Inverse Transport Problem Research Articles

Inverse Rendering for Tomographic Volumetric Additive Manufacturing

Tomographic Volumetric Additive Manufacturing (TVAM) is an emerging 3D printing technology that can create complex objects in under a minute. The key idea is to project intense light patterns onto a rotating vial of photo-sensitive resin, causing polymerization where the cumulative dose of these patterns reaches the polymerization threshold. We formulate the pattern calculation as an inverse light transport problem and solve it via physically based differentiable rendering. In doing so, we address longstanding limitations of prior work by accurately modeling and correcting for scattering in composite resins, printing in non-symmetric vials, and supporting unusual printing geometries. We also introduce an improved discretization scheme that exploits the ray tracing operation to mitigate resolution-related artifacts in prints. We demonstrate the benefits of our method in real-world experiments, where our computed patterns produce prints with an improved fidelity.

Ensemble physics informed neural networks: A framework to improve inverse transport modeling in heterogeneous domains

Modeling fluid flow and transport in heterogeneous systems is often challenged by unknown parameters that vary in space. In inverse modeling, measurement data are used to estimate these parameters. Due to the spatial variability of these unknown parameters in heterogeneous systems (e.g., permeability or diffusivity), the inverse problem is ill-posed and infinite solutions are possible. Physics-informed neural networks (PINN) have become a popular approach for solving inverse problems. However, in inverse problems in heterogeneous systems, PINN can be sensitive to hyperparameters and can produce unrealistic patterns. Motivated by the concept of ensemble learning and variance reduction in machine learning, we propose an ensemble PINN (ePINN) approach where an ensemble of parallel neural networks is used and each sub-network is initialized with a meaningful pattern of the unknown parameter. Subsequently, these parallel networks provide a basis that is fed into a main neural network that is trained using PINN. It is shown that an appropriately selected set of patterns can guide PINN in producing more realistic results that are relevant to the problem of interest. To assess the accuracy of this approach, inverse transport problems involving unknown heat conductivity, porous media permeability, and velocity vector fields were studied. The proposed ePINN approach was shown to increase the accuracy in inverse problems and mitigate the challenges associated with non-uniqueness.

Contaminant source identification in groundwater by means of artificial neural network

In a desired environmental protection system, groundwater may not be excluded. In addition to the problem of over-exploitation, in total disagreement with the concept of sustainable development, another not negligible issue concerns the groundwater contamination. Mainly, this aspect is due to intensive agricultural activities or industrialized areas. In literature, several papers have dealt with transport problem, especially for inverse problems in which the release history or the source location are identified. The innovative aim of the paper is to develop a data-driven model that is able to analyze multiple scenarios, even strongly non-linear, in order to solve forward and inverse transport problems, preserving the reliability of the results and reducing the uncertainty. Furthermore, this tool has the characteristic of providing extremely fast responses, essential to identify remediation strategies immediately. The advantages produced by the model were compared with literature studies. In this regard, a feedforward artificial neural network, which has been trained to handle different cases, represents the data-driven model. Firstly, to identify the concentration of the pollutant at specific observation points in the study area (forward problem); secondly, to deal with inverse problems identifying the release history at known source location; then, in case of one contaminant source, identifying the release history and, at the same time, the location of the source in a specific sub-domain of the investigated area. At last, the observation error is investigated and estimated. The results are satisfactorily achieved, highlighting the capability of the ANN to deal with multiple scenarios by approximating nonlinear functions without the physical point of view that describes the phenomenon, providing reliable results, with very low computational burden and uncertainty.

Parallel inverse-problem solver for time-domain optical tomography with perfect parallel scaling

This paper presents an efficient parallel radiative transfer-based inverse-problem solver for time-domain optical tomography. The radiative transfer equation provides a physically accurate model for the transport of photons in biological tissue, but the high computational cost associated with its solution has hindered its use in time-domain optical-tomography and other areas. In this paper this problem is tackled by means of a number of computational and modeling innovations, including (1) A spatial parallel-decomposition strategy with perfect parallel scaling for the forward and inverse problems of optical tomography on parallel computer systems; and, (2) A Multiple Staggered Source method (MSS) that solves the inverse transport problem at a computational cost that is independent of the number of sources employed, and which significantly accelerates the reconstruction of the optical parameters: a six-fold MSS acceleration factor is demonstrated in this paper. Finally, this contribution presents (3) An intuitive derivation of the adjoint-based formulation for evaluation of functional gradients, including the highly-relevant general Fresnel boundary conditions—thus, in particular, generalizing results previously available for vacuum boundary conditions. Solutions of large and realistic 2D inverse problems are presented in this paper, which were produced on a 256-core computer system. The combined parallel/MSS acceleration approach reduced the required computing times by several orders of magnitude, from months to a few hours.

