An inverse problem for the Helmholtz equation involving two semi-infinite fluids

An analytical method is developed for solving an inverse problem for Helmholtz's equation associated with two semi-infinite incompressible fluids of different variable refractive indices, separated by a plane interface. The unknowns of the inverse problem are: (i) the refractive indices of the two fluids, (ii) the ratio of the densities of the two fluids, and (iii) the strength of an acoustic source assumed to be situated at the interface of the two fluids. These are determined from the pressure on the interface produced by the acoustic source. The effect of the surface tension force at the interface is taken into account in this paper. The application of the proposed analytical method to solve the inverse problem is also illustrated with several examples. In particular, exact solutions of two direct problems are first derived using standard classical methods which are then used in our proposed inverse method to recover the unknowns of the corresponding inverse problems. The results are found to be in excellent agreement.

Accurate Parameter Estimation of Contaminant Transport Inverse Problem using Particle Swarm Optimization

Swarm Intelligence techniques such as particle swarm optimization (PSO) are shown to be incompetent for an accurate estimation of global solutions in several engineering applications. This problem is more severe in case of inverse optimization problems where fitness calculations are computationally expensive. In this work, a novel strategy is introduced to alleviate this problem. The proposed inverse model based on modified particle swarm optimization algorithm is applied for a contaminant transport inverse model. The inverse models based on standard-PSO and proposed-PSO are validated to estimate the accuracy of the models. The proposed model is shown to be out performing the standard one in terms of accuracy in parameter estimation. The preliminary results obtained using the proposed model is presented in this work.

Accurate Parameter Estimation of Contaminant Transport Inverse Problem using Paartaicle Swarm Optimization

Swarm Intelligence techniques such as particle swarm optimization (PSO) are shown to be incompetent for an accurate estimation of global solutions in several engineering applications. This problem is more severe in case of inverse optimization problems where fitness calculations are computationally expensive. In this work, a novel strategy is introduced to alleviate this problem. The proposed inverse model based on modified particle swarm optimization algorithm is applied for a contaminant transport inverse model. The inverse models based on standard-PSO and proposed-PSO are validated to estimate the accuracy of the models. The proposed model is shown to be out performing the standard one in terms of accuracy in parameter estimation. The preliminary results obtained using the proposed model is presented in this work.

An inverse problem for Helmholtz equation

This article is concerned with subsurface material identification for the 2-D Helmholtz equation. The algorithm is iterative in nature. It assumes an initial guess for the unknown function and obtains corrections to the guessed value. It linearizes the otherwise nonlinear problem around the background field. The background field is the field variable generated using the guessed value of the unknown function at each iteration. Numerical results indicate that the algorithm can recover a close estimate of the unknown function based on the measurements collected at the boundary.

Image reconstruction by inhomogeneous diffusion for positron emission tomography

In positron emission tomography (PET), image reconstruction is a demanding problem. Since, PET image reconstruction is an ill-posed inverse problem, new methodologies need to be developed. Although previous studies show that incorporation of spatial and median priors improves the image quality, the image artifacts such as over-smoothing and streaking are evident in the reconstructed image. In this work, we use a simple, yet powerful technique to tackle the PET image reconstruction problem. Proposed technique is based on the integration of Bayesian approach with that of finite impulse response (FIR) filter. A FIR filter is designed whose coefficients are determined based on the surface diffusion model. The resulting reconstructed image is iteratively filtered and fed back to obtain the new estimate. Experiments are performed on a simulated PET system. The results show that the proposed approach is better than recently proposed MRP algorithm in terms of image quality and normalized mean square error.

Interior Photon Absorption Based Adaptive Regularization Improves Diffuse Optical Tomography

An adaptive regularization algorithm that combines elementwise photon absorption and data misfit is proposed to stabilize the non-linear ill-posed inverse problem. The diffuse photon distribution is low near the target compared to the normal region. A Hessian is proposed based on light and tissue interaction, and is estimated using adjoint method by distributing the sources inside the discretized domain. As iteration progresses, the photon absorption near the inhomogeneity becomes high and carries more weightage to the regularization matrix. The domain's interior photon absorption and misfit based adaptive regularization method improves quality of the reconstructed Diffuse Optical Tomographic images.

