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Manuscript Title: CONTIN: a general purpose constrained regularization program for inverting noisy linear algebraic and integral equations. See also Comp. Phys. Commun. 27(1982)213.
Authors: S.W. Provencher
Program title: CONTIN (VERSION 2DP)
Catalogue identifier: AAOB_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 27(1982)229
Programming language: Fortran.
Computer: VAX 11/780.
Operating system: VMS.
RAM: 200K words
Word size: 8
Keywords: General purpose, Fit, Regularization, Inequality constraints, Ill-posed, Inverse problems, Integral equations, Quadratic programming, Deconvolution, Information content, Superresolution, Photon correlation, Constrained least square.
Classification: 4.9.

Nature of problem:
Many experiments are indirect in that the observed data are linear integral (or matrix) transforms of the quantities to be estimated. These transforms typically arise because of the imperfect impulse response of the detection system or because of the indirect nature of the experiment itself (as with Fourier transforms in diffraction and Laplace transforms in relaxation experiments). The inversion of these linear operator equations are generally ill-posed problems in that there exists a large number of possible solutions (with arbitrarily large deviations from each other) all of which fit the data to within experimental error. Therefore straightforward inversion procedures cannot be used and statistical regularization techniques are necessary.

Solution method:
A general purpose constrained regularization method finds the simplest (most parismonious) solution that is consistent with prior knowledge and the experimental data. The problem is formulated as a weighted least squares problem with an added quadratic form, the regularizor, which imposes parismony (typically smoothness) or statistical prior knowledge. Numerically stable orthogonal decomposition and quadratic programming algorithms are used to obtain the unique global solution subject to any linear equality or inequality constraints imposed by prior knowledge (e.g. nonnegativity). The regularization parameter can be automatically chosen on the basis of an F-test and confidence regions.

Restrictions:
Part of the computation time is proportional to the cube of the number of parameters used to represent the solution. Computations with no more than about 100 parameters can be done economically. This is usually more than adequate for solutions in one dimension, but not for two- or three-dimensional solutions.

Unusual features:
CONTIN has been designed to be easily adaptable to a wide variety of problems, but still easy to use. It consists of a fixed core of 53 subprograms plus simple and throughly documented "USER" subprograms that define nearly all aspects of the problem. These USER subprograms usually do not have to be changed, but they can be easily modified to define non-standard integral equations, data preprocessing, simulations, constraints, regularizors, statistical weighting, output, etc.