Time-dependent Linear Transport Research Articles

Capturing the diffusive behavior of the multiscale linear transport equations by Asymptotic-Preserving Convolutional DeepONets

In this paper, we introduce two types of novel Asymptotic-Preserving Convolutional Deep Operator Networks (APCONs) designed to solve the multiscale time-dependent linear transport equations. We observe that the vanilla physics-informed DeepONets with modified MLP may exhibit instability in maintaining the desired limiting macroscopic behavior. Therefore, this necessitates the utilization of an asymptotic-preserving loss function. Drawing inspiration from the heat kernel in the diffusion equation, we propose a new architecture called Convolutional Deep Operator Networks, which employs multiple local convolution operations instead of a global heat kernel, along with pooling and activation operations in each filter layer. Our APCON methods possess a parameter count that is independent of the grid size and are capable of capturing the diffusive behavior of the linear transport problem. Finally, we validate the effectiveness of our methods through several numerical examples.

Asymptotic-Preserving Neural Networks for Multiscale Time-Dependent Linear Transport Equations

In this paper we develop a neural network for the numerical simulation of time-dependent linear transport equations with diffusive scaling and uncertainties. The goal of the network is to resolve the computational challenges of curse-of-dimensionality and multiple scales of the problem. We first show that a standard Physics-Informed Neural Network (PINN) fails to capture the multiscale nature of the problem, hence justifies the need to use Asymptotic-Preserving Neural Networks (APNNs). We show that not all classical AP formulations are directly fit for the neural network approach. We construct a micro-macro decomposition based neural network, and also build in a mass conservation mechanism into the loss function, in order to capture the dynamic and multiscale nature of the solutions. Numerical examples are used to demonstrate the effectiveness of this APNNs.

Sparsity and level set regularization for diffuse optical tomography using a transport model in 2D

In this paper we address an inverse problem for the time-dependent linear transport equation (or radiative transfer equation) in 2D having in mind applications in diffuse optical tomography (DOT). We propose two new reconstruction algorithms which so far have not been applied to such a situation and compare their performances in certain practically relevant situations. The first of these reconstruction algorithms uses a sparsity promoting regularization scheme, whereas the second one uses a simultaneous level set reconstruction scheme for two parameters of the linear transport equation. We will also compare the results of both schemes with a third scheme which is a more traditional L2-based Landweber–Kaczmarz scheme. We focus our attention on the DOT application of imaging the human head of a neonate where the simpler diffusion approximation is not well-suited for the inversion due to the presence of a clear layer beneath the skull which is filled with ‘low-scattering’ cerebrospinal fluid. This layer, even if its location and characteristics are known a priori, poses significant difficulties for most reconstruction schemes due to its ‘wave-guiding’ property which reduces sensitivity of the data to the interior regions. A further complication arises due to the necessity to reconstruct simultaneously two different parameters of the linear transport equation, the scattering and the absorption cross-section, from the same data set. A significant ‘cross-talk’ between these two parameters is usually expected. Our numerical experiments indicate that each of the three considered reconstruction schemes do have their merits and perform differently but reasonably well when the clear layer is a priori known. We also demonstrate the behavior of the three algorithms in the particular situation where the clear layer is unknown during the reconstruction.

Derivation of a Physically Based Hybrid Technique for the Solution of Source-Driven Time-Dependent Linear Boltzmann Equations

The article presents a novel kinetic method for the solution of time-dependent source-driven linear transport problems. As will be discussed, the method is based on a multigeneration representation of linear Boltzmann equations and is derived from physically grounded considerations. The technique is said to be hybrid since it combines together collisionless transport solutions and factorization techniques. The methodology is tested on systems with different scattering properties and shows very promising performance both in terms of accuracy and computational times.

Stability for Time-Dependent Inverse Transport

This paper concerns the reconstruction of the absorption and scattering parameters in a time-dependent linear transport equation from full knowledge of the albedo operator at the boundary of a bounded domain of interest. We present optimal stability results on the reconstruction of the absorption and scattering parameters for a given error in the measured albedo operator.

