238 results on '"Guillaume Bal"'
Search Results
102. Convergence to SPDEs in Stratonovich Form
- Author
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Guillaume Bal
- Subjects
Random field ,Stochastic modelling ,Gaussian ,010102 general mathematics ,Multiplicative function ,Mathematical analysis ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,01 natural sciences ,Stochastic partial differential equation ,010104 statistics & probability ,symbols.namesake ,Amplitude ,Operator (computer programming) ,symbols ,35R60, 60H15, 35K15 ,0101 mathematics ,Mathematical Physics ,Randomness ,Mathematics - Abstract
We consider the perturbation of parabolic operators of the form $\partial_t+P(x,D)$ by large-amplitude highly oscillatory spatially dependent potentials modeled as Gaussian random fields. The amplitude of the potential is chosen so that the solution to the random equation is affected by the randomness at the leading order. We show that, when the dimension is smaller than the order of the elliptic pseudo-differential operator $P(x,D)$, the perturbed parabolic equation admits a solution given by a Duhamel expansion. Moreover, as the correlation length of the potential vanishes, we show that the latter solution converges in distribution to the solution of a stochastic parabolic equation with a multiplicative term that should be interpreted in the Stratonovich sense. The theory of mild solutions for such stochastic partial differential equations is developed. The behavior described above should be contrasted to the case of dimensions that are larger than or equal to the order of the elliptic pseudo-differential operator $P(x,D)$. In the latter case, the solution to the random equation converges strongly to the solution of a homogenized (deterministic) parabolic equation as is shown in the companion paper [2]. The stochastic model is therefore valid only for sufficiently small space dimensions in this class of parabolic problems., 21 pages
- Published
- 2009
103. Transport-Based Imaging in Random Media
- Author
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Kui Ren and Guillaume Bal
- Subjects
Physics ,Classical mechanics ,Applied Mathematics ,Energy density ,Radiative transfer ,Random media ,Monochromatic color ,High Frequency Waves ,Inverse problem - Abstract
This paper generalizes well-established derivations of the radiative transfer equation from first principles to model the energy density of time-dependent and monochromatic high frequency waves pro...
- Published
- 2008
104. Inverse transport with isotropic sources and angularly averaged measurements
- Author
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François Monard, Ian Langmore, and Guillaume Bal
- Subjects
Physics ,Control and Optimization ,medicine.diagnostic_test ,Scattering ,Mathematical analysis ,Isotropy ,Inverse ,Low frequency ,Harmonic function ,Modeling and Simulation ,Content (measure theory) ,medicine ,Discrete Mathematics and Combinatorics ,Pharmacology (medical) ,Optical tomography ,Absorption (electromagnetic radiation) ,Analysis - Abstract
We consider the reconstruction of a spatially-dependent scattering coefficient in a linear transport equation from diffusion-type measurements. In this setup, the contribution to the measurement is an integral of the scattering kernel against a product of harmonic functions, plus an additional term that is small when absorption and scattering are small. The linearized problem is severely ill-posed. We construct a regularized inverse that allows for reconstruction of the low frequency content of the scattering kernel, up to quadratic error, from the nonlinear map. An iterative scheme is used to improve this error so that it is small when the high frequency content of the scattering kernel is small.
- Published
- 2008
105. Experimental validation of a transport-based imaging method in highly scattering environments
- Author
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Dehong Liu, Guillaume Bal, Lawrence Carin, and Kui Ren
- Subjects
business.industry ,Scattering ,Applied Mathematics ,Experimental validation ,Reconstruction method ,Computer Science Applications ,Theoretical Computer Science ,Computational physics ,Set (abstract data type) ,Optics ,Signal Processing ,business ,Mathematical Physics ,Energy (signal processing) ,Mathematics - Abstract
We demonstrate the effectiveness of a transport-based reconstruction method for imaging in highly scattering environments. Experimentally measured wave energy data in the micro-wave regime are used to reconstruct extended inclusions buried in scattering media or hidden behind non-penetrable obstacles. The performance of the imaging method is illustrated under various circumstances, via a set of electromagnetic experiments.
- Published
- 2007
106. Electromagnetic Time-Reversal Source Localization in Changing Media: Experiment and Analysis
- Author
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Lawrence Carin, Dehong Liu, Guillaume Bal, S. Vasudevan, and Jeffrey L. Krolik
- Subjects
Physics ,Time inversion ,Scattering ,business.industry ,Wave propagation ,Acoustics ,Inversion (meteorology) ,symbols.namesake ,Optics ,Green's function ,Source localization ,symbols ,Electromagnetic wave scattering ,Electrical and Electronic Engineering ,business - Abstract
An experimental study is performed on electromagnetic time reversal in highly scattering environments, with a particular focus on performance when environmental conditions change. In particular, we consider the case for which there is a mismatch between the Green's function used on the forward measurement and that used for time-reversal inversion. We examine the degradation in the time-reversal image with increasing media mismatch, and consider techniques that mitigate such degradation. The experimental results are also compared with theoretical predictions for time reversal in changing media, with good agreement observed
- Published
- 2007
107. Kinetic Models for Imaging in Random Media
- Author
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Olivier Pinaud and Guillaume Bal
- Subjects
Physics ,Field (physics) ,Ecological Modeling ,Detector ,General Physics and Astronomy ,General Chemistry ,Function (mathematics) ,Object (computer science) ,Kinetic energy ,Computer Science Applications ,Modeling and Simulation ,Radiative transfer ,Statistical physics ,Diffusion (business) ,Energy (signal processing) - Abstract
We derive kinetic models for the correlations and the energy densities of wave fields propagating in random media. These models take the form of radiative transfer and diffusion equations. We use these macroscopic models to address the detection and imaging of small objects buried in highly heterogeneous media. More specifically, we quantify the influence of small objects on (i) the energy density measured at an array of detectors and (ii) the correlation between the wave field measured in the absence of the object and the wave field measured in the presence of the object. We analyze the advantages and disadvantages of such measurements as a function of the level of disorder in the random media. Numerical simulations verify the theoretical predictions.
- Published
- 2007
108. Inverse Source Problems in Transport Equations
- Author
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Alexandru Tamasan and Guillaume Bal
- Subjects
Computational Mathematics ,Radon transform ,Scattering ,Iterative method ,Applied Mathematics ,Phase space ,Mathematical analysis ,Inverse ,Inversion (meteorology) ,Inverse problem ,Convection–diffusion equation ,Analysis ,Mathematics - Abstract
This paper proposes an iterative technique to reconstruct the source term in transport equations, which account for scattering effects, from boundary measurements. In the two‐dimensional setting, the full outgoing distribution in the phase space (position and direction) needs to be measured. In three space dimensions, we show that measurements for angles that are orthogonal to a given direction are sufficient. In both cases, the derivation is based on a perturbation of the inversion of the two‐dimensional attenuated Radon transform and requires that (the anisotropic part of) scattering be sufficiently small. We present an explicit iterative procedure, which converges to the source term we want to reconstruct. Applications of the inversion procedure include optical molecular imaging, an increasingly popular medical imaging modality.
