Scalable numerical approach for the steady-state ab initio laser theory

We present an efficient and flexible method for solving the non-linear lasing equations of the steady-state ab initio laser theory. Our strategy is to solve the underlying system of partial differential equations directly, without the need of setting up a parametrized basis of constant flux states. We validate this approach in one-dimensional as well as in cylindrical systems, and demonstrate its scalability to full-vector three-dimensional calculations in photonic-crystal slabs. Our method paves the way for efficient and accurate simulations of lasing structures which were previously inaccessible.Comment: 17 pages, 8 figure

Heat flow and calculus on metric measure spaces with Ricci curvature bounded below - the compact case

We provide a quick overview of various calculus tools and of the main results concerning the heat flow on compact metric measure spaces, with applications to spaces with lower Ricci curvature bounds. Topics include the Hopf-Lax semigroup and the Hamilton-Jacobi equation in metric spaces, a new approach to differentiation and to the theory of Sobolev spaces over metric measure spaces, the equivalence of the L^2-gradient flow of a suitably defined "Dirichlet energy" and the Wasserstein gradient flow of the relative entropy functional, a metric version of Brenier's Theorem, and a new (stronger) definition of Ricci curvature bound from below for metric measure spaces. This new notion is stable w.r.t. measured Gromov-Hausdorff convergence and it is strictly connected with the linearity of the heat flow.Comment: To the memory of Enrico Magenes, whose exemplar life, research and teaching shaped generations of mathematician

Generalized reduced basis methods and n-width estimates for the approximation of the solution manifold of parametric PDEs

The set of solutions of a parameter-dependent linear partial differential equation with smooth coefficients typically forms a compact manifold in a Hilbert space. In this paper we review the generalized reduced basis method as a fast computational tool for the uniform approximation of the solution manifold. We focus on operators showing an affine parametric dependence, expressed as a linear combination of parameter-independent operators through some smooth, parameter-dependent scalar functions. In the case that the parameter-dependent operator has a dominant term in its affine expansion, one can prove the existence of exponentially convergent uniform approximation spaces for the entire solution manifold. These spaces can be constructed without any assumptions on the parametric regularity of the manifold -- only spatial regularity of the solutions is required. The exponential convergence rate is then inherited by the generalized reduced basis method. We provide a numerical example related to parametrized elliptic equations confirming the predicted convergence rate

Finite Element Methods for Elliptic Distributed Optimal Control Problems with Pointwise State Constraints

Finite element methods for a model elliptic distributed optimal control problem with pointwise state constraints are considered from the perspective of fourth order boundary value problems