Common Structure of Reduced Bases for Aggregation Kinetics Problems of Varying Dimensionality

The present paper is devoted to the investigation of the structure of linear spaces facilitating the approximation of solutions to systems of differential equations arising in problems of aggregation kinetics with particle sources and sinks, on large time intervals with good accuracy. The main parameter of the aggregation kinetics model family under consideration is the model dimension N, which corresponds to the maximum allowed particle size in the system. For large values of N the investigation of such problems becomes difficult as it requires significant computational resources. To accelerate computations we make use of the general idea of model reduction via snapshot method for basis discovery. As a result of this approach, we are able to ascertain the existence of such subspace, construct its basis and reduce the original, nonlinear high-dimensional problem to a much smaller one, allowing to perform the computations much more effectively without significant decrease in the quality of the numeric solutions. The main result of present work is the discovery of the possibility to apply the basis for problems of higher dimension N to the problem of lower dimension M > N, bypassing the need to prepare a construct a separate basis for the dimension M. In numeric experiments we show that it is sufficient to simply take first M < N components of the basis vectors for the dimension N and utilize resulting system as a basis for the new problem of dimension M. Utilizing such a basis, we are able to achieve excellent accuracy for numeric solutions of the problem of dimension M, despite the fact that the solutions of systems of dimensions M and N are not necessarily similar. The paper additionally contains several estimates for norms of solutions for the problem of irreversible aggregation with particle source. Estimates imply ineffectiveness of direct substitution of basis for dimension N into a problem of dimension M > N, and that basis expansion in this fashion requires additional effort, an observation borne out in numeric experiments.

Automated summary

The paper investigates the structure of linear spaces and proposes a model reduction method to reduce high-dimensional problems. The research discovers the possibility of applying higher dimensional bases to lower dimensional problems without significant decrease in the accuracy of numeric solutions.