- Published
- 2007
109. Homogenization in random media and effective medium theory for high frequency waves
- Author
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Guillaume Bal
- Subjects
Physics ,Wavelength ,Applied Mathematics ,Norm (mathematics) ,Mathematical analysis ,Energy density ,Discrete Mathematics and Combinatorics ,Random media ,Limiting ,High Frequency Waves ,Wave equation ,Homogenization (chemistry) - Abstract
We consider the homogenization of the wave equation with high frequency initial conditions propagating in a medium with highly oscillatory random coefficients. By appropriate mixing assumptions on the random medium, we obtain an error estimate between the exact wave solution and the homogenized wave solution in the energy norm. This allows us to consider the limiting behavior of the energy density of high frequency waves propagating in highly heterogeneous media when the wavelength is much larger than the correlation length in the medium.
- Published
- 2007
110. Fluctuations in the Homogenization of Semilinear Equations with Random Potentials
- Author
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Wenjia Jing and Guillaume Bal
- Subjects
Random field ,35B27, 35J61, 60F05 ,Applied Mathematics ,Gaussian ,Probability (math.PR) ,010102 general mathematics ,Mathematical analysis ,Asymptotic distribution ,01 natural sciences ,Homogenization (chemistry) ,010101 applied mathematics ,Nonlinear system ,symbols.namesake ,Mathematics - Analysis of PDEs ,Square-integrable function ,FOS: Mathematics ,symbols ,0101 mathematics ,Analysis ,Linear equation ,Randomness ,Mathematics - Probability ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We study the stochastic homogenization and obtain a random fluctuation theory for semilinear elliptic equations with a rapidly varying random potential. To first order, the effective potential is the average potential and the nonlinearity is not affected by the randomness. We then study the limiting distribution of the properly scaled homogenization error (random fluctuations) in the space of square integrable functions, and prove that the limit is a Gaussian distribution characterized by the homogenized solution, the Green's function of the linearized equation around the homogenized solution, and by the integral of the correlation function of the random potential. These results enlarge the scope of the framework that we have developed for linear equations to the class of semilinear equations., 19 pages
- Published
- 2015
111. Reconstruction of a fully anisotropic elasticity tensor from knowledge of displacement fields
- Author
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François Monard, Guillaume Bal, and Gunther Uhlmann
- Subjects
medicine.diagnostic_test ,Applied Mathematics ,Mathematical analysis ,Inverse problem ,Mathematics - Analysis of PDEs ,Transverse isotropy ,FOS: Mathematics ,medicine ,Elastography ,Tensor ,Algebraic number ,Elasticity (economics) ,Anisotropy ,Finite set ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We present explicit reconstruction algorithms for fully anisotropic unknown elasticity tensors from knowledge of a finite number of internal displacement fields, with applications to transient elastography. Under certain rank-maximality assumptions satified by the strain fields, explicit algebraic reconstruction formulas are provided. A discussion ensues on how to fulfill these assumptions, describing the range of validity of the approach. We also show how the general method can be applied to more specific cases such as the transversely isotropic one., 23 pages. Minor updates and additional references
- Published
- 2015
112. Displacement Reconstructions in Ultrasound Elastography
- Author
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Sébastien Imperiale, Guillaume Bal, Department of Applied Physics and Applied Mathematics [New York] (APAM), Columbia University [New York], Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris
- Subjects
medicine.medical_specialty ,Computer science ,General Mathematics ,Acoustics ,Physics::Medical Physics ,01 natural sciences ,Displacement (vector) ,030218 nuclear medicine & medical imaging ,03 medical and health sciences ,symbols.namesake ,0302 clinical medicine ,medicine ,Medical imaging ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,0101 mathematics ,Reconstruction procedure ,medicine.diagnostic_test ,business.industry ,Applied Mathematics ,Ultrasound ,Inverse problem ,010101 applied mathematics ,Interferometry ,Fourier transform ,symbols ,Elastography ,Radiology ,business - Abstract
International audience; We consider the reconstruction of internal elastic displacements from ultrasound measurements , which finds applications in the medical imaging modality called elastography. By appropriate interferometry and windowed Fourier transforms of the ultrasound measurements, we propose a reconstruction procedure of the vectorial structure of spatially varying elastic displacements in biological tissues. This provides a modeling and generalization of scalar reconstruction procedures routinely used in elastography. The proposed algorithm is justified using a single scattering approximation and local asymptotic analysis. Its validity is assessed by numerical simulations.
- Published
- 2015
113. Random Media in Inverse Problems, Theoretical Aspects
- Author
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Guillaume Bal, Olivier Pinaud, and Lenya Ryzhik
- Published
- 2015
114. Accuracy of transport models for waves in random media
- Author
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Guillaume Bal and Olivier Pinaud
- Subjects
Coupling ,Physics ,Applied Mathematics ,Monte Carlo method ,General Physics and Astronomy ,Acoustic wave ,Wave equation ,Space (mathematics) ,Computational Mathematics ,Wavelength ,Atmospheric radiative transfer codes ,Modeling and Simulation ,Radiative transfer ,Statistical physics - Abstract
This paper addresses the validity of radiative transfer equations as a model for the energy density of waves propagating in highly heterogeneous media. Comparisons between acoustic wave simulations over domains of size comparable to 500 wavelengths in two space dimensions and Monte Carlo simulations of radiative transfer equations are performed. In the so-called weak coupling regime, the agreement between the energy densities obtained by solving the wave equations and those predicted by solving the radiative transfer equations is remarkable. The domain of validity of the radiative transfer equations is assessed by looking at the fluctuations in the energy density they predict in the presence of small-volume defects in the underlying media.
- Published
- 2006
115. RECONSTRUCTION OF SINGULAR SURFACES BY SHAPE SENSITIVITY ANALYSIS AND LEVEL SET METHOD
- Author
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Guillaume Bal and Kui Ren
- Subjects
Surface (mathematics) ,Level set method ,Applied Mathematics ,Modeling and Simulation ,Vector field ,Geometry ,Sensitivity (control systems) ,Function (mathematics) ,Boundary value problem ,Inverse problem ,Diffusion (business) ,Mathematics - Abstract
We consider the reconstruction of singular surfaces from the over-determined boundary conditions of an elliptic problem. The problem arises in optical and impedance tomography, where void-like structure or cracks may be modeled as diffusion processes supported on co-dimension one surfaces. The reconstruction of such surfaces is obtained theoretically and numerically by combining a shape sensitivity analysis with a level set method. The shape sensitivity analysis is used to define a velocity field, which allows us to update the surface while decreasing a given cost function, which quantifies the error between the prediction of the forward model and the measured data. The velocity field depends on the geometry of the surface and the tangential diffusion process supported on it. The latter process is assumed to be known in this paper. The level set method is next applied to evolve the surface in the direction of the velocity field. Numerical simulations show how the surface may be reconstructed from noisy estimates of the full, or local, Neumann-to-Dirichlet map.
