Robust localization methods for passivity enforcement of linear macromodels

In this paper we solve a non-smooth convex formulation for passivity enforcement of linear macromodels using robust localization based algorithms such as the ellipsoid and the cutting plane methods. Differently from existing perturbation based techniques, we solve the formulation based on the direct ℌ∞ norm minimization through perturbation of state-space model parameters. We provide a systematic way of defining an initial set which is guaranteed to contain the global optimum. We also provide a lower bound on the global minimum, that grows tighter at each iteration and hence guarantees δ - optimality of the computed solution. We demonstrate the robustness of our implementation by generating accurate passive models for challenging examples for which existing algorithms either failed or exhibited extremely slow convergenc

The interaction of helical tip and root vortices in a wind turbine wake

Analysis of the helical vortices measured behind a model wind turbine in a water channel are reported. Phase-locked measurements using planar particle image ve- locimetry are taken behind a Glauert rotor to investigate the evolution and breakdown of the helical vortex structures. Existing linear stability theory predicts helical vortex filaments to be susceptible to three unstable modes. The current work presents tip and root vortex evolution in the wake for varying tip speed ratio and shows a breaking of the helical symmetry and merging of the vortices due to mutual inductance between the vortical filaments. The merging of the vortices is shown to be steady with rotor phase, however, small-scale non-periodic meander of the vortex positions is also ob- served. The generation of the helical wake is demonstrated to be closely coupled with the blade aerodynamics, strongly influencing the vortex properties which are shown to agree with theoretical predictions of the circulation shed into the wake by the blades. The mutual inductance of the helices is shown to occur at the same non-dimensional wake distance

Non-convex Optimization for Machine Learning

A vast majority of machine learning algorithms train their models and perform inference by solving optimization problems. In order to capture the learning and prediction problems accurately, structural constraints such as sparsity or low rank are frequently imposed or else the objective itself is designed to be a non-convex function. This is especially true of algorithms that operate in high-dimensional spaces or that train non-linear models such as tensor models and deep networks. The freedom to express the learning problem as a non-convex optimization problem gives immense modeling power to the algorithm designer, but often such problems are NP-hard to solve. A popular workaround to this has been to relax non-convex problems to convex ones and use traditional methods to solve the (convex) relaxed optimization problems. However this approach may be lossy and nevertheless presents significant challenges for large scale optimization. On the other hand, direct approaches to non-convex optimization have met with resounding success in several domains and remain the methods of choice for the practitioner, as they frequently outperform relaxation-based techniques - popular heuristics include projected gradient descent and alternating minimization. However, these are often poorly understood in terms of their convergence and other properties. This monograph presents a selection of recent advances that bridge a long-standing gap in our understanding of these heuristics. The monograph will lead the reader through several widely used non-convex optimization techniques, as well as applications thereof. The goal of this monograph is to both, introduce the rich literature in this area, as well as equip the reader with the tools and techniques needed to analyze these simple procedures for non-convex problems.Comment: The official publication is available from now publishers via http://dx.doi.org/10.1561/220000005

Simple, Efficient, and Neural Algorithms for Sparse Coding

Sparse coding is a basic task in many fields including signal processing, neuroscience and machine learning where the goal is to learn a basis that enables a sparse representation of a given set of data, if one exists. Its standard formulation is as a non-convex optimization problem which is solved in practice by heuristics based on alternating minimization. Re- cent work has resulted in several algorithms for sparse coding with provable guarantees, but somewhat surprisingly these are outperformed by the simple alternating minimization heuristics. Here we give a general framework for understanding alternating minimization which we leverage to analyze existing heuristics and to design new ones also with provable guarantees. Some of these algorithms seem implementable on simple neural architectures, which was the original motivation of Olshausen and Field (1997a) in introducing sparse coding. We also give the first efficient algorithm for sparse coding that works almost up to the information theoretic limit for sparse recovery on incoherent dictionaries. All previous algorithms that approached or surpassed this limit run in time exponential in some natural parameter. Finally, our algorithms improve upon the sample complexity of existing approaches. We believe that our analysis framework will have applications in other settings where simple iterative algorithms are used.Comment: 37 pages, 1 figur

High-contrast imaging at small separation: impact of the optical configuration of two deformable mirrors on dark holes

The direct detection and characterization of exoplanets will be a major scientific driver over the next decade, involving the development of very large telescopes and requires high-contrast imaging close to the optical axis. Some complex techniques have been developed to improve the performance at small separations (coronagraphy, wavefront shaping, etc). In this paper, we study some of the fundamental limitations of high contrast at the instrument design level, for cases that use a combination of a coronagraph and two deformable mirrors for wavefront shaping. In particular, we focus on small-separation point-source imaging (around 1 $\lambda$/D). First, we analytically or semi-analytically analysing the impact of several instrument design parameters: actuator number, deformable mirror locations and optic aberrations (level and frequency distribution). Second, we develop in-depth Monte Carlo simulation to compare the performance of dark hole correction using a generic test-bed model to test the Fresnel propagation of multiple randomly generated optics static phase errors. We demonstrate that imaging at small separations requires large setup and small dark hole size. The performance is sensitive to the optic aberration amount and spatial frequencies distribution but shows a weak dependence on actuator number or setup architecture when the dark hole is sufficiently small (from 1 to $\lesssim$ 5 $\lambda$/D).Comment: 13 pages, 18 figure

An investigation of new methods for estimating parameter sensitivities

The method proposed for estimating sensitivity derivatives is based on the Recursive Quadratic Programming (RQP) method and in conjunction a differencing formula to produce estimates of the sensitivities. This method is compared to existing methods and is shown to be very competitive in terms of the number of function evaluations required. In terms of accuracy, the method is shown to be equivalent to a modified version of the Kuhn-Tucker method, where the Hessian of the Lagrangian is estimated using the BFS method employed by the RQP algorithm. Initial testing on a test set with known sensitivities demonstrates that the method can accurately calculate the parameter sensitivity