ИСТИНА

Smoluchowski-type equations describe a distribution of particle masses in a system evolving primarily via coagulation, with additional terms describing such effects as spontaneous or collisional fragmentation, particle sources, etc. While the original equations constitute an infinite system of ODEs, for practical purposes it is necessary to somehow reduce it to a finite system, truncating the mass spectrum. For applications, however, it is highly de- sirable to set the cutoff as high as possible; some require keeping track of masses differing by over ten orders of magnitude. Presently, problems of such size are mainly solved by averaging the solution over large portions of the spectrum; a notable downside of this approach is its insensitivity to rapid changes within said regions. In this talk we consider an alternative approach to solving such extra large systems, based on the tensor train decomposition. Specifically, we show that, for a certain class of coagulation kernels, the quadratic operator featuring in the Smoluchowski-type equations admits a compact representation in the tensorised tensor train format. Assuming the initial conditions also admit such representation, we can then utilise a wide range of algorithms already developed for the tensor train decomposition to construct algorithms for both direct and implicit time integration, as well as solving stationary equations, with all operations having complexity logarithmic in the size of the retained mass range.