Full Inverse Problem Research Articles

Polarimetric inverse scattering using LSTM-aided association-learning sparse reconstruction framework

Conventional polarimetric inverse scattering is under the assumption of joint sparsity of different polarization channels. However, in practical situation, there are strong correlations among these multi-channel measurements, but the inherent dependencies are unknown; the deterministic joint sparsity assumption may cause insufficient utilization of association information among multi-polarization channels, and consequently lead to limited improvement of canonical scatterers extraction accuracy. Aimed at this problem, we propose a long short-term memory (LSTM)-aided association-learning sparse reconstruction framework for polarimetric inverse scattering, in which the dependencies among multi-channel measurements are captured via a data driven pattern. In the proposed scheme, instead of directly solving the full inverse problem, polarimetric inverse scattering with multi-attribute canonical scatterers extraction is implemented via hierarchical pattern, and consequently the required computational burden is reduced. Particularly, the LSTM network is utilized to perform atomic screening for estimating first type of attributes of canonical scatterers, and then the optimal adaptive filtering (OAF) process is carried out for estimating second and third types of attributes, followed by residual update process. Extensive experimental results demonstrate that the proposed approach is able to outperform existing algorithms in terms of reconstruction accuracy and computational speed.

Numerical Identification of External Boundary Conditions for Time Fractional Parabolic Equations on Disjoint Domains

We consider fractional mathematical models of fluid-porous interfaces in channel geometry. This provokes us to deal with numerical identification of the external boundary conditions for 1D and 2D time fractional parabolic problems on disjoint domains. First, we discuss the time discretization, then we decouple the full inverse problem into two Dirichlet problems at each time level. On this base, we develop decomposition techniques to obtain exact formulas for the unknown boundary conditions at point measurements. A discrete version of the analytical approach is realized on time adaptive mesh for different fractional order of the equations in each of the disjoint domains. A variety of numerical examples are discussed.

Kinematic Analysis of three degrees of freedom planar parallel continuum mechanisms

Parallel continuum mechanisms are a type of compliant mechanisms comprising a rigid end-effector connected to a fixed frame by flexible and slender elements in a closed-kinematic morphology, being their flexibility the source of mobility. Solving the position problem involves the static equilibrium of the whole device considering the nonlinear large deformations of slender elements. Numerical iterative solutions are already available to solve real-time simulations. However, an assembled solution, which meets both geometric conditions and static equilibrium, is needed to start the process. A comprehensive Kinematic Analysis in the classical sense is not fully established yet. The aim of the paper is to propose a procedure that solves the full inverse and forward kinematic problems for planar three degrees of freedom systems, hence obtaining the multiple solutions arising from the closed-loop structure and the multiplicity of the deformation mode of slender links. From those solutions, finding of the different aspects that comprise the workspace is possible, as well as determining singularities and instability loci.

On the inverse problem in optical coherence tomography

We examine the inverse problem of retrieving sample refractive index information in the context of optical coherence tomography. Using two separate approaches, we discuss the limitations of the inverse problem which lead to it being ill-posed, primarily as a consequence of the limited viewing angles available in the reflection geometry. This is first considered from the theoretical point of view of diffraction tomography under a weak scattering approximation. We then investigate the full non-linear inverse problem using a variational approach. This presents another illustration of the non-uniqueness of the solution, and shows that even the non-linear (strongly scattering) scenario suffers a similar fate as the linear problem, with the observable spatial Fourier components of the sample occupying a limited support. Through examples we demonstrate how the solutions to the inverse problem compare when using the variational and diffraction-tomography approaches.