Inverse Transport and Diffusion Problems in Photoacoustic Imaging with Nonlinear Absorption

Motivated by applications in imaging nonlinear optical absorption by photoacoustic tomography, we study in this work inverse coefficient problems for a semilinear radiative transport equation and its diffusion approximation with internal data that are functionals of the coefficients and the solutions to the equations. Based on the techniques of first- and second-order linearization, we derive uniqueness and stability results for the inverse problems. For uncertainty quantification purposes, we also establish the stability of the reconstruction of the absorption coefficients with respect to the change in the scattering coefficient.

Application of Markov Chain Monte Carlo Methods for Uncertainty Quantification in Inverse Transport Problems

Determination of the components of a radioactive source/shield system using the system's radiation signature is of great importance in homeland security, material safeguards, and waste management. Although significant progress has been made toward solving this inverse transport problem in recent years, work remains to be done to quantify the uncertainty in reconstructed results. In this article we apply two Markov chain Monte Carlo (MCMC) approaches, the delayed rejection adaptive metropolis (DRAM) and differential evolution adaptive metropolis (DREAM) methods, to solve inverse problems and quantify uncertainty. The DRAM method uses delayed rejection combined with global adaptation of the proposal covariance matrix. The DREAM method hybridizes MCMC sampling with the differential evolution (DE) algorithm. In numerical test cases, the DRAM and DREAM methods are shown to be superior to a first-order inverse Hessian approach for problems with noisy data and multiple unknown quantities, with DREAM converging to the posterior distribution more quickly than DRAM. The DREAM and DRAM results indicate that a full posterior distribution is required to quantify uncertainty in many inverse transport problems.

Unique Determination of Absorption Coefficients in a Semilinear Transport Equation

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 4 August 2020Accepted: 08 June 2021Published online: 16 September 2021Keywordssemilinear radiative transport equation, inverse transport problem, uniqueness and stability, quantitative photoacoustic imaging, two-photon absorptionAMS Subject Headings35R30, 78A46, 80A23, 85A25, 92C55Publication DataISSN (print): 0036-1410ISSN (online): 1095-7154Publisher: Society for Industrial and Applied MathematicsCODEN: sjmaah

Методи та інформаційні системи ідентифікації джерел радіоактивних атмосферних за-бруднень шляхом оберненого моделювання

The paper reviews the methods for identifying an unknown source of pollution by inverse mod-eling and information systems for air pollution forecasting and analysis. Several different for-eign and Ukrainian air pollution forecasting systems, such as the European Union's Nuclear Emergency Response System RODOS, have been developed on the basis of atmospheric transport models. However, the key data that determine the quality of forecasting in such sys-tems are the characteristics of the emission sources. In the case of detection of pollution from an unknown emission source, there should be performed inverse simulation. The use of the RODOS system, as well as other existing forecasting systems for such a task is possible but it requires multiple manual start of calculations of atmospheric transfer models in the reverse mode. Presented in the paper results of the application of inverse modeling methods during ra-diation incidents of the last decade demonstrate that modern methods of inverse modeling are sufficiently developed to set the task of automating inverse modeling in information systems for air pollution analysis and forecasting. Even though these methods not always can exactly identify the source of emissions due to the lack of measurements and poor conditioning of the inverse atmospheric transport problem, their application always leads to a significant reduction (by an order of magnitude or more) in the search for unknown sources compared to the detec-tion of pollutants. At present, in the existing forecasting systems the methods of inverse model-ing are only partially automated, namely for the case of known location and unknown emissions of the source of pollution. Therefore, this paper proposes the architecture of the future system for identifying unknown sources of emissions by inverse modeling.

