Well-posedness and convergence of the Lindblad master equation for a quantum harmonic oscillator with multi-photon drive and damping

We consider the model of a quantum harmonic oscillator governed by a Lindblad master equation where the typical drive and loss channels are multi-photon processes instead of single-photon ones; this implies a dissipation operator of order 2k with integer k>1 for a k-photon process. We prove that the corresponding PDE makes the state converge, for large time, to an invariant subspace spanned by a set of k selected basis vectors; the latter physically correspond to so-called coherent states with the same amplitude and uniformly distributed phases. We also show that this convergence features a finite set of bounded invariant functionals of the state (physical observables), such that the final state in the invariant subspace can be directly predicted from the initial state. The proof includes the full arguments towards the well-posedness of the corresponding dynamics in proper Banach spaces of Hermitian trace-class operators equipped with adapted nuclear norms. It relies on the Hille-Yosida theorem and Lyapunov convergence analysis.Comment: 20 pages, submitte

Bellman Error Based Feature Generation using Random Projections on Sparse Spaces

We address the problem of automatic generation of features for value function approximation. Bellman Error Basis Functions (BEBFs) have been shown to improve the error of policy evaluation with function approximation, with a convergence rate similar to that of value iteration. We propose a simple, fast and robust algorithm based on random projections to generate BEBFs for sparse feature spaces. We provide a finite sample analysis of the proposed method, and prove that projections logarithmic in the dimension of the original space are enough to guarantee contraction in the error. Empirical results demonstrate the strength of this method

Inertial manifolds and finite-dimensional reduction for dissipative PDEs

These notes are devoted to the problem of finite-dimensional reduction for parabolic PDEs. We give a detailed exposition of the classical theory of inertial manifolds as well as various attempts to generalize it based on the so-called Man\'e projection theorems. The recent counterexamples which show that the underlying dynamics may be in a sense infinite-dimensional if the spectral gap condition is violated as well as the discussion on the most important open problems are also included.Comment: This manuscript is an extended version of the lecture notes taught by the author as a part of the crash course in the Analysis of Nonlinear PDEs at Maxwell Center for Analysis and Nonlinear PDEs (Edinburgh, November, 8-9, 2012