Effectiveness and computational efficiency of absorbing boundary conditions for full-waveform inversion

Abstract. Full-waveform inversion (FWI) is a high-resolution numerical technique for seismic waves used to estimate the physical characteristics of a subsurface region. The continuous problem involves solving an inverse problem on an infinite domain, which is impractical from a computational perspective. In limited area models, absorbing boundary conditions (ABCs) are usually imposed to avoid wave reflections. Several relevant ABCs have been proposed, with extensive literature on their effectiveness on the direct wave problem. Here, we investigate and compare the theoretical and computational characteristics of several ABCs in the full inverse problem. After a brief review of the most widely used ABCs, we derive their formulations in their respective adjoint problems. The different ABCs are implemented in a highly optimized domain-specific language (DSL) computational framework, Devito, which is primarily used for seismic modelling problems. We evaluate the effectiveness, computational efficiency, and memory requirements of the ABC methods, considering from simple models to realistic ones. Our findings reveal that, even though the popular perfectly matching layers (PMLs) are effective at avoiding wave reflections at the boundaries, they can be computationally more demanding than less used hybrid ABCs. We show here that a proposed hybrid ABC formulation, with nested Higdon's boundary conditions, is the most cost-effective method among the methods considered here, for being as effective as or more effective than PML and other schemes but also for being computationally more efficient.

Extensions of the general solution families for the inverse problem of the calculus of variations for sixth- and eighth-order ordinary differential equations

New third- and fourth-order Lagrangian hierarchies are derived in this paper. The free coefficients in the leading terms satisfy the most general differential geometric criteria currently known for the existence of a variational formulation, as derived by solution of the full inverse problem of the Calculus of Variations for scalar sixth- and eighth-order ordinary differential equations (ODEs). The Lagrangians obtained here have greater freedom since they require conditions only on individual coefficients. In particular, they contain four arbitrary functions, so that some investigations based on the existing general criteria for a variational representation are particular cases of our families of models. The variational equations resulting from our generalized Lagrangians may also represent traveling waves of various nonlinear evolution equations, some of which recover known physical models. For a typical member of our generalized variational ODEs, families of regular and embedded solitary waves are also derived in appropriate parameter regimes. As usual, the embedded solitons are found to occur only on isolated curves in the part of parameter space where they exist.

Strong Solutions for PDE-Based Tomography by Unsupervised Learning

We introduce a novel neural network-based PDEs solver for forward and inverse problems. The solver is grid free, mesh free, and shape free, and the solution is approximated by a neural network. We employ an unsupervised approach such that the input to the network is a point set in an arbitrary domain, and the output is the set of the corresponding function values. The network is trained to minimize deviations of the learned function from the PDE solution and satisfy the boundary conditions. The resulting solution in turn is an explicit, smooth, differentiable function with a known analytical form. We solve the forward problem (observations given the underlying model's parameters), semi-inverse problem (model's parameters given the observations in the whole domain), and full tomography inverse problem (model's parameters given the observations on the boundary) by solving the forward and semi-inverse problems at the same time. The optimized loss function consists of few elements: fidelity term of $L_2$ norm that enforces the PDE in the weak sense, an $L_\infty$ norm term that enforces pointwise fidelity and thus promotes a strong solution, and boundary and initial conditions constraints. It further accommodates regularizers for the solution and/or the model's parameters of the differential operator. This setting is flexible in the sense that regularizers can be tailored to specific problems. We demonstrate our method on several free shape two dimensional (2D) second order systems with application to electrical impedance tomography (EIT) and diffusion equation. Unlike other numerical methods such as finite differences and finite elements, the derivatives of the desired function can be analytically calculated to any order. This framework enables, in principle, the solution of high order and high dimensional nonlinear PDEs.