Time-dependent angularly averaged inverse transport

This paper concerns the reconstruction of the absorption and scattering parameters in a time-dependent linear transport equation from knowledge of angularly averaged measurements performed at the boundary of a domain of interest. Such measurement settings find applications in medical and geophysical imaging. We show that the absorption coefficient and the spatial component of the scattering coefficient are uniquely determined by such measurements. We obtain stability results on the reconstruction of the absorption and scattering parameters with respect to the measured albedo operator. The stability results are obtained by a precise decomposition of the measurements into components with different singular behavior in the time domain.

The Asymptotic Behavior of the Solution to a Cauchy Problem Governed by a Transport Operator

The purpose of this paper is to investigate some spectral properties of a time-dependent linear transport equation with diffusive boundary conditions arising in growing cell populations. After deriving the explicit expression of the strongly continuous semigroup {U K (t ): t ≥ 0} generated by the streaming operator, we establish the strict singularity of the operator BU K (s)B, s> 0, for a wide class of collision operators B. Making use of the Weis theorem, Theorem 4.1, this enables us to estimate the essential type of the transport semigroup from which the asymptotic behavior of the solution is derived.

Spectral properties and time asymptotic behaviour of linear transport equations in slab geometry

AbstractThe goal of this paper is to analyse some spectral properties of time‐dependent monoenergetic linear transport equation with perfect reflecting boundary conditions.It is proved that the streaming operator generates a strongly continuous group and its explicit expression is derived. This enables us to investigate in details the compactness properties of the second‐order remainder term of the Dyson–Phillips expansion for a large class of collision operators. The validity of the obtained results for perfect periodic boundary conditions is also discussed. Copyright © 2001 John Wiley & Sons, Ltd.

Homogenization of a spectral equation with drift in linear transport

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.

Scattering and absorption transport sensitivity functions for optical tomography

Optical tomography is modelled as an inverse problem for the time-dependent linear transport equation. We decompose the linearized residual operator of the problem into absorption and scattering transport sensitivity functions. We show that the adjoint linearized residual operator has a similar physical meaning in optical tomography as the 'backprojection' operator in x-ray tomography. In this interpretation, the geometric patterns onto which the residuals are backprojected are given by the same absorption and scattering transport sensitivity functions which decompose the forward residual operator. Moreover, the 'backtransport' procedure, which has been introduced in an earlier paper by the author, can then be interpreted as an efficient scheme for 'backprojecting' all (filtered) residuals corresponding to one source position simultaneously into the parameter space by just solving one adjoint transport problem. Numerical examples of absorption and scattering transport sensitivity functions for various situations (including applications with voids) are presented.

Convergence in transport theory

In this work we prove the convergence of the LTS N method for the steady-state and time-dependent one-dimensional linear transport equation employing the C 0 semigroup theory and discrete schemes approach.

A transport-backtransport method for optical tomography

Optical tomography is modelled by the inverse problem of the time-dependent linear transport equation in n spatial dimensions (n = 2,3). Based on the measurements which consist of some functionals of the outgoing density at the boundary for different sources , , two coefficients of the equation, the absorption coefficient and the scattering coefficient b(x), are reconstructed simultaneously inside . Starting out from some initial guess for these coefficients, the transport-backtransport (TBT) algorithm calculates the difference between the computed and the physically given measurements for a fixed source by solving a `direct' transport problem, and then transports these residuals back into the medium by solving a corresponding adjoint transport problem. The correction to the guess is calculated from the densities of the direct and the adjoint problem inside the medium. Doing this for all source positions , , one after the other yields one sweep of the algorithm. Numerical experiments are presented for the case when n = 2. They show that the TBT-method is able to reconstruct and to distinguish between scattering and absorbing objects in the case of large mean free path (which corresponds to x-ray tomography with scattering). In the case of very small mean free path (which corresponds to optical tomography), scattering and absorbing objects are located during the early sweeps, but phantoms are built up in the reconstructed scattering coefficient at positions where an absorber is situated and vice versa.

