Deterministic global optimization using space-filling curves and multiple estimates of Lipschitz and Holder constants

In this paper, the global optimization problem $\min_{y\in S} F(y)$ with $S$ being a hyperinterval in $\Re^N$ and $F(y)$ satisfying the Lipschitz condition with an unknown Lipschitz constant is considered. It is supposed that the function $F(y)$ can be multiextremal, non-differentiable, and given as a `black-box'. To attack the problem, a new global optimization algorithm based on the following two ideas is proposed and studied both theoretically and numerically. First, the new algorithm uses numerical approximations to space-filling curves to reduce the original Lipschitz multi-dimensional problem to a univariate one satisfying the H\"{o}lder condition. Second, the algorithm at each iteration applies a new geometric technique working with a number of possible H\"{o}lder constants chosen from a set of values varying from zero to infinity showing so that ideas introduced in a popular DIRECT method can be used in the H\"{o}lder global optimization. Convergence conditions of the resulting deterministic global optimization method are established. Numerical experiments carried out on several hundreds of test functions show quite a promising performance of the new algorithm in comparison with its direct competitors.Comment: 26 pages, 10 figures, 4 table

Global optimization for structured low rank approximation

In this paper, we investigate the complexity of the numerical construction of the Hankel structured low-rank approximation (HSLRA) problem, and develop a family of algorithms to solve this problem. Briefly, HSLRA is the problem of finding the closest (in some pre-defined norm) rank r approximation of a given Hankel matrix, which is also of Hankel structure. Unlike many other methods described in the literature the family of algorithms we propose has the property of guaranteed convergence

Learning and inverse problems: from theory to solar physics applications

The problem of approximating a function from a set of discrete measurements has been extensively studied since the seventies. Our theoretical analysis proposes a formalization of the function approximation problem which allows dealing with inverse problems and supervised kernel learning as two sides of the same coin. The proposed formalization takes into account arbitrary noisy data (deterministically or statistically defined), arbitrary loss functions (possibly seen as a log-likelihood), handling both direct and indirect measurements. The core idea of this part relies on the analogy between statistical learning and inverse problems. One of the main evidences of the connection occurring across these two areas is that regularization methods, usually developed for ill-posed inverse problems, can be used for solving learning problems. Furthermore, spectral regularization convergence rate analyses provided in these two areas, share the same source conditions but are carried out with either increasing number of samples in learning theory or decreasing noise level in inverse problems. Even more in general, regularization via sparsity-enhancing methods is widely used in both areas and it is possible to apply well-known $ell_1$-penalized methods for solving both learning and inverse problems. In the first part of the Thesis, we analyze such a connection at three levels: (1) at an infinite dimensional level, we define an abstract function approximation problem from which the two problems can be derived; (2) at a discrete level, we provide a unified formulation according to a suitable definition of sampling; and (3) at a convergence rates level, we provide a comparison between convergence rates given in the two areas, by quantifying the relation between the noise level and the number of samples. In the second part of the Thesis, we focus on a specific class of problems where measurements are distributed according to a Poisson law. We provide a data-driven, asymptotically unbiased, and globally quadratic approximation of the Kullback-Leibler divergence and we propose Lasso-type methods for solving sparse Poisson regression problems, named PRiL for Poisson Reweighed Lasso and an adaptive version of this method, named APRiL for Adaptive Poisson Reweighted Lasso, proving consistency properties in estimation and variable selection, respectively. Finally we consider two problems in solar physics: 1) the problem of forecasting solar flares (learning application) and 2) the desaturation problem of solar flare images (inverse problem application). The first application concerns the prediction of solar storms using images of the magnetic field on the sun, in particular physics-based features extracted from active regions from data provided by Helioseismic and Magnetic Imager (HMI) on board the Solar Dynamics Observatory (SDO). The second application concerns the reconstruction problem of Extreme Ultra-Violet (EUV) solar flare images recorded by a second instrument on board SDO, the Atmospheric Imaging Assembly (AIA). We propose a novel sparsity-enhancing method SE-DESAT to reconstruct images affected by saturation and diffraction, without using any a priori estimate of the background solar activity

