Constrained linear inversion

Twomey, S., Comparison of Constrained Linear Inversion and an Iterative Nonlinear Algorithm Applied to the Indirect Estimation of the Particle Size Distribution, J. Comp. Phys. 18 188-200 (1975). 

Steele, H. M., and R. P. Turco, Retrieval of Aerosol Size Distributions from Satellite Extinction Spectra Using Constrained Linear Inversion, J. Geophys. Res., 102, 16737-16747 (1997a). 

A retrieval formulation developed by Backus Gilbert (1970) and used in geophysical inverse problems of the solid Earth has also found application in remote sensing of atmospheres. Although this formulation is developed from a somewhat different point of view, it is formally related to the constrained linear inversion as well as other techniques. We include a brief description here because of the additional physical insight it can provide into inversion theory. 

The constrained deconvolution algorithm produces estimates that cannot be obtained from the data by simple linear inverse filtering. This is most readily seen using the Blass-Halsey weight function as an example. 

As described in more detail below, the inverse approach generally leads to underdetermined mathematical systems that are much harder to solve than the systems encountered in forward models. Error and resolution analysis are two issues of particular importance when solving underdetermined inverse problems. First, the solved-for physical and biogeochemical parameters depend directly on the tracer data, and errors in the data propagate into errors in the solution. Second, owing to the incompleteness of information in underdetermined systems, the unknowns are usually not fully resolved. Instead, only specific linear combinations of unknowns may be well constrained by the data, while individual unknowns or other combinations of unknowns may remain poorly determined. Both, error and resolution analysis are essential for a quality assessment of the solution of underdetermined systems. 

Finally, linear equations (75b) and (79) can be solved by different methods. For example, using inverse matrices reduce Eqs. (75) and (80) to a traditional form of constrained inversion. Alternatively (Section 4.8), other numerical or computer techniques, such as, SVD, conjugated gradients, iterations, generic inversion, etc. can be used for solving Eqs. (75b) and (80). 

Given the recursive dynamic equations for a constrained chain, we will now begin the development of a linear recursive algcxithm fw A the inverse operational space inertia matrix of a single chain. First, we will define a new quantity, (A ) an inertial matrix which relates the spatial acceleration of a link and the propagated spatial contact force exoted at the tip of the same link. We may write a defining equation for this matrix (at link t) as follows ... 

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