Formulation Of Inverse Problem Research Articles

Kernel Methods for Bayesian Elliptic Inverse Problems on Manifolds

This paper investigates the formulation and implementation of Bayesian inverse problems to learn input parameters of partial differential equations (PDEs) defined on manifolds. Specifically, we stu...

Joint Acoustic Localization and Dereverberation Through Plane Wave Decomposition and Sparse Regularization

Acoustic source localization and dereverberation are formulated jointly as an inverse problem. The inverse problem consists of the approximation of the sound field measured by a set of microphones. The recorded sound pressure is matched with that of a particular acoustic model based on a collection of plane waves arriving from different directions at the microphone positions. In order to achieve meaningful results, spatial and spatio-spectral sparsity can be promoted in the weight signals controlling the plane waves. The large-scale optimization problem resulting from the inverse problem formulation is solved using a first order optimization algorithm combined with a weighted overlap-add procedure. It is shown that once the weight signals capable of effectively approximating the sound field are obtained, they can be readily used to localize a moving sound source in terms of direction of arrival (DOA) and to perform dereverberation in a highly reverberant environment. Results from simulation experiments and from real measurements show that the proposed algorithm is robust against both localized and diffuse noise exhibiting a noise reduction in the dereverberated signals.

Estimation of Spatially Distributed Thermal Properties of Heterogeneous Media with Non-Intrusive Measurement

This article deals with estimation of spatially distributed thermal properties of two-dimensional heterogeneous media from the solution of inverse heat conduction problem. The experimental procedure involves sequential heating of sample at four discrete locations on one side and recording transient temperature on the other side non-intrusively. The inverse problem formulation is carried out as parameter estimation problem and the unknown parameters are estimated through minimization of sum of squared error between measured and simulated temperatures using Levenberg–Marquardt algorithm. The huge computational time required to calculate sensitivity matrix is reduced through parallel computation strategy. Numerical estimations are carried out with synthetic temperature and it is found that the resolution and accuracy of estimated spatially distributed thermal conductivity is in better agreement with the exact distribution when compared to volumetric heat capacity even at an error level of ± 0.1 K. Transient temperature response of the fabricated heterogeneous prototype is recorded using infrared radiation camera and the same is used for estimation of unknown parameters. The estimated thermal conductivity closely mimics the actual distribution. Therefore, this simple proposed method can be directly used in thermal tomography applications for identifying the shape and size of inclusions/inhomogeneities from the estimated thermal conductivity distribution alone.

Development of combined search methods for efficiency indicator extreme in variation statement of forecasting tasks for determine characteristics of composite materials

Modified search methods for global extreme of multiparameter functions are studied, considering the structural features of efficiency criteria used in problems of forecasting residual life and durability of structures made of composite polymeric materials. Comparative and qualitative studies have been carried out and methods have been developed to select the most promising areas of search for the absolute minimum of the studied performance indicators in refined variation formulation of inverse problems of forecasting residual life of composites.

A unified weighted minimum norm solution for the reference inverse problem in EEG

A well known problem in EEG recordings deals with the unknown potential of the reference electrode. In the last years several authors presented comparisons among the most popular solutions, the global conclusion being that the traditional Average Reference (AR) and the Reference Standardization Technique (REST) are the best approximations (Nunez, 2010; Kayser and Tenke, 2010; Liu et al., 2015; Chella et al., 2016). In this work we do not aim to further compare these techniques but to support the fact that both solutions can be derived from a general inverse problem formalism for reference estimation (Hu et al., 2019; Hu et al., 2018; Salido-Ruiz et al., 2011). Using the alternative approach of least squares, our findings are consistent with the theoretical findings in Hu et al. (2019) and Hu et al. (2018) showing that the AR is the minimum norm solution, while REST is a weighted minimum norm including some approximate propagation model. AR is thus a particular case of REST, which itself uses a particular formulation of the source estimation inverse problem. With a different derivation, we provide the additional powerful evidences to reinforce the cited findings.

