Mathematical optimization approach for estimating the quantum yield distribution of a photochromic reaction in a polymer

The convolution of a series of events is often observed for a variety of phenomena such as the oscillation of a string. A photochemical reaction of a molecule is characterized by a time constant, but materials in the real world contain several molecules with different time constants. Therefore, the kinetics of photochemical reactions of the materials are usually observed with a complexity comparable with those of theoretical kinetic equations. Analysis of the components of the kinetics is quite important for the development of advanced materials. However, with a limited number of exceptions, deconvolution of the observed kinetics has not yet been mathematically solved. In this study, we propose a mathematical optimization approach for estimating the quantum yield distribution of a photochromic reaction in a polymer. In the proposed approach, time-series data of absorbances are acquired and an estimate of the quantum yield distribution is obtained. To estimate the distribution, we solve a mathematical optimization problem to minimize the difference between the input data and a model. This optimization problem involves a differential equation constrained on a functional space as the variable lies in the space of probability distribution functions and the constraints arise from reaction rate equations. This problem can be reformulated as a convex quadratic optimization problem and can be efficiently solved by discretization. Numerical results are also reported here, and they verify the effectiveness of our approach.