Soft Comput DOI 10.1007/s00500-017-2670-z METHODOLOGIES AND APPLICATION Local search methods for the solution of implicit inverse problems Elias D. Nino-Ruiz 1 · Carlos Ardila 1 · Rafael Capacho 1 © Springer-Verlag Berlin Heidelberg 2017 Abstract In this paper, we propose two local search algo- rithms based on the Tabu Search and the Simulated Annealing methods, respectively, for the solution of implicit inverse problems. In general, the proposed methods work as follows: given a noisy observation and a right-hand side function of a partial differential equation, for each model component a sub-domain is built about it and the corresponding part of observed components in such sub-domain is projected onto a space generated by a pre-defined set of local basis func- tions. For the projection, the singular value decomposition is applied to the data set in order to discard singular vectors cor- responding to small singular values in pursuance of reducing the impact of noise on the projected data. The right-hand side function is then utilized in order to estimate the quality of the projection onto such sub-space. Since different basis functions provide different spaces onto which the data can be projected, the well-known Tabu Search and Simulated Annealing methods are utilized in order to enrich the search space of the optimal set of basis functions. This is the optimal combination of basis functions which minimizes the error during the projection step. After this, the local solutions are mapped back onto the global domain from which the global solution of the inverse problem is approximated. A strength of our proposed method is that no assumption is needed over Communicated by V. Loia. B Elias D. Nino-Ruiz enino@uninorte.edu.co Carlos Ardila cardila@uninorte.edu.co Rafael Capacho jcapacho@uninorte.edu.co 1 Department of Computer Science, Universidad del Norte, Barranquilla, Colombia the measurements to be assimilated. Experimental tests are performed making use of a parabolic partial differential equa- tion and different noise levels for the data error. The results reveal that the use of the proposed implementations can pro- vide accurate estimates in a root-mean-square error sense of the reference solution with even large data errors. Keywords Reduce space · Simulated Annealing · Tabu Search · Inverse problems · Partial differential equation 1 Introduction Inverse problems are widely found in many sciences and fields (Lu et al. 2015; Florens and Simoni 2016; Cui et al. 2015). Their use ranges from parameter estimation in partial differential equations (Veitch et al. 2015; Ding et al. 2016; Xu et al. 2015) to data assimilation in the context of numerical weather forecast (Bocquet et al. 2015; Laloyaux et al. 2016; Waters et al. 2015; Ruiz et al. 2015; Ruiz and Sandu 2016). Roughly speaking, one wants to estimate a set of parameters (or model states) based on real observations. Measurements are typically corrupted by noise which can be (hopefully) associated with some probabilistic error distribution during the estimation process. However, in many cases, observa- tions are limited just to a few, and therefore, the associated error statistics can be hard to estimate (Morice et al. 2012; Hoppe et al. 2014). To overcome this situation, we propose to make use of polynomial basis functions in order to create sub- sapces onto which the noisy observation can be projected and the estimation performed. The optimal set of basis functions can be estimated making use of well-known metaheuristics in the context of combinatorial optimization such as Tabu Search and Simulated Annealing, and in this manner, no assumptions are needed over either the observational error 123