Separable nonlinear least squares fitting with linear bound constraints and its application in magnetic resonance spectroscopy data quantification

• Published in: Journal of computational and applied mathematics Vol. 203; no. 1; pp. 264 - 278
• Main Authors: Sima, Diana M., Van Huffel, Sabine
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.06.2007; Elsevier
• Subjects: Applied sciences; Exact sciences and technology; Magnetic resonance spectroscopy data quantification; Mathematical programming; Mathematics; Nonlinear least squares; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Operational research and scientific management; Operational research. Management science; Sciences and techniques of general use; Variable projection; 92C55; Nonlinear least squares; 90C30; Magnetic resonance spectroscopy data quantification; 65F99; Variable projection; Fitting; Least squares problem; Magnetic resonance; Minimization; Nonlinear problems; Quadratic programming; Nuclear magnetic resonance imaging; Constrained optimization; Numerical analysis; Applied mathematics; Least squares method; Magnetic resonance spectroscopy data quantification; Nonlinear least squares; Variable projection; 90C30;92C55;65F99; Jacobian function; Spectroscopy data quantification
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2006.03.025

An application in magnetic resonance spectroscopy quantification models a signal as a linear combination of nonlinear functions. It leads to a separable nonlinear least squares fitting problem, with linear bound constraints on some variables. The variable projection (VARPRO) technique can be applied to this problem, but needs to be adapted in several respects. If only the nonlinear variables are subject to constraints, then the Levenberg–Marquardt minimization algorithm that is classically used by the VARPRO method should be replaced with a version that can incorporate those constraints. If some of the linear variables are also constrained, then they cannot be projected out via a closed-form expression as is the case for the classical VARPRO technique. We show how quadratic programming problems can be solved instead, and we provide details on efficient function and approximate Jacobian evaluations for the inequality constrained VARPRO method.