Solution of underdetermined systems of equations with gridded a priori constraints

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.

Time‐space modeling of the dynamics of Santorini volcano (Greece) during the 2011–2012 unrest

“…Modeling of the intrusion using GPS data was based on the assumption of spherical magma sources (Mogi models), which are described by well‐known equations [ Mogi , ] and are valid under certain conditions [ McTigue , ; Pascal et al ., ]. The overall analysis was based on the TOPINV (Topological Inversion) algorithm, first validated for the adjustment of common geodetic networks and explicitly analyzed in Saltogianni and Stiros [, , ] and in Stiros and Saltogianni []. This algorithm permits simultaneous inversion of all n unknown variables in a redundant system of highly nonlinear equations for modeling single and double Mogi sources on the basis of a quasi‐deterministic approach (“scanning” a n ‐dimensional grid of all possible values of the unknown variable allowing an error margin) without any need to a priori constrain certain of its components.…”

Target-depth estimation in active sonar: Cramer–Rao bounds for a bilinear sound-speed profile

“…To increase the probability to have an intersection between the uncertainty ellipses, two scale factors, k 1 and k 2 , are introduced, to expand the uncertainty margins. 22 The uncertainties of the measurements can be re-written as follows:…”