A Comparision of Time Marching Methods Coupled with a Higher-Order TVD Scheme for Time-Dependent Advection Equation.

Time marching methods coupled with a third order upwind TVD scheme are compared from the viewpoints of accuracy, memory requirement and computational cost. The methods examined are the second and the third order Runge-Kutta presented by Shu and Osher, the explicit Euler, the second order Adams-Bashforth, the implicit Euler and the Crank-Nicolson methods. Benchmark problems on time-dependent linear advective transport are solved using these methods. As a result, the Shu and Osher's Runge-Kutta methods of the second and the third order are recommended. The explicit Euler method suffers from oversteepening of the gradient due to numerical diffusion with a negative coefficient. The Adams-Bashforth method is inferior to the Runge-Kutta method in accuracy and memory requirement. The Crank-Nicolson method produces solutions as accurate as the Runge-Kutta method, but requires much CPU time due to the nonlinear feature of the TVD flux limiter.

Linear transport equation with specular reflection boundary condition

In this paper, the time dependent linear transport equation with anisotropic, energy dependent, non-uniform bounded convex media and specular reflection boundary condition is considered. The existence, uniqueness and stability of its solution for initial value are proved. The spectral properties of the time independent linear transport operator are studied.

Time dependent linear transport via green's functions

Abstract The direct problem for time-dependent energy transport in a two dimensional stratified planar slab is considered. An obliquely incident planar field penetrates the slab at one boundary, and the internal energy intensity distribution is determined. The intensity is represented by the Green's function, so that the problem may be solved in terms of a canonical solution. The representation is simple, and leads to a straightforward numerical algorithm. This paper is the first step in the development of a more comprehensive theory of time-dependent transport via the Green's function representation.

A Fokker-Planck equation for growing cell populations.

Closed form solutions are obtained for a Fokker-Planck model for cell growth as a function of maturation velocity and degree of maturation. For reproduction rules where daughter cells inherit their parent's maturation velocity the complete solution is derived in terms of Airy functions. For more complicated reproduction rules partial results are obtained. Emphasis is given to the relationship of these problems to time dependent linear transport theory.

Abstract time-dependent transport equations

We consider the abstract time-dependent linear transport equation as an initial-boundary value evolution problem in the Banach spaces L p , 1 ⩽ p < ∞, or on a space of measures on a (possibly time-dependent) kinetic phase space. Existence, uniqueness, dissipativity, and positivity results are proved for very general, possibly time-dependent, transport operators and boundary conditions. When the phase space, boundary conditions, and transport operator are independent of time, corresponding results are obtained for the associated semigroup.

On the spectral decomposition of the transport operator with anisotropic scattering and periodic boundary conditions

Abstract The time-dependent linear monoenergetic transport equation with linearly anisotropic scattering is considered in slab geometry With periodic boundary conditions. Adapting a result of Friedrichs, it is shown explicitly that the spectral theorem holds for the corresponding operator, for all but a set of zero measure in the space of parameters.

Positivity in time dependent linear transport theory

positivity in time dependent linear transport theory

Time dependent linear transport Ii∗ properties of weak solutions

Abstract In this paper properties of weak solutions of the time-dependent linear transport equation posed in a multidimensional rectangular parallelepiped with partially reflecting walls are established. Nonnegative constituent functions are shown to guarantee that a weak solution is nonnegative. This implies a monotonicity principle which is established. Sufficient continuity properties of the constituent functions are shown to guarantee that a weak solution will be uniformly continuous in each of its variables independently of the others on compact subsets of the interior of its domain of definition. Continuity with respect to time follows from continuity with respect to space; but continuity with respect to velocity seems to require continuity with respect to space as a precondition.