Welcome to the web-page of the Croatian Science Foundation Project Grant #6268.
Low rank approximation techniques are the backbone of many modern data driven applications. Such applications have utilization ranging from the analysis of the safety and robustness of engineering problems, over stress testing in financial applications to general big data questions in applications of societal relevance. We will study both the theoretical as well as practical aspects of this problem. As prototypes we will consider dynamical systems in the presence of uncertainty. The quantities of interest are the mean, the variance and the exceedance probability for the solution field. The main challenge in the study of these problems is to beat the so-called curse of the dimensionalty (exponential increase in computing effort with the increase of the number of uncertain parameters). We will concentrate on the family of parameter dependent partial differential equations and we will develop and analyze robust and efficient numerical methods to tackle them. We will also study the modal problem and the frequency response problem, which are used to analyze dynamical systems. Both problems have a core computational task which is the computational analysis of the resolvent function. These problems fall into the category of modern data driven applications, even though we sample the system by simulation and not measurement. Main tools which we employ are randomized sampling in the image space of a matrix loosely based on maxvol algorithm and its variants and model order reduction by low rank compression. We also aim to use this project to lay foundation for a research group in Croatia which will study mathematical foundations of data driven applications (including those that fall into the area of machine learning). In the first instance, we will leverage our experience in numerical linear algebra and computational operator theory in the context of the study of multi parameter eigenvalue problems in thermoacustics and photonics.