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20
Adaptive hybrid FEM/FDM methods for inverse scattering problems
 Department of Mathematics; Chalmers University of Technology & Goteborg University
, 2002
"... This thesis is devoted to adaptive hybrid finite element / finite difference methods for an inverse scattering problem for the timedependent acoustic wave equation in 2D and 3D, where we seek to reconstruct an unknown sound velocity c(x) from measured wavereflection data. ..."
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Cited by 24 (19 self)
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This thesis is devoted to adaptive hybrid finite element / finite difference methods for an inverse scattering problem for the timedependent acoustic wave equation in 2D and 3D, where we seek to reconstruct an unknown sound velocity c(x) from measured wavereflection data.
Reconstruction of dielectrics from experimental data via a hybrid globally convergent/adaptive inverse algorithm, Inverse Problems, accepted for publication
, 2010
"... The validity of a synthesis of a globally convergent numerical method with the adaptive FEM technique for a coefficient inverse problem is verified on time resolved experimental data. Refractive indices, locations and shapes of dielectric abnormalities are accurately imaged. 1 ..."
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Cited by 15 (6 self)
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The validity of a synthesis of a globally convergent numerical method with the adaptive FEM technique for a coefficient inverse problem is verified on time resolved experimental data. Refractive indices, locations and shapes of dielectric abnormalities are accurately imaged. 1
Adaptive finite volume methods for distributed nonsmooth parameter identification
, 2007
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Discretization error analysis and adaptive meshing algorithms for fluorescence diffuse optical tomography: Part II
 IEEE Trans. Med. Imag
"... Abstract—For imaging problems in which numerical solutions need to be computed for both the inverse and the underlying forward problems, discretization can be a major factor that determines the accuracy of imaging. In this work, we analyze the effect of discretization on the accuracy of fluorescence ..."
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Cited by 4 (1 self)
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Abstract—For imaging problems in which numerical solutions need to be computed for both the inverse and the underlying forward problems, discretization can be a major factor that determines the accuracy of imaging. In this work, we analyze the effect of discretization on the accuracy of fluorescence diffuse optical tomography. We model the forward problem by a pair of diffusion equations at the excitation and emission wavelengths and consider a finite element discretization method for the numerical solution of the forward problem. For the inverse problem, we use an optimization framework which allows incorporation of a priori information in the form of zeroth and firstorder Tikhonov regularization terms. Next, we convert the inverse problem into a variational problem and use Galerkin projection to discretize the inverse problem. Following the discretization, we analyze the error in reconstructed images due to the discretization of the forward and inverse problems and present two theorems which point out the factors that may lead to high error such as the mutual dependence of the forward and inverse problems, the number of sources and detectors, their configuration and their positions with respect to fluorophore concentration, and the formulation of the inverse problem. Finally, we demonstrate the results and implications of our error analysis by numerical experiments. In the second part of the paper, we apply our results to design novel adaptive discretization algorithms. Index Terms—Adaptive meshing algorithms, error analysis, fluorescence diffuse optical tomography. I.
Discretization Error Analysis and Adaptive Meshing Algorithms for Fluorescence Diffuse Optical Tomography: Part II
"... Abstract—In the first part of this work, we analyze the effect of discretization on the accuracy of fluorescence diffuse optical tomography (FDOT). Our error analysis provides two new error estimates which present a direct relationship between the error in the reconstructed fluorophore concentration ..."
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Cited by 3 (1 self)
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Abstract—In the first part of this work, we analyze the effect of discretization on the accuracy of fluorescence diffuse optical tomography (FDOT). Our error analysis provides two new error estimates which present a direct relationship between the error in the reconstructed fluorophore concentration and the discretization of the forward and inverse problems. In this paper, based on these error estimates, we develop two new adaptive mesh generation algorithms for the numerical solutions of the forward and inverse problems in FDOT, with the objective of error reduction in the reconstructed optical images due to discretization while keeping the size of the discretized forward and inverse problems within the allowable limits. We present threedimensional numerical simulations to demonstrate the improvements in accuracy, resolution and detectability of small heterogeneities in reconstructed images provided by the use of the new adaptive mesh generation algorithms. Finally, we compare our algorithms both analytically and numerically with the existing conventional adaptive mesh generation algorithms. Index Terms—Adaptive meshing algorithms, error analysis, fluorescence diffuse optical tomography. I.
1 BLIND BACKSCATTERING EXPERIMENTAL DATA COLLECTED IN THE FIELD AND AN APPROXIMATELY GLOBALLY CONVERGENT INVERSE ALGORITHM
"... An approximately globally convergent numerical method for a 1D Coefficient Inverse Problem for a hyperbolic PDE is applied to image dielectric constants of targets from blind experimental data. The data were collected in the field by the Forward Looking Radar of the US Army Research Laboratory. A p ..."