Novel Lagrangian hierarchies, generalized variational ODE’s, and families of regular and embedded solitary waves

Hierarchies of Lagrangians of degree two, each only partly determined by the choice of leading terms and with some coefficients remaining free, are considered. The free coefficients they contain satisfy the most general differential geometric criterion currently known for the existence of a Lagrangian and variational formulation, and derived by solution of the full inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations (ODEs) respectively. However, our Lagrangians have significantly greater freedom since our existence conditions are for individual coefficients in the Lagrangian. In particular, the classes of Lagrangians derived here have four arbitrary or free functions, including allowing the leading coefficient in the resulting variational ODEs to be arbitrary, and with models based on the earlier general criteria for a variational representation being special cases. For different choices of leading coefficients, the resulting variational equations could also represent traveling waves of various nonlinear evolution equations, some of which recover known physical models. Families of regular and embedded solitary waves are derived for some of these generalized variational ODEs in appropriate parameter regimes, with the embedded solitons occurring only on isolated curves in the part of parameter space where they exist. Future work will involve higher order Lagrangians, the resulting equations of motion, and their solitary wave solutions.

Efficient inversion strategies for estimating optical properties with Monte Carlo radiative transport models.

.Significance: Indirect imaging problems in biomedical optics generally require repeated evaluation of forward models of radiative transport, for which Monte Carlo is accurate yet computationally costly. We develop an approach to reduce this bottleneck, which has significant implications for quantitative tomographic imaging in a variety of medical and industrial applications.Aim: Our aim is to enable computationally efficient image reconstruction in (hybrid) diffuse optical modalities using stochastic forward models.Approach: Using Monte Carlo, we compute a fully stochastic gradient of an objective function for a given imaging problem. Leveraging techniques from the machine learning community, we then adaptively control the accuracy of this gradient throughout the iterative inversion scheme to substantially reduce computational resources at each step.Results: For example problems of quantitative photoacoustic tomography and ultrasound-modulated optical tomography, we demonstrate that solutions are attainable using a total computational expense that is comparable to (or less than) that which is required for a single high-accuracy forward run of the same Monte Carlo model.Conclusions: This approach demonstrates significant computational savings when approaching the full nonlinear inverse problem of optical property estimation using stochastic methods.

Bayesian Inversion for Layered Spherical Symmetric Earth Conductivity Model from Global Magnetic Data

In the Bayesian perspective, inference on model parameters from observed data is performed by calculating the likelihood of the data given prior model parameters, i.e. to estimate the posterior probability of model parameters. With the advent of computational resources, there are increasing interests in resolving full non-linear inverse problems using global approach. Although the current trends are geared towards algorithms to efficiently explore the model space, we employed the classical “pure” Monte Carlo method to resolve the inverse problem in the global scale induction study. Observatory and satellite magnetic data are used to provide insight on the deep mantle conductivity. In this case, layered (1D) spherical symmetric conductivity model can be considered as adequate to represent the Earth’s conductivity variation with depth. Model parameters (resistivities and thicknesses) with uniform probabilities over predefined intervals are drawn as samples of the model space. Reliable posterior estimates are derived from a large number of samples which are still manageable with the current PC technology. Relatively small uncertainties of the posterior estimates suggest that the Monte Carlo method is adequately sampled the model space with a small number of model parameters. Our results are consistent with a monotonic increase of conductivity with depth with a marked inflexion at about 700-900 km, while discontinuities at 410 km and 660 km known from seismic and petrology data seem unresovable directly from EM data.

Optoacoustic image reconstruction: the full inverse problem with variable bases

Optoacoustic imaging was for a long time concerned with the reconstruction of energy density or optical properties. In this work, we present the full inverse problem with respect to optical absorption and diffusion as well as speed of sound and mass density. The inverse problem is solved by an iterative gradient-based optimization procedure. Since the ill-conditioning increases with the number of sought parameters, we propose two approaches to improve the conditioning. The first approach is based on the reduction of the size of the basis for the parameter spaces, that evolves according to the particular characteristics of the solution, while maintaining the flexibility of element-wise parameter selection. The second approach is a material identification technique that incorporates prior knowledge of expected material types and uses the acoustical gradients to identify materials uniquely. We present numerical studies to illustrate the properties and functional principle of the proposed methods. Significant convergence speed-ups are gained by the two approaches countering ill-conditioning. Additionally, we show results for the reconstruction of a mouse brain from in vivo measurements.

