Numerical studies of the optimization of the first eigenvalue of the heat diffusion in inhomogeneous media
In this paper, we study optimization of the first eigenvalue of the heat equation with spatially nonuniform conductivity on a bounded domain under several constraints for the conductivity. We consider this problem in various boundary conditions and various type of topology of domains. As a result, we numerically observe several common criteria of the conductivity for optimizing eigenvalues in terms of corresponding eigenfunctions, which are independent of topology of domains and boundary conditions. The geometric characterization of optimizers are also numerically observed.
We consider an extended Merton's problem of optimal consumption and investment in continuous-time with stochastic volatility. The wealth process of the investor is approximated by a particular weak Ito-Taylor approximation called Euler scheme. It is shown that the optimal control of the value function generated by the Euler scheme is an e-optimal control of the original problem of maximizing total expected discounted HARA utility from consumption.