Linear discrete inverse problems

Thus, we can see that in the case of a linear discrete inverse problem the operator equation (3.1) is reduced to the matrix equation (3.2). To solve this equation we have to use some formulae and rules from matrix algebra, described in Appendix E. 

JW. Enquire, WE. Deeds, and CV. Dodd. Alternating current distribution between planar conductors. Journal of Applied Physics, 41(10) 3983-3991, September 1970. C. De Mol M. Bertero and E.R. Pike. Linear inverse problems with discrete data. li. stability and regularization. Inverse Problems, 4 pp. 573-594, 1987. 

We have demonstrated in the previous sections that the solution of a gravity or magnetic field inverse problem in discrete form is reduced to a linear matrix equation. Therefore, the full arsenal of solutions to the linear inverse problem, developed in Part... 

To reduce the requirements of long experimental data records and improve the kernel estimation accuracy, least-squares methods also can be used to solve the classical linear inverse problem described earlier in Equation 13.6, where the parameter vector 9 includes all discrete kernel values of the finite Volterra model of Equation 13.17, which is Knear in these unknown parameters (i.e., kernel values). Least-squares methods also can be used in connection with orthogonal expansions of the kernels to reduce the number of unknown parameters, as outlined below. Note that solution of this inverse problem via OLS requires inversion of a large square matrix with dimensions [(M -I- f -I- 1) /((M -F 1) / )], where M is... 

Hansen, P.C., Rank-Deficient and Discrete Ill-Posed Problems Numerical Aspects of Linear Inversion, SIAM, Philadelphia, 1998. 

Hansen, C., 1998, Rank-deficient and discrete ill-posed problems. Numerical aspects of linear inversion Department of mathematical modeling. Technical University of Denmark, Lyngby, 247 pp. 

This is the primary means of obtaining information about the canopy source distribution of a scalar from atmospheric concentration measurements. A formal discrete solution is found by matrix inversion of Et]. (17), choosing the number of source layers (m) to be ecjual to the number of concentration measurements (n) so that D j is a scjuare matrix. However, this solution provides no redundancy in concentration information, and therefore no possibility for smoothing measurement errors in the concentration profile, which can cause large errors in the inferred source profile. A simple means of overcoming this problem is to include redundant concentration information, and then find the sources , which produce the best fit to the measured concentrations c, by maximum-likelihood estimation. By minimizing the squared error between measured values and concentrations predicted by Eq. (17), 4>j is found (Raupach, 1989b) to be the solution of m linear ec[uations... 

One approach to the problem of retrieval of gas profiles by inversion is linearization and iteration. The unknown profile is expressed in terms of a set of discrete parameters these may consist of the values of the gas mole fraction at the quadrature points used in the numerical integrations required to calculate the radiance... 

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