Tensor train versus Monte Carlo for the multicomponent Smoluchowski coagulation equation

摘要: In this paper we present a novel numerical algorithm for the space-homogeneous multicomponent (multidimensional) Smoluchowski coagulation equation, number of components is considered as dimensionality. The new methodology based on classical finite-difference predictor-corrector scheme. straightforward implementation scheme, however, one would have to compute and store prohibitively many values grid function at nodes multidimensional grid. We propose use special low-parametric representations functions well kernel. corresponding arrays are approximated by low-rank tensor-train decompositions reducing them combinations small low-dimensional arrays, eventually matrices which can fast algorithms linear algebra. Instead O ( N 2 d ) operations in method that requires only log ? operations, where per axis space components. work accelerate time-scheme trapezoidal rule computation integral operators. Thus, accuracy h + , step time step.