Numerical Linear Algebra: Theory and Applications

Abstract

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Microwave thermometry with potential application in non-invasive monitoring of hyperthermia

“…These figures show that the optimal λ was found on the interval λ ∈ [0.46, 0.48] on all refined meshes. Then the AFEM computations were performed in the software package WavES [34] with β = 0.5 in (3.11) which allows to refine meshes locally at places where the inclusion was located; see Figures 4,5. In the transverse plane, Figure 4 compares the original reconstructions on the unrefined initial FEM mesh with that of AFEM on locally adapted meshes at each time step. One can see from these plots how the density of the mesh changes around the border of the target and how this change becomes more and more pronounced as the contrast grows bigger and bigger due to thermal changes of the target as time goes by.…”

“…Although the method of normal equations is the fastest method for solving the linear least-squares problems (LLSP), it is not as accurate as QR or SVD decompositions. For instance, the method of normal equations is applied to the solution of LLSP when the condition number of the matrix A is not large [5]. Alternatively, using SVD of the matrix A = UΣV T and substituting it into (2.6), one can obtain a much more robust solution for m as follows:…”

“…and m * is the "ideal" exact solution which cannot be found in real experiments but we assume that it exists; see details in [1,2,18,20]. In [1,5], it is shown that condition (2.8) is satisfied when the following iterative update of the regularization parameters λ k is considered:…”

A General Method to Compute Numerical Dispersion Error

“…Defining the discrete differential operator as A, and splitting it into the Hermitian, D, and skew-Hermitian, C, parts [23],…”

High Level Design of a Flexible PCA Hardware Accelerator Using a New Block-Streaming Method

“…The first one is the floating-point version, which uses the built-in HLS function for SVD computation. The second one is the fixed-point version, which uses the general two-sided Jacobi method for EVD computation [22,23]. The details of this method for computing EVD is shown in Algorithm 4.…”