Structurally tailored 3D anisotropic controlled-source electromagnetic resistivity inversion with cross-gradient criterion and simultaneous model calibration

Geologic interpretation of 3D anisotropic resistivity models from conventional marine controlled-source electromagnetic (CSEM) data inversion faces difficulties in low-resistivity contrast sediments and structurally complex environments that typify the new frontiers for hydrocarbon exploration. Currently, the typically reconstructed horizontal resistivity [Formula: see text] and vertical resistivity [Formula: see text] models often have conflicting depth structures that are difficult to explain in terms of subsurface geology, and the resulting resistivities may not be close to the true formation resistivities required for estimating reservoir parameters. We have investigated the concept that an objective geologically oriented or structurally tailored inversion can be achieved by requiring that the cross-product of the gradient of horizontal resistivity and the gradient of the vertical resistivity is equal to zero at significant geologic boundaries. We incorporate this boundary-shape criterion in our 3D inverse problem formulations, implemented within nonlinear model-space and conjugate-gradient contexts, for cases in which a priori calibration data from wells and/or seismically derived subsurface boundaries are available and for cases in which these are lacking. The resulting fit-for-purpose solutions serve to better analyze the peculiarity of a given data set. We applied these algorithms to synthetic and field CSEM data sets representing a fold-thrust environment with low-resistivity and low-contrast sediments. The resulting [Formula: see text] and [Formula: see text] models from cross-gradient joint inversion of synthetic data of appropriate frequency bandwidth without a priori information are structurally similar and consistent with the test models, whereas those from the inversions of band-limited field data are consistent with the available seismic and resistivity well-log data. This particular approach will thus be useful for lithologic correlation in frontier regions with limited a priori information using broadband CSEM data. For these band-limited field data, we found that the anisotropic bulk resistivities of the low-contrast sediments are better determined by incorporating a priori calibration data from triaxial resistivity logs and seismic horizons.

Direct error in constitutive equation formulation for inverse problems in time harmonic viscoelasticity

Direct error in constitutive equation formulation for inverse problems in time harmonic viscoelasticity

Inverse problem on conservation laws

The explicit formulation of the general inverse problem on conservation laws is presented for the first time. Within this problem, one aims to derive the general form of systems of differential equations that admit a prescribed set of conservation laws. The particular cases of the inverse problem on first integrals of ordinary differential equations and on conservation laws for evolution equations are studied. We also solve the inverse problem on conservation laws for differential equations admitting an infinite dimensional space of zeroth-order conservation-law characteristics. This particular case is further studied in the context of conservative first-order parameterization schemes for the two-dimensional incompressible Euler equations. We exhaustively classify conservative first-order parameterization schemes for the eddy-vorticity flux that lead to a class of closed, averaged Euler equations possessing conservation of generalized circulation, generalized momentum and energy.

Parameter estimation and mathematical modeling of the diffusion process of a benzoic acid infused Fricke gel dosimeter

This work presents a mathematical model implementation for studying the diffusion process of ferric ions in post-irradiated Fricke gel dosimeters. Fricke gel samples were manufactured in layers for their analysis by optical transmission imaging and benzoic acid was incorporated to improve their radiosensitivity. The developed method consists on designing the dosimeters with a suitable composition for their optical analysis which allows accurate measurements in a wide dose range with noticeable linear responses with an average linear correlation coefficient of r2=0.993. The developed algorithm calculates the diffusion coefficient by means of an inverse problem formulation. Moreover, the proposed method is capable of modeling time-dependent physical properties and evolution over time of the irradiated dosimeters accurately and comparable with experimentally measured data. The obtained results evince the complexity of the diffusion phenomena in this type of materials and the capacity of the proposed method to provide a reliable quantification of the diffusion coefficient as a scalar approximation of the diffusion tensor. The obtained diffusion coefficient varies from 1.21 to 1.61 mm2/h with uncertainties lower than 2%. The model is also transferable to other diffusion driven systems and the numerical solving method can also be adapted to solve diffusion calculation based on other theoretical approaches.

Direct error in constitutive equation formulation for inverse problems in time harmonic elastography

Shear wave imaging techniques allow the evaluation of rigidity and viscosity of tissues locally within a material. From an inverse problem perspective, the approach is quite attractive insofar as it provides a densely sampled displacement field in the interior of the object from which to invert for material properties. We consider several challenges related to elastic wave inverse problems arising in acoustic radiation force imaging. First, we validate an axisymmetric viscoelasticity model suitable for some applications of acoustic radiation force imaging. Second, we consider reconstructing lateral displacement components from measured axial displacement components. Finally, we present a new variational formulation, the direct error in constitutive equation formulation, for inverse problems in time harmonic viscoelastic wave propagation with full-field data. The formulation relies on minimizing the error in the constitutive equation with a momentum equation constraint. Numerical results on model problems show that the formulation is capable of handling discontinuous and noisy strain fields and also converging with mesh refinement for continuous and discontinuous material property distributions. Applications to MRE and ARFI measured wave data are considered.

