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Register a new userA Conjugate-Gradient Approach to the Parameter Estimation Problem of Magnetic Resonance Advection Imaging
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Added by
Simon Hubmer
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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We consider the inverse problem of estimating the spatially varying pulse wave velocity in blood vessels in the brain from dynamic MRI data, as it appears in the recently proposed imaging technique of Magnetic Resonance Advection Imaging (MRAI). The problem is formulated as a linear operator equation with a noisy operator and solved using a conjugate gradient type approach. Numerical examples on experimental data show the usefulness and advantages of the developed algorithm in comparison to previously proposed methods.
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In this paper, we extend to the block case, the a posteriori bound showing superlinear convergence of Conjugate Gradients developed in [J. Comput. Applied Math., 48 (1993), pp. 327- 341], that is, we obtain similar bounds, but now for block Conjugate Gradients. We also present a series of computational experiments, illustrating the validity of the bound developed here, as well as the bound from [SIAM Review, 47 (2005), pp. 247-272] using angles between subspaces. Using these bounds, we make some observations on the onset of superlinearity, and how this onset depends on the eigenvalue distribution and the block size.
The trace of a matrix function f(A), most notably of the matrix inverse, can be estimated stochastically using samples< x,f(A)x> if the components of the random vectors x obey an appropriate probability distribution. However such a Monte-Carlo sampling suffers from the fact that the accuracy depends quadratically of the samples to use, thus making higher precision estimation very costly. In this paper we suggest and investigate a multilevel Monte-Carlo approach which uses a multigrid hierarchy to stochastically estimate the trace. This results in a substantial reduction of the variance, so that higher precision can be obtained at much less effort. We illustrate this for the trace of the inverse using three different classes of matrices.
This paper proposes a novel, rigorous and simple Fourier-transformation approach to study resonances in a perfectly conducting slab with finite number of subwavelength slits of width $hll 1$. Since regions outside the slits are variable separated, by Fourier transforming the governing equation, we could express field in the outer regions in terms of field derivatives on the aperture. Next, in each slit where variable separation is still available, wave field could be expressed as a Fourier series in terms of a countable basis functions with unknown Fourier coefficients. Finally, by matching field on the aperture, we establish a linear system of infinite number of equations governing the countable Fourier coefficients. By carefully asymptotic analysis of each entry of the coefficient matrix, we rigorously show that, by removing only a finite number of rows and columns, the resulting principle sub-matrix is diagonally dominant so that the infinite dimensional linear system can be reduced to a finite dimensional linear system. Resonance frequencies are exactly those frequencies making the linear system rank-deficient. This in turn provides a simple, asymptotic formula describing resonance frequencies with accuracy ${cal O}(h^3log h)$. We emphasize that such a formula is more accurate than all existing results and is the first accurate result especially for slits of number more than two to our best knowledge. Moreover, this asymptotic formula rigorously confirms a fact that the imaginary part of resonance frequencies is always ${cal O}(h)$ no matter how we place the slits as long as they are spaced by distances independent of width $h$.
State estimation aims at approximately reconstructing the solution $u$ to a parametrized partial differential equation from $m$ linear measurements, when the parameter vector $y$ is unknown. Fast numerical recovery methods have been proposed based on reduced models which are linear spaces of moderate dimension $n$ which are tailored to approximate the solution manifold $mathcal{M}$ where the solution sits. These methods can be viewed as deterministic counterparts to Bayesian estimation approaches, and are proved to be optimal when the prior is expressed by approximability of the solution with respect to the reduced model. However, they are inherently limited by their linear nature, which bounds from below their best possible performance by the Kolmogorov width $d_m(mathcal{M})$ of the solution manifold. In this paper we propose to break this barrier by using simple nonlinear reduced models that consist of a finite union of linear spaces $V_k$, each having dimension at most $m$ and leading to different estimators $u_k^*$. A model selection mechanism based on minimizing the PDE residual over the parameter space is used to select from this collection the final estimator $u^*$. Our analysis shows that $u^*$ meets optimal recovery benchmarks that are inherent to the solution manifold and not tied to its Kolmogorov width. The residual minimization procedure is computationally simple in the relevant case of affine parameter dependence in the PDE. In addition, it results in an estimator $y^*$ for the unknown parameter vector. In this setting, we also discuss an alternating minimization (coordinate descent) algorithm for joint state and parameter estimation, that potentially improves the quality of both estimators.
In this paper, we consider the imaging problem of terahertz (THz) tomography, in particular as it appears in non-destructive testing. We derive a nonlinear mathematical model describing a full THz tomography experiment, and consider linear approximations connecting THz tomography with standard computerized tomography and the Radon transform. Based on the derived models we propose different reconstruction approaches for solving the THz tomography problem, which we then compare on experimental data obtained from THz measurements of a plastic sample.
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