The Proper Orthogonal Decomposition for Dimensionality Reduction in Mode-Locked Lasers and Optical Systems

Abstract

The onset of multipulsing, a ubiquitous phenomenon in laser cavities, imposes a fundamental limit on the maximum energy delivered per pulse. Managing the nonlinear penalties in the cavity becomes crucial for increasing the energy and suppressing the multipulsing instability. A proper orthogonal decomposition (POD) allows for the reduction of governing equations of a mode-locked laser onto a low-dimensional space. The resulting reduced system is able to capture correctly the experimentally observed pulse transitions. Analysis of these models is used to explain the sequence of bifurcations that are responsible for the multipulsing instability in the master mode-locking and the waveguide array mode-locking models. As a result, the POD reduction allows for a simple and efficient way to characterize and optimize the cavity parameters for achieving maximal energy output.

Model Order Reduction of Electrical Machines With Multiple Inputs

“…Assume that this system is defined by the following partial differential equation over some domain of interest [8] ( , , ,..., , )…”

Experimental design for estimating unknown groundwater pumping using genetic algorithm and reduced order model

“…This set is then stored in X 2 R N n xn s , where N n is the number of nodes (or ODE's) in the full model and n s is the total number of snapshots taken. A number of methods can be used to construct the orthonormal basis of the reduced space [Siade et al, 2010;Shlizerman et al, 2012]. We choose Singular Value Decomposition (SVD) for its computational simplicity compared to eigenvalue decomposition.…”

Reduced order modelling for dynamic simulations: LDRD feasibility study final report

“…This data is then used to construct a lower dimensional basis using techniques such as the proper orthogonal decomposition (POD, also known as principle component analysis or the Karhunen-Loéve decomposition) [1,2,3] or the reduced basis method (RBM) [1,2,5,6]. In the POD approach, for instance, the N snapshots are reduced into a small subspace by computing the singular-value-decomposition (SVD) of…”

“…Currently, the most popular, and computationally effective, reduced order models project a high-fidelity mathematical model into a lower dimensional subspace to produce a dynamical ODE system with many fewer degrees a freedom: this smaller system forms the fast running code (see, for example [1,2,3]). Such model order reduction techniques have been successfully applied to a number of specific areas; however, their practical application to high-fidelity, nonlinear partial-differential (PDE) and differential-algebraic (DAE) equation based simulations has been limited by a number of open research problems.…”