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Cited by 3 (1 self)
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An approximately globally convergent numerical method for a 1D Coefficient Inverse Problem for a hyperbolic PDE is applied to image dielectric constants of targets from blind experimental data. The data were collected in the field by the Forward Looking Radar of the US Army Research Laboratory. A posteriori analysis has revealed that computed and tabulated values of dielectric constants are in a good agreement. Convergence analysis is presented.
Convergence of an adaptive finite element method for distributed flux reconstruction
"... We shall establish the convergence of an adaptive conforming finite element method for the reconstruction of the distributed flux in a diffusion system. The adaptive method is based on a posteriori error estimators for the distributed flux, state and costate variables. The sequence of discrete solut ..."
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Cited by 2 (1 self)
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We shall establish the convergence of an adaptive conforming finite element method for the reconstruction of the distributed flux in a diffusion system. The adaptive method is based on a posteriori error estimators for the distributed flux, state and costate variables. The sequence of discrete solutions produced by the adaptive algorithm is proved to converge to the true triplet satisfying the optimality conditions in the energy norm and the corresponding error estimator converges to zero asymptotically.
A globally convergent numerical method and adaptivity for a hyperbolic coefficient inverse problem
, 2009
"... A globally convergent numerical method for a multidimensional Coefficient Inverse Problem for a hyperbolic equation is presented.
It is shown that this technique provides a good starting point for the socalled finite element adaptive method (adaptivity).
This leads to a natural twostage numerica ..."
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A globally convergent numerical method for a multidimensional Coefficient Inverse Problem for a hyperbolic equation is presented.
It is shown that this technique provides a good starting point for the socalled finite element adaptive method (adaptivity).
This leads to a natural twostage numerical procedure, which synthesizes both these methods.
Numerical examples are presented.
Discretization Error Analysis and 1 Adaptive Meshing Algorithms for Fluorescence Diffuse Optical Tomography in the Presence of Measurement Noise
, 2010
"... Quantitatively accurate Fluorescence Diffuse Optical Tomographic (FDOT) image reconstruction is a computationally demanding problem that requires repeated numerical solutions of two coupled partial differential equations and an associated inverse problem. Recently, adaptive finite element methods ha ..."
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Quantitatively accurate Fluorescence Diffuse Optical Tomographic (FDOT) image reconstruction is a computationally demanding problem that requires repeated numerical solutions of two coupled partial differential equations and an associated inverse problem. Recently, adaptive finite element methods have been explored to reduce the computation requirements of the FDOT image reconstruction. However, existing approaches ignore the ubiquitous presence of noise in boundary measurements. In this paper, we analyze the effect of finite element discretization on the FDOT forward and inverse problems in the presence of measurement noise and develop novel adaptive meshing algorithms for FDOT that take into account noise statistics. We formulate the FDOT inverse problem as an optimization problem in the maximum a posteriori framework to estimate the fluorophore concentration in a bounded domain. We use the MeanSquareError (MSE) between the exact solution and the discretized solution as a figure of merit to evaluate the image reconstruction accuracy, and derive an upper bound on the MSE which depends on the forward and inverse problem discretization parameters, noise statistics, a priori information of fluorophore concentration, source and detector geometry, as well as background optical properties. Next, we use this error bound to develop adaptive meshing algorithms for the FDOT forward and inverse problems to reduce the MSE due to discretization in the reconstructed images. Finally, we present a set of numerical simulations to illustrate the practical advantages of our adaptive meshing algorithms for FDOT image reconstruction.
A Posteriori Error Estimates for the Coupling Equations of Scalar Conservation Laws
, 2009
"... In this paper we prove a posteriori L2(L2) and L∞(H −1) residual based error estimates for a finite element method for the onedimensional time dependent coupling equations of two scalar conservation laws. The underlying discretization scheme is Characteristic Galerkin method which is the particul ..."
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In this paper we prove a posteriori L2(L2) and L∞(H −1) residual based error estimates for a finite element method for the onedimensional time dependent coupling equations of two scalar conservation laws. The underlying discretization scheme is Characteristic Galerkin method which is the particular variant of the Streamline diffusion finite element method for δ = 0. Our estimate contains certain strong stability factors related to the solution of an associated linearized dual problem combined with the Galerkin orthogonality of the finite element method. The stability factor measures the stability properties of the linearized dual problem. We compute the stability factors for some examples by solving the dual problem numerically.