A spectral nodal method for the adjoint [formula omitted] neutral particle transport equations in [formula omitted]-geometry: Application to direct and inverse multigroup source-detector problems

In this work, we apply an adjoint technique to solve two-dimensional direct and inverse source-detector transport problems. In order to obtain the neutral-particle importance maps, we have developed the extended adjoint spectral Green’s function constant-nodal (SGF†-CN) method to numerically solve multigroup adjoint transport problems in the discrete ordinates formulation. Described here is the methodology to obtain the discretized SGF†-CN equations and the adjoint partial one-node block inversion scheme used to iteratively solve these equations. Then, the Levenberg–Marquardt method is applied to minimize a least-squares difference in order to identify the multigroup sources. To illustrate the accuracy of the present methodology, we present and discuss numerical results to two typical model problems.

Applications of kinetic tools to inverse transport problems

We show that the inverse problems for a class of kinetic equations can be solved by classical tools in PDE analysis including energy estimates and the celebrated averaging lemma. Using these tools, we give a unified framework for the reconstruction of the absorption coefficient for transport equations in the subcritical and critical regimes. Moreover, we apply this framework to obtain, to the best of our knowledge, the first result in a nonlinear setting. We also extend the result of recovering the scattering coefficient in Choulli and Stefanov (1998 Osaka J. Math. 36 87–104) from 3D to 2D strictly convex domains.

Inverse transport problem in fluorescence ultrasound modulated optical tomography with angularly averaged measurements

We consider an inverse transport problem in fluorescence ultrasound modulated optical tomography (fUMOT) with angularly averaged illuminations and measurements. We study the uniqueness and stability of the reconstruction of the absorption coefficient and the quantum efficiency of the fluorescent probes. Reconstruction algorithms are proposed and numerical validations are performed. This paper is an extension of Li et al (2019 SIAM J. Appl. Math. 79 356–76), where a diffusion model for this problem was considered.

The Adjoint Discrete Ordinates Solution Applied to Direct and Inverse Neutral Particle X, Y-Geometry Source-Detector Problems

Solving the adjoint transport equation allows obtaining maps of the neutral-particle importance to an objective function, e.g., a detector response. In source-detector problems, the adjoint interior source is defined as the macroscopic absorption cross section of the material the detector is made of. Determining the intensity of a source of particles in each energy group (causes) is possible by solving an inverse source-detector problem, given the source location and the detector response (effects). In this work, we present the application of an adjoint technique to solve energy multigroup X,Y-geometry direct and inverse transport problems. In order to obtain the importance maps, we have extended the adjoint spectral Green's function constant-nodal (SGF*-CN) method to numerically solve energy multigroup adjoint transport problems in the discrete ordinates formulation. Described here are the methodology to obtain the discretized SGF*-CNequations and the adjoint partial one-node block inversion scheme used to iteratively solve these equations. We present numerical results to two test problems to illustrate the accuracy of the present methodology.

Lipschitz stability for a semi-linear inverse stochastic transport problem

Abstract This paper is addressed to a semi-linear stochastic transport equation with three unknown parameters. It is proved that the initial displacement, the terminal state and the random term in diffusion are uniquely determined by the state on partial boundary and a Lipschitz stability of the inverse problem is established. The main tool we employ is a global Carleman estimate for stochastic transport equations.

Global Lipschitz stability for a fractional inverse transport problem by Carleman estimates

We consider a fractional radiative transport equation, where the time derivative is of half order in the Caputo sense. By establishing Carleman estimates, we prove the global Lipschitz stability in determining the coefficients of the one-dimensional time-fractional radiative transport equation of half-order.

Solving the inverse transport problem in intergalactic space and determining the energy spectrum and composition of extragalactic sources of ultrahigh-energy particles

The propagation of ultrahigh-energy nuclei in an expanding Universe filled with background electromagnetic radiation is considered. A numerical method for solving the inverse problem for the equation of cosmic-ray transport is developed that allows the spectrum of sources to be determined from the cosmic-ray spectrum observed near the Earth. The spectra of injected protons and nuclei in extragalactic sources are found by assuming that they are functions of the magnetic rigidity of particles. The data from observations obtained in the Auger experiment are used.

