Adaptive truncation of matrix decompositions and efficient estimation of NMR relaxation distributions

Abstract

The two most successful methods of estimating the distribution of nuclear magnetic resonance relaxation times from two dimensional data are data compression followed by application of the Butler-Reeds-Dawson algorithm, and a primal-dual interior point method using preconditioned conjugate gradient. Both of these methods have previously been presented using a truncated singular value decomposition of matrices representing the exponential kernel. In this paper it is shown that other matrix factorizations are applicable to each of these algorithms, and that these illustrate the different fundamental principles behind the operation of the algorithms. These are the rank-revealing QR (RRQR) factorization and the LDL factorization with diagonal pivoting, also known as the Bunch-Kaufman-Parlett factorization. It is shown that both algorithms can be improved by adaptation of the truncation as the optimization process progresses, improving the accuracy as the optimal value is approached. A variation on the interior method viz, the use of barrier function instead of the primal-dual approach, is found to offer considerable improvement in terms of speed and reliability. A third type of algorithm, related to the algorithm known as Fast iterative shrinkage-thresholding algorithm, is applied to the problem. This method can be efficiently formulated without the use of a matrix decomposition.

Identification of extracellular nanoparticle subsets by nuclear magnetic resonance

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“…The algorithm based on adaptive truncation of matrix decompositions was used for Laplace inversion. 32 The value of parameter a scaling the weight of the Tikhonov regularization was adjusted in a standard way by running the a loop.…”

“…The milk signal is decaying much slower than the EKC and RCC signals, indicating larger average particle size in the former case. The decay curves were converted into diffusion coefficient (D) distributions by the inverse Laplace transform (ILT) 32,37,38. Fig.…”

Ultrafast Laplace NMR to study metal–ligand interactions in reversible polarisation transfer from parahydrogen

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Nuclear magnetic resonance footprint of Wharton Jelly mesenchymal stem cells death mechanisms and distinctive in‐cell biophysical properties in vitro

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