Position analysis in planar parallel continuum mechanisms

Parallel continuum mechanisms are closed, kinematic, chained compliant mechanisms that are mobile due to flexible and slender elements. Large deformations and the force-moment equilibrium require a more complex modeling of the kinematic analysis than in their rigid-link counterparts. Although real-time solutions to the position problem using numerical iterative methods are available, a comprehensive kinematic analysis of such mechanisms has not yet been conducted. The first step to this end is a procedure that solves the full forward and inverse kinematic problems, and is proposed in this paper for planar systems to allow for an easier interpretation of their kinematic characteristics. This procedure provides the means to obtain multiple solutions arising from the closed-loop structure and the multiplicity of the buckling mode of slender links.

Inverse acoustic scattering problem in half-space with anisotropic random impedance

We study an inverse acoustic scattering problem in half-space with a probabilistic impedance boundary value condition. The Robin coefficient (surface impedance) is assumed to be a Gaussian random function with a pseudodifferential operator describing the covariance. We measure the amplitude of the backscattered field averaged over the frequency band and assume that the data is generated by a single realization of λ. Our main result is to show that under certain conditions the principal symbol of the covariance operator of λ is uniquely determined. Most importantly, no approximations are needed and we can solve the full non-linear inverse problem. We concentrate on anisotropic models for the principal symbol, which leads to the analysis of a novel anisotropic spherical Radon transform and its invertibility.

Object‐level SAR imaging method with canonical scattering characterisation and inter‐subdictionary interferences mitigation

Under the assumption of ideal-point scattering model, conventional developed synthetic aperture radar (SAR) imaging techniques often ignore frequency/angle dependence, which may result in scattering estimation inaccurate and object-level information loss. To solve the problem, in this study, the authors propose an object-level SAR imaging method by virtual of sparse representations and canonical shape feature models. Specifically, the representation basis vectors are combined with parametric scattering models for characterising angle/frequency dependence, so that they can accurately capture physically relevant scattering geometry information. To acquire high-quality object-level SAR image and mitigate the inter-subdictionary interferences, the authors propose a regularisation algorithm to take advantage of the inherent prior information including the sparsity of coefficient vector and the interaction nature between different spatial locations. The advantages of the proposed method can be summarised as follows: (i) by using the location-wise coordinate descent strategy, it can save memory cost and also reduce computation complexity in comparison with the direct solving the full inverse problem; and (ii) it can effectively acquire more parsimonious representation in the number of spatial locations and canonical scattering type information of strong scattering points such as top-hats and trihedral. Finally, experimental results are provided to demonstrate the validity of the proposed method.

A Hybrid Segmentation and D-Bar Method for Electrical Impedance Tomography

The Regularized D-bar method for Electrical Impedance Tomography provides a rigorous mathematical approach for solving the full nonlinear inverse problem directly, i.e. without iterations. It is based on a low-pass filtering in the (nonlinear) frequency domain. However, the resulting D-bar reconstructions are inherently smoothed leading to a loss of edge distinction. In this paper, a novel approach that combines the rigor of the D-bar approach with the edge-preserving nature of Total Variation regularization is presented. The method also includes a data-driven contrast adjustment technique guided by the key functions (CGO solutions) of the D-bar method. The new TV-Enhanced D-bar Method produces reconstructions with sharper edges and improved contrast while still solving the full nonlinear problem. This is achieved by using the TV-induced edges to increase the truncation radius of the scattering data in the nonlinear frequency domain thereby increasing the radius of the low pass filter. The algorithm is tested on numerically simulated noisy EIT data and demonstrates significant improvements in edge preservation and contrast which can be highly valuable for absolute EIT imaging.

