Machine Learning for Multi-stage Selection of Numerical Methods*

In various areas of numerical analysis, there are several possible algorithms for solving a prob-lem. In such cases, each method potentially solves the problem, but the runtimes can widely differ, and breakdown is possible. Also, there is typically no governing theory for finding the best method, or the theory is in essence uncomputable. Thus, the choice of the optimal method is in practice determined by experimentation and ‘numerical folklore’. However, a more systematic approach is needed, for instance since such choices may need to be made in a dynamic context such as a time-evolving system. Thus we formulate this as a classification problem: assign each numerical problem to a class corresponding to the best method for solving that problem. What makes this an interesting problem for Machine Learning, is the large number of classes, and their relationships. A method is a combination of (at least) a preconditioner and an itera-tive scheme, making the total number of methods the product of these individual cardinalities. Since this can be a very large number, we want to exploit this structure of the set of classes, and find a way to classify the components of a method separately. We have developed various techniques for such multi-stage recommendations, using auto-matic recognition of super-clases. These techniques are shown to pay off very well in our application area of iterative linear system solvers. We present the basic concepts of our recommendation strategy, and give an overview of the software libraries that make up the Salsa (Self-Adapting Large-scale Solver Architecture) project. 1