- Published
- 2006
116. Kinetics of scalar wave fields in random media
- Author
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Guillaume Bal
- Subjects
Field (physics) ,Applied Mathematics ,General Physics and Astronomy ,Differential operator ,Kinetic energy ,Computational Mathematics ,Classical mechanics ,Correlation function ,Modeling and Simulation ,Phase space ,Wave function ,Asymptotic expansion ,Scalar field ,Mathematics - Abstract
This paper concerns the derivation of kinetic models for high frequency scalar wave fields propagating in random media. The kinetic equations model the propagation in the phase space of the energy density of a wave field or the correlation function of two wave fields propagating in two possibly different media. Dispersive effects due to, e.g. spatial and temporal discretizations, which are modeled as non-local pseudo-differential operators, are taken into account. The derivation of the models is based on a multiple-scale asymptotic expansion of the spatio-temporal Wigner transform of two scalar wave fields.
- Published
- 2005
117. Ray transforms in hyperbolic geometry
- Author
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Guillaume Bal
- Subjects
Mathematics(all) ,Field (physics) ,Geodesic ,Applied Mathematics ,General Mathematics ,Hyperbolic geometry ,Ray transform ,010102 general mathematics ,Coordinate system ,Mathematical analysis ,Complexification ,01 natural sciences ,010101 applied mathematics ,Transport equation ,symbols.namesake ,Riemann–Hilbert problem ,Euclidean geometry ,symbols ,Vector field ,0101 mathematics ,Mathematics - Abstract
We derive explicit inversion formulae for the attenuated geodesic and horocyclic ray transforms of functions and vector fields on two-dimensional manifolds equipped with the hyperbolic metric. The inversion formulae are based on a suitable complexification of the associated vector fields so as to recast the reconstruction as a Riemann–Hilbert problem. The inversion formulae have a very similar structure to their counterparts in Euclidean geometry and may therefore be amenable to efficient discretizations and numerical inversions. An important field of application is geophysical imaging when absorption effects are accounted for.
- Published
- 2005
118. Time-reversal-based detection in random media
- Author
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Guillaume Bal and Olivier Pinaud
- Subjects
Diffusion (acoustics) ,Diffusion equation ,business.industry ,Applied Mathematics ,Statistical parameter ,Context (language use) ,Acoustic wave ,Noise (electronics) ,Computer Science Applications ,Theoretical Computer Science ,Background noise ,Optics ,Signal Processing ,Statistical physics ,business ,Mathematical Physics ,Realization (probability) ,Mathematics - Abstract
We consider the detection and imaging of inclusions buried in highly heterogeneous media. We assume that only the statistical properties of the heterogeneous media can be observed and that the wave energy density may be modelled by macroscopic equations. The detection and imaging capabilities hinge on ensuring that the measured data are statistically stable, which means that they depend only on the macroscopic statistical parameters of the random media and not on the microscopic statistical realization. In this paper, the macroscopic model is a diffusion equation. In this context, we construct statistical tests to detect inclusions based on macroscopic diffusion measurements and perform asymptotic expansions to image their location and volume. We show that time-reversal measurements enjoy a much larger signal-to-noise ratio in the presence of background noise than do direct wave energy measurements. This is a direct consequence of the enhanced refocusing properties that characterize time reversed waves propagating in heterogeneous media. Finally, we present numerical simulations of acoustic waves propagating in heterogeneous two-dimensional media. The numerical simulations illustrate which factors contribute to 'noise' in the measured data and how they affect the detection and imaging capabilities.
- Published
- 2005
119. Atmospheric concentration profile reconstructions from radiation measurements
- Author
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Guillaume Bal and Kui Ren
- Subjects
Spectral power distribution ,business.industry ,Applied Mathematics ,Radiation ,Inverse problem ,Computer Science Applications ,Theoretical Computer Science ,Computational physics ,Atmosphere of Earth ,Optics ,Signal Processing ,Radiance ,Uniqueness ,Asymptotic expansion ,business ,Reconstruction procedure ,Mathematical Physics ,Mathematics - Abstract
We consider the reconstruction of vertical concentration profiles of atmospheric gases from a spectral distribution of radiation measured from a space-borne infrared spectrometer. Under some separability assumptions of the gases' spectral absorption coefficients, we obtain uniqueness results on the reconstruction of concentration profiles from (multiple-wavenumber) radiance measurements and provide an explicit reconstruction procedure. We show that the reconstruction is a severely ill-posed problem. To address the reconstruction of localized layers, such as ozone or dust layers, we model the reconstruction of strong localized variations in the concentration profiles by using asymptotic expansions in the layer thickness. Assuming the background is known, we obtain that the location as well as the product of the concentration variability within the layer multiplied and the thickness of the layer may be reconstructed from moderately noisy data. The reconstructions of both the concentration and the thickness of the layer require more accurate data.
- Published
- 2004
120. Reconstructions in impedance and optical tomography with singular interfaces
- Author
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Guillaume Bal
- Subjects
Surface (mathematics) ,medicine.diagnostic_test ,Applied Mathematics ,Operator (physics) ,Mathematical analysis ,Boundary (topology) ,Geometry ,Computer Science Applications ,Theoretical Computer Science ,Diffusion process ,Signal Processing ,medicine ,Tomography ,Optical tomography ,Diffusion (business) ,Material properties ,Mathematical Physics ,Mathematics - Abstract
Singular layers modelled by a tangential diffusion process supported on an embedded closed surface (of co-dimension 1) have found applications in tomography problems. In optical tomography they may model the propagation of photons in thin clear layers, which are known to hamper the use of classical diffusion approximations. In impedance tomography they may be used to model thin regions of very high conductivity profile. In this paper we show that such surfaces can be reconstructed from boundary measurements (more precisely, from a local Neumann-to-Dirichlet operator) provided that the material properties between the measurement surface and the embedded surface are known. The method is based on the factorization technique introduced by Kirsch. Once the location of the surface is reconstructed, we show under appropriate assumptions that the full tangential diffusion process and the material properties in the region enclosed by the surface can also uniquely be determined.
- Published
- 2004
121. Time splitting for wave equations in random media
- Author
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Lenya Ryzhik and Guillaume Bal
- Subjects
Numerical Analysis ,Spacetime ,Computer simulation ,Field (physics) ,Scattering ,Applied Mathematics ,Numerical analysis ,Geometry ,Wave equation ,Mathematical theory ,Computational Mathematics ,Modeling and Simulation ,Convergence (routing) ,Statistical physics ,Analysis ,Mathematics - Abstract
Numerical simulation of high frequency waves in highly heterogeneous media is a challenging problem. Resolving the fine structure of the wave field typically requires extremely small time steps and spatial meshes. We show that capturing macroscopic quantities of the wave field, such as the wave energy density, is achievable with much coarser discretizations. We obtain such a result using a time splitting algorithm that solves separately and successively propagation and scattering in the simplified regime of the parabolic wave equation in a random medium. The mathematical theory of the convergence and statistical properties of the algorithm is based on the analysis of the Wigner transforms in random media. Our results provide a step toward understanding time and space discretizations that are needed in order for the numerical algorithm to capture the correct macroscopic statistics of the wave energy density in a random medium.
