Adjustment of highly non-linear redundant systems of equations using a numerical, topology-based approach

Abstract

The adjustment of systems of highly non-linear, redundant equations f j , deriving from observations of certain geophysical processes and geodetic data cannot be based on conventional least-squares techniques, and is based on various numerical inversion techniques. Still these techniques lead to solutions trapped in local minima, to correlated estimates and to solution with poor error control. To overcome these problems, we propose an alternative numerical-topological approach inspired by lighthouse beacon navigation, usually used in 2-D, low-accuracy applications. In our approach, an m-dimensional grid G of points around the real solution (an m-dimensional vector) is at first specified. Then, for each equation f j an uncertainty " j is assigned to the corresponding measurement l j , and the sets A j G of the grid points which satisfy the condition jf j l j j < " j are detected. This process is repeated for all equations, and the common section A of the sets of grid points A j is defined. From this set of grid points, which define a space including the real solution, we compute its center of weight, which corresponds to an estimate of the solution, and its variance-covariance matrix. An optimal solution can be obtained through optimization of the uncertainty " j in each observation. The efficiency of the overall process was assessed in comparison with conventional least squares adjustment.

Fault slip source models for the 2014Mw6.9 Samothraki‐Gökçeada earthquake (North Aegean Trough) combining geodetic and seismological observations

“…For computational reasons, the number of grid points in G is typically limited to 10 10 , and once possible solutions are identified, the algorithm is repeated adopting finer, nested grids around the possible solutions. The details of this algorithm and its assessment using synthetic data are analyzed in Saltogianni and Stiros [, ] and in Saltogianni et al . [].…”

Health monitoring sensor placement optimization for Canton Tower using immune monkey algorithm

“…Therefore, IMA not only can keep random global search ability, but also avoid local premature convergence, which is more suitable for sensor placement optimization.Numerical studies have been carried out to assess and also demonstrate the efficacy of the proposed IMA by considering the Canton Tower with or without the antenna mast. The results presented in this paper clearly suggest that the proposed IMA is effective in suggesting the optimal sensor configuration for the given structure, while the IMA monotonically converges to the global optimal solution, the original MA oscillates around its local optimal solution.In addition, what need to be noted is that except for heuristic algorithms, rather deterministic algorithms, which may also take into account measurement errors have been recently proposed, and they can solve (i.e., invert) highly non‐linear systems of redundant equations and lead to solutions not trapped to local minima , which can be studied in the future.…”

Solution of underdetermined systems of equations with gridded a priori constraints

“…4 ; Figure 5 c). The centre of gravity of their intersection practically coincides with the Best Linear Unbiased Estimator (BLUE-type estimate) of M. This approach explains the function of the TOPINV algorithm (Saltogianni and Stiros 2012a , b ; Harvey 2013 ). The variance (quality, uncertainty) of the estimator, however depends on the prior information for point M, i.e.…”

“…A topological inversion technique for the solution of redundant systems of non-linear equations with n unknowns has recently been presented by Saltogianni and Stiros ( 2012b ; 2013 ) and was further assessed by Harvey ( 2013 ). This technique, thereafter called TOPINV (from Topological Inversion, or TGS, Topological Grid Search), exploits the power of modern computers and is based on the principle of intersection of geometric loci in the R n space.…”

“…The TOPINV (or TGS) algorithm is fully explained in Saltogianni and Stiros ( 2012b ; 2013 ) and is summarized in this section.…”