Phonon relaxation time reconstruction from transient thermal grating experiments and comparison with density functional theory predictions

Phonon relaxation time and free path distributions are reconstructed from experimental measurements on a two-dimensional transient thermal grating and compared with density functional theory (DFT) results for silicon. The reconstruction is performed using the inverse problem formulation of Forghani et al. [Phys. Rev. B 94, 155439 (2016)]. The discrepancies observed between reconstructed and DFT results are analyzed in terms of the ability of each set of data to reproduce the experimental temperature relaxation profiles; the reconstructed data are found to predict temperature profiles in closer agreement with the experimental data than the DFT results, possibly due to discrepancies between the actual material and the idealized model studied in the DFT calculations. The reconstructed phonon properties accurately predict temperature relaxation profiles at grating length scales smaller than those spanned by the experimental data. This is a very important feature since in a variety of experimental setups, including the one providing the data in the present study, measurements are not available at all scales spanned by the material free paths.

Estimation of probability distributions of parameters using aggregate population data: analysis of a CAR T-cell cancer model.

In this effort we explain fundamental formulations for aggregate data inverse problems requiring estimation of probability distribution parameters. We use as a motivating example a class of CAR T-call cancer models in mice. After ascertaining results on model stability and sensitivity with respect to parameters, we carry out first elementary computations on the question how much data is needed for successful estimation of probability distributions.

The development and application of generalized models long-term forecasting of the residual resource composite materials and structures at the impact of extreme climate factors of the Arctic zone

The problem of developing effective methods for predicting the residual life of composite materials and structures under the influence of extreme environmental factors characteristic of the Arctic zone’s sharp continental climate is investigated. Part of the problem is the solution to the problem of effective construction solutions, delivering the absolute minimum of multi-parameter performance criteria associated with the task of remaining life prediction of composites. The formulated problems are investigated within the framework of refined variational formulations of inverse prediction problems. Comparative qualitative researches are carried out and methods of the choice of the most perspective directions of search of an absolute minimum of the investigated indicators of efficiency are developed. The results of comparative computational experiments on the effective solution of predicting the residual life of composites in the variational formulation are presented.

A framework for solving meta inverse problems: experimental design and application to an acoustic source problem

The selection of optimal measurement locations in remote sensing or imaging algorithms is of large practical interest in many applications. The target is usually to choose a measurement setup that best resolves some particular quantity of interest. This work describes a general framework for selecting such an optimal setup within a given set Q of possible setups for the formulation and solution of the meta inverse problem. The work shows that it is crucial to incorporate the basic ingredients which are usually part of the inversion process. In particular, it takes care of the nature and the size of the measurement error, the choice of the regularization scheme which is employed for the inverse problem, and the prior knowledge on solutions. The basic idea of the framework is to minimize the errors associated with the reconstruction of a given quantity of interest. Five functional layers which reflect the structure of the meta inverse problem are introduced. Further, with framework adaption, an iterative algorithm is formulated to solve the meta inverse problem at each iterative step in order to obtain improved reconstructions of the inverse problem. Using the initial reconstructions as input for meta inversion, the framework adaption algorithm does not require prior knowledge of the source distribution. The feasibility of the framework adaption algorithm is illustrated by using it to solve the inverse acoustic source problem.

Variation Formulations of Inverse Problems in Forecasting the Residual Life of Composites

An approach is proposed on the basis of more precise formulations of the forecasting problem and developed within the framework of variation principles for the solution of inverse problems, into which precise estimation of forecast solutions is incorporated for the first time, thus providing the possibility to obtain new scientifically substantiated results and achieve a required precision of forecasting. An original model is constructed to forecast the residual life and longevity of composites and describe the processes occurring in polymer composite materials and structures under the simultaneous influence of several destabilizing physical factors at the physical level. The application of a newly formulated principle of multiplicity for forecasting models and the first constructed multiparametric models of optimal complexity within the refined variation formulation of inverse forecasting problems of the residual life has appreciably improved the precision of solution for inverse forecasting problems and enhanced the time interval within which the time dependence of the residual life and longevity of composites can be predicted at a specified maximum permissible precision of forecasting. The proposed approach can be used to solve a broad range of problems of forecasting the residual life, longevity, and strength of composites.