- Published
- 2004
122. Fast numerical inversion of the attenuated Radon transform with full and partial measurements
- Author
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Guillaume Bal, Philippe Moireau, Department of Applied Physics and Applied Mathematics [New York] (APAM), and Columbia University [New York]
- Subjects
02 engineering and technology ,01 natural sciences ,Theoretical Computer Science ,symbols.namesake ,Optics ,0202 electrical engineering, electronic engineering, information engineering ,Medical imaging ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,0101 mathematics ,ComputingMilieux_MISCELLANEOUS ,Mathematical Physics ,Mathematics ,Radon transform ,business.industry ,Applied Mathematics ,Numerical analysis ,Inverse problem ,Binary logarithm ,[INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation ,Computer Science Applications ,Exponential function ,010101 applied mathematics ,Fourier transform ,Attenuation coefficient ,Signal Processing ,symbols ,020201 artificial intelligence & image processing ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,business ,Algorithm - Abstract
We propose a numerical method to simulate and invert the two-dimensional attenuated Radon transform (AtRT) from full (360 ◦ ) or partial (180 ◦ ) measurements. The method is based on an extension of the fast slant stack algorithm developed for the Radon transform. We show that the algorithm offers robust and fast inversion of the AtRT for a wide class of synthetic sources and absorptions. The complexity of the fast algorithm to compute the AtRT of a n × n image and perform the reconstruction from the AtRT data is O(Nn 2 log n) operations, with N the number of Fourier modes necessary to accurately represent the absorption map. The algorithm is applied to the reconstruction of the exponential Radon transform, where the absorption coefficient is constant, and of the AtRT when only 180 ◦ measurements are available. The reconstruction from partial measurements is based on an iterative scheme introduced recently in Bal (2004 Inverse Problems 20 399– 419). Single-photon emission computed tomography is an important medical imaging technique based on the inversion of the AtRT.
- Published
- 2004
123. Optical tomography for small volume absorbing inclusions
- Author
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Guillaume Bal
- Subjects
Well-posed problem ,Diffusion equation ,medicine.diagnostic_test ,Applied Mathematics ,Mathematical analysis ,Boundary (topology) ,Domain (mathematical analysis) ,Computer Science Applications ,Theoretical Computer Science ,Signal Processing ,medicine ,Optical tomography ,Asymptotic expansion ,Absorption (electromagnetic radiation) ,Finite set ,Mathematical Physics ,Mathematics - Abstract
We present the asymptotic expansion of the solution to a diffusion equation with a finite number of absorbing inclusions of small volume. We use the first few terms in this expansion measured at the domain boundary to reconstruct the absorption parameters of the inclusions and certain geometrical characteristics. We demonstrate theoretically and numerically that the number of inclusions, their location and their capacity can be reconstructed in a stable way even from moderately noisy data. The reconstruction of the absorption parameter, which is important in optical tomography to discriminate between healthy and unhealthy tissues, requires us however to have far less noisy data. Since the reconstruction of absorption maps from boundary measurements is an extremely ill posed problem, the method of asymptotic expansions of small volume inclusions provides a useful framework to decide which information can be reconstructed from boundary measurements with a given noise level.
- Published
- 2003
124. Time Reversal and Refocusing in Random Media
- Author
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Leonid Ryzhik and Guillaume Bal
- Subjects
Diffraction ,Time reversal signal processing ,Optics ,Scattering ,business.industry ,Applied Mathematics ,Wave field ,Random media ,Beat (acoustics) ,Heavy traffic approximation ,business ,Image resolution ,Mathematics - Abstract
In time reversal acoustics experiments, a signal is emitted from a localized source, recorded at an array of receivers, time reversed, and finally reemitted into the medium. A celebrated feature of time reversal experiments is that the refocusing of the reemitted signals at the location of the initial source is improved when the medium is heterogeneous. Contrary to intuition, multiple scattering enhances the spatial resolution of the refocused signal and allows one to beat the diffraction limit obtained in homogeneous media. This paper presents a quantitative explanation of time reversal and other more general refocusing phenomena for general classical waves in heterogeneous media. The theory is based on the asymptotic analysis of the Wigner transform of wave fields in the high frequency limit. Numerical experiments complement the theory.
- Published
- 2003
125. SELF-AVERAGING IN TIME REVERSAL FOR THE PARABOLIC WAVE EQUATION
- Author
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George Papanicolaou, Leonid Ryzhik, and Guillaume Bal
- Subjects
Self-averaging ,Wave packet ,010102 general mathematics ,Mathematical analysis ,FOS: Physical sciences ,Markov process ,Nonlinear Sciences - Chaotic Dynamics ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Parabolic wave equation ,Modeling and Simulation ,Phase space ,symbols ,Wigner distribution function ,Chaotic Dynamics (nlin.CD) ,0101 mathematics ,Convection–diffusion equation ,Martingale (probability theory) ,Mathematics - Abstract
We analyze the self-averaging properties of time-reversed solutions of the paraxial wave equation with random coefficients, which we take to be Markovian in the direction of propagation. This allows us to construct an approximate martingale for the phase space Wigner transform of two wave fields. Using a prioriL2-bounds available in the time-reversal setting, we prove that the Wigner transform in the high frequency limit converges in probability to its deterministic limit, which is the solution of a transport equation.
- Published
- 2002
126. Particle Transport through Scattering Regions with Clear Layers and Inclusions
- Author
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Guillaume Bal
- Subjects
Physics ,Coupling ,Numerical Analysis ,Photon ,Physics and Astronomy (miscellaneous) ,business.industry ,Scattering ,Applied Mathematics ,Space (mathematics) ,Heavy traffic approximation ,Particle transport ,Computer Science Applications ,Computational physics ,Computational Mathematics ,Optics ,Modeling and Simulation ,Diffusion (business) ,Spectroscopy ,business - Abstract
This paper introduces generalized diffusion models for the transport of particles in scattering media with nonscattering inclusions. Classical diffusion is known as a good approximation of transport only in scattering media. Based on asymptotic expansions and the coupling of transport and diffusion models, generalized diffusion equations with nonlocal interface conditions are proposed which offer a computationally cheap, yet accurate, alternative to solving the full phase-space transport equations. The paper shows which computational model should be used depending on the size and shape of the nonscattering inclusions in the simplified setting of two space dimensions. An important application is the treatment of clear layers in near-infrared (NIR) spectroscopy, an imaging technique based on the propagation of NIR photons in human tissues.
