Stability Remarks on Discretized Multi-dimensional Diffusion Process Models and its Application to Model Reduction

This paper presents some stability remarks on the discretized diffusion process models. The motivation arises from our observation of stability preserving property in terms of a model reduction procedure of the 1D model. Before the stability analysis of the non-reduced order model, the state-space model is extended to multi-dimensional cases in a systematic manner. This formulation and the corresponding stability analysis are the first non-trivial contributions here. Then we clarify the fact behind the stability preserving property. As a consequence, one can employ arbitrary size of reduced order model based on the techniques such as the principal component analysis without any concern for the stability. The power of this reduction method is demonstrated via numerical examples for the 1D and 2D cases.