Brown University Graduate School Dissertation Defense Information
Scientific Computing Seminar
Abstract: Estimating the entries of a relaxation, or system, matrix is today among the most important computational problems in NMR spectrography. The reason is that a relaxation matrix is related to the distances between different nuclei of a given protein whose 3D structure determination is essential for either medical, chemical or pharmacological reasons.
This talk describes a new method I developed for a reliable estimation of the system matrix and the initial conditions from the set of noisy measurements. The method avoids the ill-conditioning associated with a multiexponential fitting, or the ambiguities associated with evaluating the logarithm of a nonpositive definite matrix.
The method is based on a nonlinear least squares criterion that incorporates the system relaxation dynamics and the initial conditions as constraints. To avoid multiple minima and speed up the convergence to the global one, I formulate and solve two simpler linear least squares problems, one for the matrizant and the other for the initial conditions.
The method has been extensively tested on a variety of 2x2 systems for which simulated data have been generated with 50% error. It has also been successfully tested on severely corrupted measurements of the amide groups in the 15-N-labeled protein oxidized flavodoxin. The conclusion is that if there are eight or more equally spaced measurements spanning the intervals around the maximum and the inflection points of the so called off-diagonal curve (in a symmetrical 2x2 system), then the relative errors in the matrix entries are generally less then the relative errors in the measurements.
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