- Published
- 2002
127. Wave transport for a scalar model of the Love waves
- Author
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Leonid Ryzhik and Guillaume Bal
- Subjects
Physics ,Wave propagation ,Applied Mathematics ,General Physics and Astronomy ,Internal wave ,Ion acoustic wave ,Physics::Geophysics ,Computational Mathematics ,symbols.namesake ,Love wave ,Classical mechanics ,Modeling and Simulation ,symbols ,Astrophysics::Earth and Planetary Astrophysics ,Gravity wave ,Rayleigh wave ,Mechanical wave ,Longitudinal wave - Abstract
We study transport for a scalar model of Love waves. These waves arise in the propagation of seismic waves whose energy is concentrated in the vicinity of the earth surface. We derive radiative transfer equations from first principles for the angularly resolved energy density of the Love waves in a simplified acoustic model. We consider a rough top surface with weak fluctuations at the scale of the wavelength. The transport equation accounts for the multiple scattering of the Love waves and their scattering into volume waves. We also analyze a diffusive regime when energy is universally distributed over various modes of the Love waves.
- Published
- 2002
128. Radiative transport limit for the random Schrödinger equation
- Author
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George Papanicolaou, Leonid Ryzhik, and Guillaume Bal
- Subjects
Physics ,Random potential ,Applied Mathematics ,010102 general mathematics ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Nonlinear Sciences - Chaotic Dynamics ,01 natural sciences ,Radiative transport ,Schrödinger equation ,010101 applied mathematics ,symbols.namesake ,Phase space ,symbols ,Wigner distribution function ,0101 mathematics ,Martingale (probability theory) ,Mathematical Physics ,Mathematical physics - Abstract
We give a detailed mathematical analysis of the radiative transport limit for the average phase space density of solutions of the Schroedinger equation with time dependent random potential. Our derivation is based on the construction of an approximate martingale for the random Wigner distribution.
- Published
- 2002
129. Capillary–gravity wave transport over spatially random drift
- Author
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Tom Chou and Guillaume Bal
- Subjects
Physics ,Capillary wave ,Slowly varying envelope approximation ,Wave propagation ,Applied Mathematics ,General Physics and Astronomy ,Mechanics ,Computational Mathematics ,Wavelength ,Classical mechanics ,Surface wave ,Modeling and Simulation ,Wind wave ,Wave vector ,Envelope (waves) - Abstract
We derive transport equations for the propagation of water wave action in the presence of subsurface random flows. Using the Wigner distribution W(x, k ,t )to represent the envelope of the wave amplitude at position x, time t contained in high frequency waves with wave vector k/e (where e is a small parameter compared to a characteristic distance of propagation), we describe surface wave transport over flows consisting of two length scales; one varying slowly on the wavelength scale, the other varying on a scale comparable to the wavelength. Both static underlying flows and time-varying underlying flows are considered. The spatially rapidly varying but weak surface flows augment the characteristic equations with scattering terms that are explicit functions of the correlations of the random surface currents. These scattering terms depend parametrically on the magnitudes and directions of the smoothly varying drift and are shown to give rise to a Doppler-coupled scattering mechanism. Conservation of wave action (CWA), typically derived for drift varying over long distances, is extended to systems with flow that varies on small length scales of order the surface wavelength. Our results provide a formal set of equations to analyze transport of surface wave action, intensity, energy, and wave scattering as a function of the smoothly varying drifts and the correlation functions of the random, highly oscillating surface flows. © 2002 Elsevier Science B.V. All rights reserved.
- Published
- 2002
130. Coupling of transport and diffusion models in linear transport theory
- Author
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Guillaume Bal and Yvon Maday
- Subjects
Numerical Analysis ,Diffusion equation ,Applied Mathematics ,Domain decomposition methods ,Heavy traffic approximation ,Boltzmann equation ,Computational Mathematics ,Modeling and Simulation ,Phase space ,Linear transport theory ,Statistical physics ,Convection–diffusion equation ,Analysis ,Linear equation ,Mathematics - Abstract
This paper is concerned with the coupling of two models for the propagation of particles in scattering media. The first model is a linear transport equation of Boltzmann type posed in the phase space (position and velocity). It accurately describes the physics but is very expensive to solve. The second model is a diffusion equation posed in the physical space. It is only valid in areas of high scattering, weak absorption, and smooth physical coefficients, but its numerical solution is much cheaper than that of transport. We are interested in the case when the domain is diffusive everywhere except in some small areas, for instance non-scattering or oscillatory inclusions. We present a natural coupling of the two models that accounts for both the diffusive and non-diffusive regions. The interface separating the models is chosen so that the diffusive regime holds in its vicinity to avoid the calculation of boundary or interface layers. The coupled problem is analyzed theoretically and numerically. To simplify the presentation, the transport equation is written in the even parity form. Applications include, for instance, the treatment of clear or spatially inhomogeneous regions in near-infra-red spectroscopy, which is increasingly being used in medical imaging for monitoring certain properties of human tissues.
- Published
- 2002
131. Imaging of anisotropic conductivities from current densities in two dimensions
- Author
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François Monard, Chenxi Guo, and Guillaume Bal
- Subjects
Geometrical optics ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Open set ,02 engineering and technology ,01 natural sciences ,010101 applied mathematics ,Elliptic curve ,Mathematics - Analysis of PDEs ,Bounded function ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,Nabla symbol ,Tensor ,Boundary value problem ,0101 mathematics ,Anisotropy ,Analysis of PDEs (math.AP) ,Mathematics ,Mathematical physics - Abstract
We consider the imaging of anisotropic conductivity tensors $\gamma=(\gamma_{ij})_{1\leq i,j\leq 2}$ from knowledge of several internal current densities $\mathcal{J}=\gamma\nabla u$ where $u$ satisfies a second order elliptic equation $\nabla\cdot(\gamma\nabla u)=0$ on a bounded domain $X\subset\mathbb{R}^2$ with prescribed boundary conditions on $\partial X$. We show that $\gamma$ can be uniquely reconstructed from four {\em well-chosen} functionals $\mathcal{J}$ and that noise in the data is differentiated once during the reconstruction. The inversion procedure is local in the sense that (most of) the tensor $\gamma(x)$ can be reconstructed from knowledge of the functionals $\mathcal{J}$ in the vicinity of $x$. We obtain the existence of an open set of boundary conditions on $\partial X$ that guaranty stable reconstructions by using the technique of complex geometric optics (CGO) solutions. The explicit inversion procedure is presented in several numerical simulations, which demonstrate the influence of the choice boundary conditions on the stability of the reconstruction. This problem finds applications in the medical imaging modality called Current Density Imaging or Magnetic Resonance Electrical Impedance Tomography.
