Research

Digital Image Processing

Digital images are an instrument of analysis in many scientific fields such as robotics, medicine, astronomy, study of documents, etc ... In many cases, due to degradation such as the aging of the documents or the blurring of the acquisition instruments, you can not conduct a proper analysis of these images. Restoration techniques are necessary for improving the quality and the elimination of degradation.

The inverse visual problems appear to have a ill posed nature, that does not guarantee the existence, uniqueness and stability of the solution. For this reason regularization techniques are used for imposing constraints on a priori knowledge that we have about the solution.

Numerical Linear Algebra

Computational linear algebra deals with solutions of linear systems, decomposition to singular values and linear least squares problem. For each of these problems there exist efficient algorithms, which could produce weak solutions as the dimension of the problem increases. Thus, it is necessary to develop new algorithms that take advantage of the problem's structure (e.g. methods based on Krylov spaces).

Research is also interested in more specific problems like solutions of matrix equations, computation of matrix maps and solutions of nonlinear eigenvalue problems.

Genetic and Evolutionary Algorithms

Many optimization systems exist in the nature. Evolution, for example, is a system that ensures the survival of individuals best adapted to the environment in which they live. More complex systems such as ant colonies are an example of aggregation of hundreds of thousands of individuals, whose global behavior (search for food, defense ...) is determined by actions of a single agent who apparently have nothing to do with the overall result.

Computational mathematics creates models that simulate the behavior of these systems to solve complex computational problems. These models are generally applied to the search of minimum paths on graphs, but also to the improvement of digital images.

Computational Complexity

Computational mathematics deals principally with the analysis of computational problems. A computational problem consists of a set of instances one has to find an answer to. A calculation model (e.g. Turing machines) has to be fixed in order to classify different computational problems according to their hardeness. Once fixed it, it is possible to define an algorithm to solve the problem.

The cost of an algorithm depends on the resources required for its computation, such as the computation time and the memory space. The complexity of a problem according to the fixed calculation model is defined as the computational cost of the best algorithm that solves it. Usually it is not possible to give a precise classification of a computational problem, therefore are known just lower bounds for the complexity, namely the minimum computing resources to solve the problem, or upper bounds, often represented by the cost of better known algorithm that solves the problem.

One of the major open problems of mathematics is to determine the proper hierarchy of the computational classes, such as the problem of P = NP.