A global optimization method using a random walk on a topological map and local variational inversions

This paper presents an improved variational method suitable for inverting a problem associated with integral constrains. The method allows a global minimization. We minimized a cost function representing the mismatch between the measurements and the output of a numerical model, to which we added a restoring term to a background. A way to process the covariance matrix associated with the above-weighted quadratic background is to model the control vectors using probabilistic principal component analysis (PPCA). The use of PPCA presents difficulties in the case of a large dataset representing the overall variability of the control space. We therefore developed a method based on a topological map model, which allows partition of the dataset into subsets more suited to the PPCA approach and thus leading to a local inversion by the variational method. A random walk based on a Markov chain was used to find the most appropriate subsets of the topological map by taking into account a priori information on the unknown vector. This random walk on a topological map allows us to reduce the number of subsets able to give the optimal solution and thus to achieve a better performance at a lower cost. An example of the application of this method to the shallow water acoustic tomography inverse problem, showing its effectiveness, is presented.