- Published
- 2014
132. Ultrasound-modulated bioluminescence tomography
- Author
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Guillaume Bal and John C. Schotland
- Subjects
Physics ,Diffusion equation ,business.industry ,Scattering ,Optical measurements ,Ultrasound ,FOS: Physical sciences ,Physics::Optics ,Iterative reconstruction ,Image Enhancement ,Molecular Imaging ,Inverse source problem ,Optics ,Image Interpretation, Computer-Assisted ,Luminescent Measurements ,Tomography, Optical ,Bioluminescence ,Tomography ,business ,Algorithms ,Ultrasonography ,Physics - Optics ,Optics (physics.optics) - Abstract
We propose a method to reconstruct the density of a luminescent source in a highly-scattering medium from ultrasound modulated optical measurements. Our approach is based on the solution to a hybrid inverse source problem for the diffusion equation., Comment: Supplementary information included
- Published
- 2014
133. Homogenization of Parabolic Equations with Large Time-dependent Random Potential
- Author
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Guillaume Bal and Yu Gu
- Subjects
Statistics and Probability ,Spatial variable ,Random potential ,Weak convergence ,Applied Mathematics ,Probability (math.PR) ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,Homogenization (chemistry) ,Parabolic partial differential equation ,010104 statistics & probability ,Mathematics - Analysis of PDEs ,Modeling and Simulation ,FOS: Mathematics ,0101 mathematics ,Randomness ,Brownian motion ,Mathematics - Probability ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
This paper concerns the homogenization problem of a parabolic equation with large, time-dependent, random potentials in high dimensions $d\geq 3$. Depending on the competition between temporal and spatial mixing of the randomness, the homogenization procedure turns to be different. We characterize the difference by proving the corresponding weak convergence of Brownian motion in random scenery. When the potential depends on the spatial variable macroscopically, we prove a convergence to SPDE., 23 pages, to appear in SPA
- Published
- 2014
134. Reconstruction of constitutive parameters in isotropic linear elasticity from noisy full-field measurements
- Author
-
Sébastien Imperiale, Guillaume Bal, Cédric Bellis, François Monard, Department of Applied Physics and Applied Mathematics [New York] (APAM), Columbia University [New York], Matériaux et Structures (M&S), Laboratoire de Mécanique et d'Acoustique [Marseille] (LMA ), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine (M3DISIM), Laboratoire de mécanique des solides (LMS), École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Department of Mathematics [Seattle], University of Washington [Seattle], École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Mines Paris - PSL (École nationale supérieure des mines de Paris)
- Subjects
Quantitative parameter identification ,Applied Mathematics ,Mathematical analysis ,Linear elasticity ,Isotropy ,Stability (probability) ,Regularization (mathematics) ,Computer Science Applications ,Theoretical Computer Science ,Mathematics - Analysis of PDEs ,Signal Processing ,Isotropic solid ,FOS: Mathematics ,Numerical differentiation ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Direct integration of a beam ,Uniqueness ,Internal data ,Elastography ,Lamé system ,Mathematical Physics ,Analysis of PDEs (math.AP) ,Mathematics ,Regularization method - Abstract
Within the framework of linear elasticity we assume the availability of internal full-field measurements of the continuum deformations of a non-homogeneous isotropic solid. The aim is the quantitative reconstruction of the associated moduli. A simple gradient system for the sought constitutive parameters is derived algebraically from the momentum equation, whose coefficients are expressed in terms of the measured displacement fields and their spatial derivatives. Direct integration of this system is discussed to finally demonstrate the inexpediency of such an approach when dealing with noisy data. Upon using polluted measurements, an alternative variational formulation is deployed to invert for the physical parameters. Analysis of this latter inversion procedure provides existence and uniqueness results while the reconstruction stability with respect to the measurements is investigated. As the inversion procedure requires differentiating the measurements twice, a numerical differentiation scheme based on an ad hoc regularization then allows an optimally stable reconstruction of the sought moduli. Numerical results are included to illustrate and assess the performance of the overall approach., Comment: 23 pages, 7 figures
- Published
- 2014
135. Fourier analysis of diamond discretization¶in particle transport
- Author
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Guillaume Bal
- Subjects
Free particle ,Algebra and Number Theory ,Discretization ,Numerical analysis ,Mathematical analysis ,Scalar (mathematics) ,Computational Mathematics ,symbols.namesake ,Fourier analysis ,symbols ,Dissipative system ,Boundary value problem ,Hyperbolic partial differential equation ,Mathematics - Abstract
This paper analyzes, theoretically and numerically, the diamond discretization (DD) of free particle transport equations. DD is one of the most commonly used discretizations used in particle transport theory, with applications ranging from the propagation of neutrons in nuclear physics to that of near-infra-red photons in medical imaging. Based on the theory of scalar hyperbolic equations in the Fourier domain, we show that DD performs very well when the boundary data are smooth and the absorption coefficient small. However, because high frequency modes are not damped at all by DD even in the dissipative case, several unphysical modes appear when the boundary conditions are not so smooth. These spurious oscillations are analyzed carefully and displayed numerically.
- Published
- 2001
136. Theoretical and numerical analysis of polarization for time-dependent radiative transfer equations
- Author
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Guillaume Bal and Miguel Moscoso
- Subjects
Physics ,Radiation ,Numerical analysis ,Monte Carlo method ,Monte Carlo method for photon transport ,Atomic and Molecular Physics, and Optics ,Computational physics ,symbols.namesake ,Classical mechanics ,Atmospheric radiative transfer codes ,Photon transport in biological tissue ,Radiative transfer ,symbols ,Stokes parameters ,Convection–diffusion equation ,Spectroscopy - Abstract
We consider the matrix-valued radiative transfer equations for the Stokes parameters for the propagation of light through turbulent atmospheres. A Monte Carlo method is introduced to solve the time dependent matrix-valued radiative transfer equations in 3D geometry. The Monte Carlo method is based on a probabilistic representation of the radiative transfer equations involving an augmented scalar transport equation where the polarization parameters are independent variables. The linear moments of the augmented transport equation with respect to the polarization parameters solve the matrix-valued radiative transfer equations. We show how polarization and depolarization effects develop in time for isotropic and unpolarized point sources, considered for concreteness in spherical and half-space geometries. We analyze in detail the creation of polarization by single- and multiple-scattering effects.
- Published
- 2001
137. DIFFUSION APPROXIMATION OF RADIATIVE TRANSFER EQUATIONS IN A CHANNEL
- Author
-
Guillaume Bal
- Subjects
Physics ,Diffusion equation ,Applied Mathematics ,General Physics and Astronomy ,Transportation ,Statistical and Nonlinear Physics ,Mechanics ,Heavy traffic approximation ,Seismic wave ,Physics::Geophysics ,Classical mechanics ,Photon transport in biological tissue ,Radiative transfer ,Wavenumber ,Diffusion (business) ,Photon diffusion ,Mathematical Physics - Abstract
We address the propagation of elastic waves generated by an earthquake in the earth crust modeled by a channel separated from the atmosphere and the mantel by two horizontal interfaces. Geophysical studies have shown the validity of radiative transfer in this frequency regime to describe the phase space energy density of seismic waves. For long times and large distances, radiative transfer in weakly absorbing media can be approximated by a diffusion equation. However, the thickness of the crust is of the order of the transport mean free path, the average distance it takes for waves with a wavenumber v to be scattered into another wavenumber v′ by interaction with the inhomogeneous underlying medium. Hence there cannot be diffusion in the vertical direction. This paper shows that diffusion is still valid in the following sense. The radiative transfer solution factors asymptotically in the limit of vanishing mean free paths as the product of a two-dimensional diffusion term in the horizontal directions and ...
