the necessary condition is known to fail under certain circumstances for sufficiently largeamplitude shocks; both are readily evaluable numerically. The precise conditions under and the nature in which transition from stability to instability occurs are outstanding open questions in the theory.
A review of geometric transformations for nonrigid body registration
 IEEE TRANSACTIONS ON MEDICAL IMAGING
, 2007
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