Accelerated gradient based diffuse optical tomographic image reconstruction

Purpose: Fast reconstruction of interior optical parameter distribution using a new approach called Broyden-based model iterative image reconstruction (BMOBIIR) and adjoint Broyden-based MOBIIR (ABMOBIIR) of a tissue and a tissue mimicking phantom from boundary measurement data in diffuse optical tomography (DOT). Methods: DOT is a nonlinear and ill-posed inverse problem. Newton-based MOBIIR algorithm, which is generally used, requires repeated evaluation of the Jacobian which consumes bulk of the computation time for reconstruction. In this study, we propose a Broyden approach-based accelerated scheme for Jacobian computation and it is combined with conjugate gradient scheme (CGS) for fast reconstruction. The method makes explicit use of secant and adjoint information that can be obtained from forward solution of the diffusion equation. This approach reduces the computational time many fold by approximating the system Jacobian successively through low-rank updates. Results: Simulation studies have been carried out with single as well as multiple inhomogeneities. Algorithms are validated using an experimental study carried out on a pork tissue with fat acting as an inhomogeneity. The results obtained through the proposed BMOBIIR and ABMOBIIR approaches are compared with those of Newton-based MOBIIR algorithm. The mean squared error and execution time are used as metrics for comparing the results of reconstruction. Conclusions: We have shown through experimental and simulation studies that Broyden-based MOBIIR and adjoint Broyden-based methods are capable of reconstructing single as well as multiple inhomogeneities in tissue and a tissue-mimicking phantom. Broyden MOBIIR and adjoint Broyden MOBIIR methods are computationally simple and they result in much faster implementations because they avoid direct evaluation of Jacobian. The image reconstructions have been carried out with different initial values using Newton, Broyden, and adjoint Broyden approaches. These algorithms work well when the initial guess is close to the true solution. However, when initial guess is far away from true solution, Newton-based MOBIIR gives better reconstructed images. The proposed methods are found to be stable with noisy measurement data. (C) 2011 American Association of Physicists in Medicine. DOI: 10.1118/1.3531572]

Linearization and Reconstruction of Non-linear Diffuse Optical Tomographic Image

Diffuse optical tomography (DOT) is one of the ways to probe highly scattering media such as tissue using low-energy near infra-red light (NIR) to reconstruct a map of the optical property distribution. The interaction of the photons in biological tissue is a non-linear process and the phton transport through the tissue is modelled using diffusion theory. The inversion problem is often solved through iterative methods based on nonlinear optimization for the minimization of a data-model misfit function. The solution of the non-linear problem can be improved by modeling and optimizing the cost functional. The cost functional is f(x) = x(T)Ax - b(T)x + c and after minimization, the cost functional reduces to Ax = b. The spatial distribution of optical parameter can be obtained by solving the above equation iteratively for x. As the problem is non-linear, ill-posed and ill-conditioned, there will be an error or correction term for x at each iteration. A linearization strategy is proposed for the solution of the nonlinear ill-posed inverse problem by linear combination of system matrix and error in solution. By propagating the error (e) information (obtained from previous iteration) to the minimization function f(x), we can rewrite the minimization function as f(x; e) = (x + e)(T) A(x + e) - b(T)(x + e) + c. The revised cost functional is f(x; e) = f(x) + e(T)Ae. The self guided spatial weighted prior (e(T)Ae) error (e, error in estimating x) information along the principal nodes facilitates a well resolved dominant solution over the region of interest. The local minimization reduces the spreading of inclusion and removes the side lobes, thereby improving the contrast, localization and resolution of reconstructed image which has not been possible with conventional linear and regularization algorithm.

Convergence analysis of the Newton algorithm and a pseudo-time marching scheme for diffuse correlation tomography

We propose a self-regularized pseudo-time marching scheme to solve the ill-posed, nonlinear inverse problem associated with diffuse propagation of coherent light in a tissuelike object. In particular, in the context of diffuse correlation tomography (DCT), we consider the recovery of mechanical property distributions from partial and noisy boundary measurements of light intensity autocorrelation. We prove the existence of a minimizer for the Newton algorithm after establishing the existence of weak solutions for the forward equation of light amplitude autocorrelation and its Frechet derivative and adjoint. The asymptotic stability of the solution of the ordinary differential equation obtained through the introduction of the pseudo-time is also analyzed. We show that the asymptotic solution obtained through the pseudo-time marching converges to that optimal solution provided the Hessian of the forward equation is positive definite in the neighborhood of optimal solution. The superior noise tolerance and regularization-insensitive nature of pseudo-dynamic strategy are proved through numerical simulations in the context of both DCT and diffuse optical tomography. (C) 2010 Optical Society of America.

Agents Based Algorithms for Design Parameter Estimation in Contaminant Transport Inverse Problems

A considerable amount of work has been dedicated on the development of analytical solutions for flow of chemical contaminants through soils. Most of the analytical solutions for complex transport problems are closed-form series solutions. The convergence of these solutions depends on the eigen values obtained from a corresponding transcendental equation. Thus, the difficulty in obtaining exact solutions from analytical models encourages the use of numerical solutions for the parameter estimation even though, the later models are computationally expensive. In this paper a combination of two swarm intelligence based algorithms are used for accurate estimation of design transport parameters from the closed-form analytical solutions. Estimation of eigen values from a transcendental equation is treated as a multimodal discontinuous function optimization problem. The eigen values are estimated using an algorithm derived based on glowworm swarm strategy. Parameter estimation of the inverse problem is handled using standard PSO algorithm. Integration of these two algorithms enables an accurate estimation of design parameters using closed-form analytical solutions. The present solver is applied to a real world inverse problem in environmental engineering. The inverse model based on swarm intelligence techniques is validated and the accuracy in parameter estimation is shown. The proposed solver quickly estimates the design parameters with a great precision.