Solution to the Full Vectorial 3-D Inverse Source Problem by Multilevel Fast Multipole Method Inspired Hierarchical Disaggregation

A 3-D imaging approach based on a multilevel fast multipole method inspired hierarchical disaggregation of the broadband plane wave spectrum is introduced to solve the full vectorial electromagnetic inverse source problem based on the knowledge of the radiated fields. In terms of complexity, this algorithm is identical to fast Fourier transform accelerated 3-D imaging techniques. In terms of applicability, the hierarchical disaggregation is advantageous for commonly encountered spherical <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$k$</tex> </formula> -space data. Since the method can be directly combined with the near-field fast multipole translations, both far- and near-field data can be handled conveniently. The investigation includes an analysis of the inherent point spread function and numerical results.

Multichannel Three-Dimensional SOLA Inversion for Local Helioseismology

Inversions for local helioseismology are an important and necessary step for obtaining three-dimensional maps of various physical quantities in the solar interior. Frequently, the full inverse problems that one would like to solve prove intractable because of computational constraints. Due to the enormous seismic data sets that already exist and those forthcoming, this is a problem that needs to be addressed. To this end, we present a very efficient linear inversion algorithm for local helioseismology. It is based on a subtractive optimally localized averaging (SOLA) scheme in the Fourier domain, utilizing the horizontal-translation invariance of the sensitivity kernels. In Fourier space the problem decouples into many small problems, one for each horizontal wave vector. This multi-channel SOLA method is demonstrated for an example problem in time-distance helioseismology that is small enough to be solved both in real and Fourier space. We find that both approaches are successful in solving the inverse problem. However, the multi-channel SOLA algorithm is much faster and can easily be parallelized.

Group kinematic motion decoupling in parallel mechanisms

Parallel mechanisms with six degrees of freedom with full and partial group kinematic motion decoupling are considered. For mechanisms with full group kinematic motion decoupling, direct and inverse kinematic problems are solved in explicit form.

A note on sparse reconstruction methods

In this paper we discuss some aspects of sparse reconstruction techniques for inverse problems, which recently became popular due to several superior properties compared to linear reconstructions. We briefly review the standard sparse reconstructions based on l1-minimization of coefficients with respect to an orthonormal basis, and also some recently proposed improvements based on Bregman iterations and inverse scale space techniques. For the latter we provide uniqueness results not available before for inverse problems with sparsity constraints.In order to gain further understanding of sparse reconstruction techniques we provide a detailed analysis in the singular basis for the operator describing the inverse problem. This allows to compute analytic expressions for the reconstructions and highlight certain features. We also show that a very classical linear reconstruction technique, the truncated singular value decomposition is indeed equivalent to a sparse reconstruction technique with data-dependent weights.Finally we touch the question whether it pays off to use sparse reconstruction schemes directly for the full inverse problem or if simple two-step schemes, consisting of a linear inversion and subsequent shrinkage, can potentially yield results of similar quality.

Application of the representer method for parameter estimation in numerical reservoir models

A data assimilation method was applied to estimate poorly known parameters (permeabilities) in a numerical reservoir model. Most variational methods for data assimilation are based on the assumption that the model is perfect except for the poorly known parameters. The representer method allows also for model errors, i.e. for uncertainties in the state variables (pressures and saturations). The method is based on minimizing a cost functional, assuming all the errors and parameters to be multivariate Gaussian random variables with given mean and covariances. The uncertain parameters and variables are expanded into a finite sum of basis functions called representers, and the gradients of the cost functional are obtained with an adjoint method. This approach gives an optimal parametrization in the sense that the final result is equal to the solution of the full inverse problem. The method was tested on a simple one-dimensional model to simulate two-phase (oil-water) flow through a heterogeneous reservoir. The results show that the method is able to provide an acceptable estimate of the permeability field. We used pressure measurements from a small number of observation wells in between the injection and production wells, but the representer method could be used equally well to assimilate data from other sources. The method appears to be a promising data assimilation tool for applications in reservoir engineering.