Abstract
The TOPINV, Topological Inversion algorithm (or TGS, Topological Grid Search) initially developed for the inversion of highly non-linear redundant systems of equations, can solve a wide range of underdetermined systems of non-linear equations. This approach is a generalization of a previous conclusion that this algorithm can be used for the solution of certain integer ambiguity problems in Geodesy.The overall approach is based on additional (a priori) information for the unknown variables. In the past, such information was used either to linearize equations around approximate solutions, or to expand systems of observation equations solved on the basis of generalized inverses. In the proposed algorithm, the a priori additional information is used in a third way, as topological constraints to the unknown n variables, leading to an Rn grid containing an approximation of the real solution.The TOPINV algorithm does not focus on point-solutions, but exploits the structural and topological constraints in each system of underdetermined equations in order to identify an optimal closed space in the Rn containing the real solution. The centre of gravity of the grid points defining this space corresponds to global, minimum-norm solutions. The rationale and validity of the overall approach are demonstrated on the basis of examples and case studies, including fault modelling, in comparison with SVD solutions and true (reference) values, in an accuracy-oriented approach.
Highlights
Redundant systems of non-linear equations with n variables and m measurements (m > n) are frequent in various fields of science and engineering, but there does not exist a unique or general method for their solution
In this article we present a generalization of the idea of Harvey (2013) that TOPINV can cover a wide range of underdetermined systems of equations observed in engineering and geophysics
Alternative solution of underdetermined systems of equations Among the possible solutions for the system of Figure 3, an unconstrained Single Value Decomposition (SVD) would lead to a solution with equal diagonals, a solution characterized by minimum norm in the differences of the coordinates of the corner points
Summary
Redundant systems of non-linear equations with n variables and m measurements (m > n) are frequent in various fields of science and engineering, but there does not exist a unique or general method for their solution. TOPINV (TGS) introduces a third type of additional/a priori information that can be used for the solution of systems of equations This information corresponds to constraining the expected solution of each of the unknown variables to a certain range of possible values and defining a grid G in the Rn space. Alternative solution of underdetermined systems of equations Among the (infinite) possible solutions for the system of Figure 3, an unconstrained SVD would lead to a solution with equal diagonals, a solution characterized by minimum norm in the differences of the coordinates of the corner points This solution, requires linearization of the non-linear equations, and this requires additional (a priori) information (or constraints) for the unknown variables (coordinates). This process was repeated four times, and the final solution S was identified, and the estimates of the 9 variables, along with their variances, were computed
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