- Published
- 2001
138. Homogenization of a spectral equation with drift in linear transport
- Author
-
Guillaume Bal
- Subjects
Computational Mathematics ,Control and Optimization ,Diffusion equation ,Factorization ,Control and Systems Engineering ,Homogeneous ,Mathematical analysis ,Linear transport theory ,Neutron ,Particle density ,Homogenization (chemistry) ,Eigenvalues and eigenvectors ,Mathematics - Abstract
This paper deals with the homogenization of a spectral equation posed in a periodic domain in linear transport theory. The particle density at equilibrium is given by the unique normalized positive eigenvector of this spectral equation. The corresponding eigenvalue indicates the amount of particle creation necessary to reach this equilibrium. When the physical parameters satisfy some symmetry conditions, it is known that the eigenvectors of this equation can be approximated by the product of two term. The first one solves a local transport spectral equation posed in the periodicity cell and the second one a homogeneous spectral diffusion equation posed in the entire domain. This paper addresses the case where these symmetry conditions are not fulfilled. We show that the factorization remains valid with the diffusion equation replaced by a convection-diffusion equation with large drift. The asymptotic limit of the leading eigenvalue is also modified. The spectral equation treated in this paper can model the stability of nuclear reactor cores and describe the distribution of neutrons at equilibrium. The same techniques can also be applied to the time-dependent linear transport equation with drift, which appears in radiative transfer theory and which models the propagation of acoustic, electromagnetic, and elastic waves in heterogeneous media.
- Published
- 2001
139. A Posteriori Learning for Quasi‐Geostrophic Turbulence Parametrization
- Author
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Hugo Frezat, Julien Le Sommer, Ronan Fablet, Guillaume Balarac, and Redouane Lguensat
- Subjects
parametrization ,machine learning ,turbulence ,quasi‐geostrophic ,Physical geography ,GB3-5030 ,Oceanography ,GC1-1581 - Abstract
Abstract The use of machine learning to build subgrid parametrizations for climate models is receiving growing attention. State‐of‐the‐art strategies address the problem as a supervised learning task and optimize algorithms that predict subgrid fluxes based on information from coarse resolution models. In practice, training data are generated from higher resolution numerical simulations transformed in order to mimic coarse resolution simulations. By essence, these strategies optimize subgrid parametrizations to meet so‐called a priori criteria. But the actual purpose of a subgrid parametrization is to obtain good performance in terms of a posteriori metrics which imply computing entire model trajectories. In this paper, we focus on the representation of energy backscatter in two‐dimensional quasi‐geostrophic turbulence and compare parametrizations obtained with different learning strategies at fixed computational complexity. We show that strategies based on a priori criteria yield parametrizations that tend to be unstable in direct simulations and describe how subgrid parametrizations can alternatively be trained end‐to‐end in order to meet a posteriori criteria. We illustrate that end‐to‐end learning strategies yield parametrizations that outperform known empirical and data‐driven schemes in terms of performance, stability, and ability to apply to different flow configurations. These results support the relevance of differentiable programming paradigms for climate models in the future.
- Published
- 2022
- Full Text
- View/download PDF
140. SPATIALLY VARYING DISCRETE ORDINATES METHODS IN XY-GEOMETRY
- Author
-
Guillaume Bal
- Subjects
Coupling ,symbols.namesake ,Linear transport equation ,Ordinate ,Discrete Ordinates Method ,Discretization ,Applied Mathematics ,Modeling and Simulation ,Mathematical analysis ,symbols ,Gaussian quadrature ,Mathematics - Abstract
We consider the coupling of angular discretizations of the two-dimensional linear transport equation. We show the well-posedness of the coupled problem and give an error estimate. The angular discretization is based on the discrete ordinates method. It involves a quadrature rule specifically designed for our coupling. The theory uses the integral formulation of transport and will be demonstrated on simplified geometries.
- Published
- 2000
141. Wave transport along surfaces with random impedance
- Author
-
Leonid Ryzhik, Guillaume Bal, George Papanicolaou, and Valentin Freilikher
- Subjects
Physics ,Random surface ,Surface wave ,Mean free path ,Scattering ,Quantum mechanics ,Spectral density ,Heavy traffic approximation ,Electrical impedance ,Leakage (electronics) ,Computational physics - Abstract
We study transport and diffusion of classical waves in two-dimensional disordered systems and in particular surface waves on a flat surface with randomly fluctuating impedance. We derive from first principles a radiative transport equation for the angularly resolved energy density of the surface waves. This equation accounts for multiple scattering of surface waves as well as for their decay because of leakage into volume waves. We analyze the dependence of the scattering mean free path and of the decay rate on the power spectrum of fluctuations. We also consider the diffusion approximation of the surface radiative transport equation and calculate the angular distribution of the energy transmitted by a strip of random surface impedance.
- Published
- 2000
142. Polarization effects of seismic waves on the basis of radiative transport theory
- Author
-
Guillaume Bal and Miguel Moscoso
- Subjects
Physics ,Scattering ,Monte Carlo method ,Polarization (waves) ,Seismic wave ,Physics::Geophysics ,Computational physics ,Geophysics ,Classical mechanics ,Geochemistry and Petrology ,Lithosphere ,Radiative transfer ,Scattering theory ,Seismogram - Abstract
Summary Radiative transfer theory provides a good framework for the study of multiple scattering in the randomly inhomogeneous lithosphere. Envelopes of high-frequency seismograms (mainly S coda waves) of local earthquakes have been synthesized on the basis of this theory, and inversions for some Earth parameters such as intrinsic attenuation, scattering attenuation and degree of non-isotropic scattering have been carried out. However, a scalar model has often been assumed because of its mathematical relative simplicity. The simplification amounts to neglecting the polarized nature of the underlying motion. This approach is only valid for long lapse times when S waves become unpolarized because of high-order scattering, and cannot be justified by only assuming that the source is unpolarized. We show that incoming unpolarized S waves can be up to 80 per cent polarized after single scattering. Depolarization of S waves after multiple scattering is studied by a Monte Carlo method. We show that the scattering of S waves off different kinds of inhomogeneities gives rise to different polarization and depolarization patterns. Consequently, polarization should provide valuable information for the understanding of the physics of wave motion and the properties of the Earth’s lithosphere.
- Published
- 2000
143. Inverse problems for homogeneous transport equations: II. The multidimensional case
- Author
-
Guillaume Bal
- Subjects
Applied Mathematics ,Signal Processing ,Mathematical Physics ,Computer Science Applications ,Theoretical Computer Science - Published
- 2000
144. Inverse problems for homogeneous transport equations: I. The one-dimensional case
- Author
-
Guillaume Bal
- Subjects
Series (mathematics) ,Applied Mathematics ,Mathematical analysis ,Degenerate energy levels ,Boundary (topology) ,Inverse problem ,Computer Science Applications ,Theoretical Computer Science ,law.invention ,Invertible matrix ,law ,Bounded function ,Signal Processing ,Inverse scattering problem ,Linear independence ,Mathematical Physics ,Mathematics - Abstract
A companion paper by Bal (Bal G 2000 Inverse Problems 16 997) and this paper are parts I and II of a series dealing with the reconstruction from boundary measurements of the scattering operator of homogeneous linear transport equations. This part II deals with the case of convex bounded domains in dimensions higher than one. We distinguish the analysis of smooth boundaries from that of boundaries with discontinuities such as corners. We propose a reconstruction in the case of degenerate symmetric scattering operators and show the well-posedness of the inverse problem. The proof of well-posedness is based on a decomposition of angular moments of the transport solution into unbounded and bounded components. This decomposition allows us to show the linear independence of a sufficiently large number of angular moments of the transport solution that are used to construct an invertible system for the scattering coefficients to be reconstructed.
