Tensor completion in hierarchical tensor representations

Compressed sensing extends from the recovery of sparse vectors from undersampled measurements via efficient algorithms to the recovery of matrices of low rank from incomplete information. Here we consider a further extension to the reconstruction of tensors of low multi-linear rank in recently introduced hierarchical tensor formats from a small number of measurements. Hierarchical tensors are a flexible generalization of the well-known Tucker representation, which have the advantage that the number of degrees of freedom of a low rank tensor does not scale exponentially with the order of the tensor. While corresponding tensor decompositions can be computed efficiently via successive applications of (matrix) singular value decompositions, some important properties of the singular value decomposition do not extend from the matrix to the tensor case. This results in major computational and theoretical difficulties in designing and analyzing algorithms for low rank tensor recovery. For instance, a canonical analogue of the tensor nuclear norm is NP-hard to compute in general, which is in stark contrast to the matrix case. In this book chapter we consider versions of iterative hard thresholding schemes adapted to hierarchical tensor formats. A variant builds on methods from Riemannian optimization and uses a retraction mapping from the tangent space of the manifold of low rank tensors back to this manifold. We provide first partial convergence results based on a tensor version of the restricted isometry property (TRIP) of the measurement map. Moreover, an estimate of the number of measurements is provided that ensures the TRIP of a given tensor rank with high probability for Gaussian measurement maps.Comment: revised version, to be published in Compressed Sensing and Its Applications (edited by H. Boche, R. Calderbank, G. Kutyniok, J. Vybiral

Optimization by Record Dynamics

Large dynamical changes in thermalizing glassy systems are triggered by trajectories crossing record sized barriers, a behavior revealing the presence of a hierarchical structure in configuration space. The observation is here turned into a novel local search optimization algorithm dubbed Record Dynamics Optimization, or RDO. RDO uses the Metropolis rule to accept or reject candidate solutions depending on the value of a parameter akin to the temperature, and minimizes the cost function of the problem at hand through cycles where its `temperature' is raised and subsequently decreased in order to expediently generate record high (and low) values of the cost function. Below, RDO is introduced and then tested by searching the ground state of the Edwards-Anderson spin-glass model, in two and three spatial dimensions. A popular and highly efficient optimization algorithm, Parallel Tempering (PT) is applied to the same problem as a benchmark. RDO and PT turn out to produce solution of similar quality for similar numerical effort, but RDO is simpler to program and additionally yields geometrical information on the system's configuration space which is of interest in many applications. In particular, the effectiveness of RDO strongly indicates the presence of the above mentioned hierarchically organized configuration space, with metastable regions indexed by the cost (or energy) of the transition states connecting them.Comment: 14 pages, 12 figure

Design of a Low‐Cost Permanent Synchronous Machine for Isolated Wind Conversion Systems

The chapter deals with the theoretical analysis of two configurations of low‐cost permanent synchronous generator (PMSG), suitable for small rating, direct driven applications, such as small‐ and microscale wind power plants. The first structure is a permanent magnet claw‐pole synchronous generator (PMCPSG) to be used in an isolated microwind power plants with installed power around few hundred Watts. A permanent magnet synchronous machine with outer rotor (PMSMOR) is the second presented structure, suitable for small wind system with installed power between 2 and 5 kW. In order to obtain the optimal value of the main geometric dimensions of the generators, an optimization procedure, based on Hooke‐Jeeves method, was implemented for all the considered structures

Taming Landau singularities in QCD perturbation theory: The analytic approach 2.0

The aim of this topical article is to outline the fundamental ideas underlying the recently developed Fractional Analytic Perturbation Theory (FAPT) of QCD and present its main calculational tools together with key applications. For this, it is first necessary to review previous methods to apply QCD perturbation theory at low spacelike momentum scales, where the influence of the Landau singularities becomes inevitable. Several concepts are considered and their limitations are pointed out. The usefulness of FAPT is discussed in terms of two characteristic hadronic quantities: the perturbatively calculable part of the pion's electromagnetic form factor in the spacelike region and the Higgs-boson decay into a $b\bar b$ pair in the timelike region. In the first case, the focus is on the optimization of the prediction with respect to the choice of the renormalization scheme and the dependence on the renormalization and the factorization scales. The second case serves to show that the application of FAPT to this reaction reaches already at the four-loop level an accuracy of the order of 1%, avoiding difficulties inherent in the standard perturbative expansion. The obtained results are compared with estimates from fixed-order and contour-improved QCD perturbation theory. Using the brand-new Higgs mass value of about 125 GeV, measured at the Large Hadron Collider (CERN), a prediction for $\Gamma_{H\to b\bar{b}}=2.4 \pm 0.15 {\rm MeV}$ is extracted.Comment: v3: 23 pages, 7 figures, Invited topical article published in Particles and Nuclei with update using the CERN Higgs discovery. Abridged version presented as plenary talk at International Conference on Renormalization Group and Related Topics (RG 2008), Dubna, Russia, September 1 - 5, 2008. v4 typo in Eq. (3) correcte

Saliency Ratio and Power Factor of IPM Motors Optimally Designed for High Efficiency and Low Cost Objectives

This paper uses formal mathematical optimization techniques based on parametric finite-element-based computationally efficient models and differential evolution algorithms. For constant-power applications, in the novel approach described, three concurrent objective functions are minimized: material cost, losses, in order to ensure high efficiency, and the difference between the rated and the characteristic current, aiming to achieve very high constant-power flux-weakening range. Only the first two objectives are considered for constant-torque applications. Two types of interior permanent magnet rotors in a single- and double-layer V-shaped configuration are considered, respectively. The stator has the typical two slots per pole and phase distributed winding configuration. The results for the constant-torque design show that, in line with expectations, high efficiency and high power factor machines are more costly, and that the low-cost machines have poorer efficiency and power factor and most importantly, and despite a common misconception, the saliency ratio may also be lower in this case. For constant-power designs, the saliency ratio can be beneficial. Nevertheless, despite a common misconception, when cost is considered alongside performance as an objective, a higher saliency ratio does not necessarily improve the power factors of motors suitable for ideal infinite flux weakening

Conic Optimization Theory: Convexification Techniques and Numerical Algorithms

Optimization is at the core of control theory and appears in several areas of this field, such as optimal control, distributed control, system identification, robust control, state estimation, model predictive control and dynamic programming. The recent advances in various topics of modern optimization have also been revamping the area of machine learning. Motivated by the crucial role of optimization theory in the design, analysis, control and operation of real-world systems, this tutorial paper offers a detailed overview of some major advances in this area, namely conic optimization and its emerging applications. First, we discuss the importance of conic optimization in different areas. Then, we explain seminal results on the design of hierarchies of convex relaxations for a wide range of nonconvex problems. Finally, we study different numerical algorithms for large-scale conic optimization problems.Comment: 18 page