Spherical Harmonics Decomposition in inverse acoustic methods involving spherical arrays

Inverse methods for acoustic source mapping have gathered the attention of the beamforming community during the last years. Indeed, they provide higher accuracy in both source localization and strength estimation with respect to Conventional Beamforming (CB). One of the main drawbacks of the current formulations is the need of a regularization strategy for tackling the ill-posedness of the problem. Very often, Tikhonov regularization is exploited to face this issue, but different methods for estimating the regularization factor associated to the Tikhonov formulation may lead to different regularization levels and, therefore, to different results. This paper presents a way to face this problem when dealing with spherical arrays. The new approach proposed by the authors exploits Spherical Harmonics Decomposition (SHD) of complex pressure data at microphone locations. SHD performs a spatial filtering that reduces the effect of noise and causes an intrinsic stabilization of the numerical problem associated to the inverse problem formulation. When the source-receiver propagation model is appropriate to describe the acoustic environment in which the test takes place and noise is not spoiling excessively measurement data, the SHD approach is sufficient to obtain a regularized solution. If these conditions are not satisfied, SHD can be exploited as a pre-processing step in a twofold procedure also involving classical Tikhonov regularization. In this paper the SHD approach is tested in the Generalized Inverse Beamforming (GIBF) formulation. Classical Tikhonov approach, in which the regularization factor is estimated using the Generalized Cross-Validation (GCV), L-curve functions and Bayesian regularization, is presented as a way to enhance data processed by SHD. A sensitivity analysis of the approach to measurement noise and source-receiver relative positions is presented on simulated data. Results on experimental data are presented and discussed for both a simplified test case and an application to a real car cabin.

An Axially Variant Kernel Imaging Model Applied to Ultrasound Image Reconstruction

Existing ultrasound deconvolution approaches unrealistically assume, primarily for computational reasons, that the convolution model relies on a spatially invariant kernel and circulant boundary conditions. We discard both restrictions and introduce an image formation model applicable to ultrasound imaging and deconvolution based on an axially varying kernel, that accounts for arbitrary boundary conditions. Our model has the same computational complexity as the one employing spatially invariant convolution and has negligible memory requirements. To accommodate state-of-the-art deconvolution approaches when applied to a variety of inverse problem formulations, we also provide an equally efficient adjoint expression of our model. Simulation results confirm the tractability of our model for the deconvolution of large images. Moreover, the quality of reconstruction using our model is superior to that obtained using spatially invariant convolution.

Direct error in constitutive equation formulation for inverse heat conduction problem

SummaryWe present a mixed numerical formulation that handles discontinuities well for scalar hyperbolic partial differential equations. The formulation is based on a least‐square error in the constitutive equation. It is motivated by scalar inverse diffusion problems with interior data and applies to convection of a passive scalar in a discontinuous compressible flow field. We motivate the need for a mixed formulation by discretizing using an irreducible finite element method and discuss some of the limitations of that approach. We then develop and prove that the mixed formulation is well posed and verify that it works for problems with continuous and discontinuous thermal conductivity distributions.

Identification and optimization of unbalance parameters in rotor-bearing systems

The paper describes the identification and optimization of unbalance parameters in rotor-bearing systems. Two methods are proposed for the identification of the unbalance characteristics: the first is based on modal expansion combined with the use of optimization algorithms, while the second relates to the use of modal expansion technique applied to the inverse problem. In this work, the modal expansion technique is used to overcome the issue related to the use of a reduced number of measuring points. The equivalent unbalance forces can then be estimated by expanding the modal displacements into generalized coordinates of the equations of motion of the rotor system. An error due to the use of a modal expansion is however inevitable, and to solve the issue we propose the adoption of an inverse problem formulation to avoid the computation of the displacements at each measurement point. The axial location of the unbalance must be however known in advance, if the inverse problem approach is used to identify the unbalance parameters. We therefore propose in this work an integrated modal expansion/inverse problem methodology combined with an optimization procedure. The technique allows to identify the axial location of the unbalance, its magnitude and phase. Simulation and experimental investigations are carried out to verify the validity of the proposed methods in a double-disk rotor-bearing system. The results show that identification and optimization procedure for the integrated modal expansion/inverse problem approach provides more accurate predictions than the ones given by the pure modal expansion method.

An iterative finite-element algorithm for solving two-dimensional nonlinear inverse heat conduction problems

It is often useful to determine temperature and heat flux in multidimensional solid domains of arbitrary shape with inaccessible boundaries. In this study, an effective algorithm for solving boundary inverse heat conduction problems (IHCPs) is implemented: transient temperatures on inaccessible boundaries are estimated from redundant simulated measurements on accessible boundaries. A nonlinear heat equation is considered, where some of the material properties are dependent on temperature. The IHCP is reformulated as an optimization problem. The resulting functional is iteratively minimized using a conjugate gradient method together with an adjoint (dual) problem approach. The associated partial differential equations are solved using the finite-element package FEniCS. Tikhonov regularization is introduced to mitigate the ill-posedness of the IHCP. The accuracy of the implemented algorithm is assessed by comparing the solutions to the IHCP with the correct temperature values, on the inaccessible boundaries. The robustness of our method is tested by adding Gaussian noise to the initial conditions and redundant boundary data in the inverse problem formulation. A mesh independence study is performed.