Inverse Green element evaluation of source strength and concentration in groundwater contaminant transport

Evaluating pollution sources from limited concentration data of their contaminant plumes in groundwater systems is an inverse problem that presents computational challenges because of its ill-posedness. In this work, the Green element method (GEM) is used to solve inverse contaminant transport problems in 2-D orthotropic homogeneous aquifers. The GEM discretization of the differential equation produces an over-determined, ill-conditioned global matrix that is decomposed by the singular value decomposition method and solved by the least square method with Tikhonov regularization. Five test cases of pollution sources of unknown strengths are used to evaluate the performance of GEM. The results indicate that the current methodology is capable of correctly predicting the strength of pollution sources and the historical concentration plumes that they produce. However, the computation accuracy is influenced by the location of the observation points in relation to the source, data errors of observed concentrations, the transport mode, and the value of the Courant number used in the simulations. Data errors at observation points influence more significantly the GEM prediction of the pollution source strength than the concentration plume, while numerical artefacts of dispersion and oscillations observed in direct solutions of advection-dominant transport cases are also evident in inverse modelling.

Radiation source reconstruction based on a covariance matrix adaptation evolution strategy and the adjoint flux and forward source inner product evaluation method

A novel approach to solve inverse photon transport problems is presented and used to characterize a shielded source based on gamma measurements taken outside the shielding. The iterations on source characteristics are controlled by a covariance matrix volution strategy and the evaluation of each of the trial sources calculates the inner product of the adjoint flux and the trial source, for each detector and relevant energy group. This approach uses a combination of cross sections evaluated for both coarse energy groups and discrete energies. The impact of including information from photon scattering on reconstruction accuracy is studied. Using a 2D Cartesian system, we show that a successful characterization is possible and that the inclusion of scattered photons can significantly improve results, especially for determining the source strength and shielding thickness at the same time. It is further demonstrated that the source reconstruction is sensitive to any discrepancies between the source evaluation deterministic method and the simulated detector measurements used for validation.

Merging parallel tempering with sequential geostatistical resampling for improved posterior exploration of high-dimensional subsurface categorical fields

The sequential geostatistical resampling (SGR) algorithm is a Markov chain Monte Carlo (MCMC) scheme for sampling from possibly non-Gaussian, complex spatially-distributed prior models such as geologic facies or categorical fields. In this work, we highlight the limits of standard SGR for posterior inference of high-dimensional categorical fields with realistically complex likelihood landscapes and benchmark a parallel tempering implementation (PT-SGR). Our proposed PT-SGR approach is demonstrated using synthetic (error corrupted) data from steady-state flow and transport experiments in categorical 7575- and 10,000-dimensional 2D conductivity fields. In both case studies, every SGR trial gets trapped in a local optima while PT-SGR maintains an higher diversity in the sampled model states. The advantage of PT-SGR is most apparent in an inverse transport problem where the posterior distribution is made bimodal by construction. PT-SGR then converges towards the appropriate data misfit much faster than SGR and partly recovers the two modes. In contrast, for the same computational resources SGR does not fit the data to the appropriate error level and hardly produces a locally optimal solution that looks visually similar to one of the two reference modes. Although PT-SGR clearly surpasses SGR in performance, our results also indicate that using a small number (16–24) of temperatures (and thus parallel cores) may not permit complete sampling of the posterior distribution by PT-SGR within a reasonable computational time (less than 1–2 weeks).

Using a derivative-free optimization method for multiple solutions of inverse transport problems

Identifying unknown components of an object that emits radiation is an important problem for national and global security. Radiation signatures measured from an object of interest can be used to infer object parameter values that are not known. This problem is called an inverse transport problem. An inverse transport problem may have multiple solutions and the most widely used approach for its solution is an iterative optimization method. This paper proposes a stochastic derivative-free global optimization algorithm to find multiple solutions of inverse transport problems. The algorithm is an extension of a multilevel single linkage (MLSL) method where a mesh adaptive direct search (MADS) algorithm is incorporated into the local phase. Numerical test cases using uncollided fluxes of discrete gamma-ray lines are presented to show the performance of this new algorithm.

Inverse transport problems in quantitative PAT for molecular imaging

Fluorescence photoacoustic tomography (fPAT) is a molecular imaging modality that combines photoacoustic tomography with fluorescence imaging to obtain high-resolution imaging of fluorescence distributions inside heterogeneous media. The objective of this work is to study inverse problems in the quantitative step of fPAT where we intend to reconstruct physical coefficients in a coupled system of radiative transport equations using internal data recovered from ultrasound measurements. We derive uniqueness and stability results on the inverse problems and develop some efficient algorithms for image reconstructions. Numerical simulations based on synthetic data are presented to validate the theoretical analysis. The results we present here complement these in Ren K and Zhao H (2013 SIAM J. Imaging Sci. 6 2024–49) on the same problem but in the diffusive regime.