Estimation of soil hydraulic properties and their uncertainty: comparison between laboratory and field experiment

Often the soil hydraulic parameters are obtained by the inversion of measured data (e.g. soil moisture, pressure head, and cumulative infiltration, etc.). However, the inverse problem in unsaturated zone is ill-posed due to various reasons, and hence the parameters become non-unique. The presence of multiple soil layers brings the additional complexities in the inverse modelling. The generalized likelihood uncertainty estimate (GLUE) is a useful approach to estimate the parameters and their uncertainty when dealing with soil moisture dynamics which is a highly non-linear problem. Because the estimated parameters depend on the modelling scale, inverse modelling carried out on laboratory data and field data may provide independent estimates. The objective of this paper is to compare the parameters and their uncertainty estimated through experiments in the laboratory and in the field and to assess which of the soil hydraulic parameters are independent of the experiment. The first two layers in the field site are characterized by Loamy sand and Loamy. The mean soil moisture and pressure head at three depths are measured with an interval of half hour for a period of 1 week using the evaporation method for the laboratory experiment, whereas soil moisture at three different depths (60, 110, and 200 cm) is measured with an interval of 1 h for 2 years for the field experiment. A one-dimensional soil moisture model on the basis of the finite difference method was used. The calibration and validation are approximately for 1 year each. The model performance was found to be good with root mean square error (RMSE) varying from 2 to 4 cm(3) cm(-3). It is found from the two experiments that mean and uncertainty in the saturated soil moisture (theta(s)) and shape parameter (n) of van Genuchten equations are similar for both the soil types. Copyright (C) 2010 John Wiley & Sons, Ltd.

Swam Intelligent Based Inverse Model for Laboratory Double Reservoir Diffusion Experiments

Theoretical approaches are of fundamental importance to predict the potential impact of waste disposal facilities on ground water contamination. Appropriate design parameters are, in general, estimated by fitting the theoretical models to a field monitoring or laboratory experimental data. Double-reservoir diffusion (Transient Through-Diffusion) experiments are generally conducted in the laboratory to estimate the mass transport parameters of the proposed barrier material. These design parameters are estimated by manual parameter adjusting techniques (also called eye-fitting) like Pollute. In this work an automated inverse model is developed to estimate the mass transport parameters from transient through-diffusion experimental data. The proposed inverse model uses particle swarm optimization (PSO) algorithm which is based on the social behaviour of animals for finding their food sources. Finite difference numerical solution of the transient through-diffusion mathematical model is integrated with the PSO algorithm to solve the inverse problem of parameter estimation.The working principle of the new solver is demonstrated by estimating mass transport parameters from the published transient through-diffusion experimental data. The estimated values are compared with the values obtained by existing procedure. The present technique is robust and efficient. The mass transport parameters are obtained with a very good precision in less time

A numerical study of variable coefficient elliptic Cauchy problem via projection method

In this paper, we investigate a numerical method for the solution of an inverse problem of recovering lacking data on some part of the boundary of a domain from the Cauchy data on other part for a variable coefficient elliptic Cauchy problem. In the process, the Cauchy problem is transformed into the problem of solving a compact linear operator equation. As a remedy to the ill-posedness of the problem, we use a projection method which allows regularization solely by discretization. The discretization level plays the role of regularization parameter in the case of projection method. The balancing principle is used for the choice of an appropriate discretization level. Several numerical examples show that the method produces a stable good approximate solution.

Minimal residual method provides optimal regularization parameter for diffuse optical tomography

The inverse problem in the diffuse optical tomography is known to be nonlinear, ill-posed, and sometimes under-determined, requiring regularization to obtain meaningful results, with Tikhonov-type regularization being the most popular one. The choice of this regularization parameter dictates the reconstructed optical image quality and is typically chosen empirically or based on prior experience. An automated method for optimal selection of regularization parameter that is based on regularized minimal residual method (MRM) is proposed and is compared with the traditional generalized cross-validation method. The results obtained using numerical and gelatin phantom data indicate that the MRM-based method is capable of providing the optimal regularization parameter. (C) 2012 Society of Photo-Optical Instrumentation Engineers (SPIE). DOI: 10.1117/1.JBO.17.10.106015]

Multidimensional data reconstruction for two color fluorescence microscopy

We propose an iterative data reconstruction technique specifically designed for multi-dimensional multi-color fluorescence imaging. Markov random field is employed (for modeling the multi-color image field) in conjunction with the classical maximum likelihood method. It is noted that, ill-posed nature of the inverse problem associated with multi-color fluorescence imaging forces iterative data reconstruction. Reconstruction of three-dimensional (3D) two-color images (obtained from nanobeads and cultured cell samples) show significant reduction in the background noise (improved signal-to-noise ratio) with an impressive overall improvement in the spatial resolution (approximate to 250 nm) of the imaging system. Proposed data reconstruction technique may find immediate application in 3D in vivo and in vitro multi-color fluorescence imaging of biological specimens. (C) 2012 American Institute of Physics. http://dx.doi.org/10.1063/1.4769058]