- Published
- 2000
145. Probabilistic Theory of Transport Processes with Polarization
- Author
-
Leonid Ryzhik, George Papanicolaou, and Guillaume Bal
- Subjects
Physics ,Variables ,Turbulence ,Applied Mathematics ,media_common.quotation_subject ,Monte Carlo method ,Probabilistic logic ,Polarization (waves) ,symbols.namesake ,Photon transport in biological tissue ,symbols ,Stokes parameters ,Statistical physics ,Convection–diffusion equation ,media_common - Abstract
We derive a probabilistic representation for solutions of matrix-valued transport equations that account for polarization effects. Such equations arise in radiative transport for the Stokes parameters that model the propagation of light through turbulent atmospheres. They also arise in radiative transport for seismic wave propagation in the earth's crust. The probabilistic representation involves an augmented scalar transport equation in which the polarization parameters become independent variables. Our main result is that the linear moments of the augmented transport equation with respect to the polarization variables are the solution of the matrix-valued transport equation. The augmented scalar transport equation is well suited to analyzing the hydrodynamic regime of small mean free paths. It is also well suited to getting approximate solutions by Monte Carlo simulation.
- Published
- 2000
146. Diffusion Approximation of Radiative Transfer Problems with Interfaces
- Author
-
Leonid Ryzhik and Guillaume Bal
- Subjects
Physics ,Mathematical optimization ,Interface (Java) ,Applied Mathematics ,Radiative transfer ,Energy density ,Acoustic wave ,Mechanics ,Diffusion (business) ,Classification of discontinuities ,Heavy traffic approximation ,Term (time) - Abstract
We derive the diffusion approximation of transport equations with discontinuities at interfaces. The transport equations model the energy density of acoustic waves. The waves are reflected and transmitted at the interface between different media,which leads to discontinuities of the energy density across the interface. The diffusion approximation, which is valid inside each region is not correct at the vicinity of the interface. However, using interface layer analysis, we prove that the transport solution can be approximated by a diffusion term plus an interface layer which decays exponentially fast. We derive systematically the correct form of the interface conditions for this diffusion term.
- Published
- 2000
147. Transport theory for acoustic waves with reflection and transmission at interfaces
- Author
-
George Papanicolaou, Joseph B. Keller, Leonid Ryzhik, and Guillaume Bal
- Subjects
Applied Mathematics ,Mathematical analysis ,General Physics and Astronomy ,Acoustic wave ,Computational Mathematics ,Classical mechanics ,Singularity ,Transmission (telecommunications) ,Modeling and Simulation ,Reflection (physics) ,Wigner distribution function ,Boundary value problem ,Born approximation ,Smoothing ,Mathematics - Abstract
Transport theoretic boundary conditions are derived for acoustic wave reflection and transmission at a rough interface with small random fluctuations. The Wigner distribution is used to go from waves to energy transport in the high frequency limit, and the Born expansion is used to calculate the effect of the random rough surface. The smoothing method is also used to remove the grazing angle singularity due to the Born approximation. The results are presented in a form that is convenient both for theoretical analysis and for numerical computations.
- Published
- 1999
148. Diffusive energy scattering from weakly random surfaces
- Author
-
George Papanicolaou, Leonid Ryzhik, and Guillaume Bal
- Subjects
Physics ,Physical acoustics ,Statistical and Nonlinear Physics ,Acoustic wave ,Ion acoustic wave ,Computational physics ,symbols.namesake ,Classical mechanics ,Reflection (physics) ,symbols ,Acoustic wave equation ,Rayleigh wave ,Mechanical wave ,Mathematical Physics ,Longitudinal wave - Abstract
We derive transport theoretic boundary conditions for acoustic wave reflection at a weakly rough boundary in an inhomogeneous half space. We use the Wigner distribution to go from waves to energy transport in the high frequency limit. We generalize known results on the reflection of acoustic plane waves in a homogeneous medium. We analyze higher order corrections, which include a enhanced backscattering effect in the back direction.
- Published
- 1999
149. Homogenization of the criticality spectral equation in neutron transport
- Author
-
Grégoire Allaire and Guillaume Bal
- Subjects
Numerical Analysis ,Neutron transport ,Diffusion equation ,Applied Mathematics ,Mathematical analysis ,Nuclear reactor ,Homogenization (chemistry) ,law.invention ,Computational Mathematics ,Criticality ,law ,Neutron flux ,Modeling and Simulation ,Convection–diffusion equation ,Analysis ,Eigenvalues and eigenvectors ,Mathematics - Abstract
We address the homogenization of an eigenvalue problem for the neutron transport equation in a periodic heterogeneous domain, modeling the criticality study of nuclear reactor cores. We prove that the neutron flux, corresponding to the first and unique positive eigenvector, can be factorized in the product of two terms, up to a remainder which goes strongly to zero with the period. One term is the first eigenvector of the transport equation in the periodicity cell. The other term is the first eigenvector of a diffusion equation in the homogenized domain. Furthermore, the corresponding eigenvalue gives a second order corrector for the eigenvalue of the heterogeneous transport problem. This result justifies and improves the engineering procedure used in practice for nuclear reactor cores computations.
- Published
- 1999
150. First-Order Corrector for the Homogenization of the Criticality Eigenvalue Problem in the Even Parity Formulation of the Neutron Transport
- Author
-
Guillaume Bal
- Subjects
Neutron transport ,Diffusion equation ,Applied Mathematics ,Mathematical analysis ,Homogenization (chemistry) ,Computational Mathematics ,Boundary layer ,symbols.namesake ,Dirichlet boundary condition ,symbols ,Periodic boundary conditions ,Neutron ,Analysis ,Eigenvalues and eigenvectors ,Mathematics - Abstract
We consider the homogenization of the criticality eigenvalue problem for the even parity flux of neutron transport in a domain with isotropic and periodically oscillating coefficients. We prove that the neutron density is factored in the product of two terms. The first one describes local behavior of the density at the cell level. It is a solution of a heterogeneous transport problem with periodic boundary conditions. The second term gives global behavior on the whole domain. It satisfies a homogeneous diffusion equation posed on the whole domain with Dirichlet boundary conditions. We also give the asymptotic analysis of the corresponding eigenvalues. This expansion gives rise to errors of the order of the cell size. It does not account for neutron leakage at the boundary of the core and yields unacceptable errors in practice. We derive a more accurate expansion of the eigenelements in the case of a symmetric and cubic domain. The analysis of a boundary layer allows us to derive modified boundary conditio...
- Published
- 1999
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