Proceedings of the XIII Global Optimization Workshop: GOW'16

[Excerpt] Preface: Past Global Optimization Workshop shave been held in Sopron (1985 and 1990), Szeged (WGO, 1995), Florence (GO’99, 1999), Hanmer Springs (Let’s GO, 2001), Santorini (Frontiers in GO, 2003), San José (Go’05, 2005), Mykonos (AGO’07, 2007), Skukuza (SAGO’08, 2008), Toulouse (TOGO’10, 2010), Natal (NAGO’12, 2012) and Málaga (MAGO’14, 2014) with the aim of stimulating discussion between senior and junior researchers on the topic of Global Optimization. In 2016, the XIII Global Optimization Workshop (GOW’16) takes place in Braga and is organized by three researchers from the University of Minho. Two of them belong to the Systems Engineering and Operational Research Group from the Algoritmi Research Centre and the other to the Statistics, Applied Probability and Operational Research Group from the Centre of Mathematics. The event received more than 50 submissions from 15 countries from Europe, South America and North America. We want to express our gratitude to the invited speaker Panos Pardalos for accepting the invitation and sharing his expertise, helping us to meet the workshop objectives. GOW’16 would not have been possible without the valuable contribution from the authors and the International Scientific Committee members. We thank you all. This proceedings book intends to present an overview of the topics that will be addressed in the workshop with the goal of contributing to interesting and fruitful discussions between the authors and participants. After the event, high quality papers can be submitted to a special issue of the Journal of Global Optimization dedicated to the workshop. [...

Mathematical Challenges in Electron Microscopy

Development of electron microscopes first started nearly 100 years ago and they are now a mature imaging modality with many applications and vast potential for the future. The principal feature of electron microscopes is their resolution; they can be up to 1000 times more powerful than a visible light microscope and resolve even the smallest atoms. Furthermore, electron microscopes are also sensitive to many material properties due to the very rich interactions between electrons and other matter. Because of these capabilities, electron microscopy is used in applications as diverse as drug discovery, computer chip manufacture, and the development of solar cells. In parallel to this, the mathematical field of inverse problems has also evolved dramatically. Many new methods have been introduced to improve the recovery of unknown structures from indirect data, typically an ill-posed problem. In particular, sparsity promoting functionals such as the total variation and its extensions have been shown to be very powerful for recovering accurate physical quantities from very little and/or poor quality data. While sparsity-promoting reconstruction methods are powerful, they can also be slow, especially in a big-data setting. This trade-off forms an eternal cycle as new numerical tools are found and more powerful models are developed. The work presented in this thesis aims to marry the tools of inverse problems with the problems of electron microscopy: bringing state-of-the-art image processing techniques to bear on challenges specific to electron microscopy, developing new optimisation methods for these problems, and modelling new inverse problems to extend the capabilities of existing microscopes. One focus is the application of a directional total variation to overcome the limited angle problem in electron tomography, another is the proposal of a new inverse problem for the reconstruction of 3D strain tensor fields from electron microscopy diffraction data. The remaining contributions target numerical aspects of inverse problems, from new algorithms for non-convex problems to convex optimisation with adaptive meshes.Cantab Capital Institute for Mathematics of Informatio

International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022

Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library

Well-Distributed Sequences: Number Theory, Optimal Transport, and Potential Theory

The purpose of this dissertation will be to examine various ways of measuring how uniformly distributed a sequence of points on compact manifolds and finite combinatorial graphs can be, providing bounds and novel explicit algorithms to pick extremely uniform points, as well as connecting disparate branches of mathematics such as Number Theory and Optimal Transport. Chapter 1 sets the stage by introducing some of the fundamental ideas and results that will be used consistently throughout the thesis: we develop and establish Weyl\u27s Theorem, the definition of discrepancy, LeVeque\u27s Inequality, the Erdős-Turán Inequality, Koksma-Hlawka Inequality, and Schmidt\u27s Theorem about Irregularities of Distribution. Chapter 2 introduces the Monge-Kantorovich transport problem with special emphasis on the Benamou-Brenier Formula (from 2000) and Peyre\u27s inequality (from 2018). Chapter 3 explores Peyre\u27s Inequality in further depth, considering how specific bounds on the Wasserstein distance between a point measure and the uniform measure may be obtained using it, in particular in terms of the Green\u27s function of the Laplacian on a manifold. We also show how a smoothing procedure can be applied by propagating the heat equation on probability mass in order to get stronger bounds on transport distance using well-known properties of the heat equation. In Chapter 4, we turn to the primary question of the thesis: how to select points on a space which are as uniformly distributed as possible. We consider various diverse approaches one might attempt: an ergodic approach iterating functions with good mixing properties; a dyadic approach introduced in a 1975 theorem of Kakutani on proportional splittings on intervals; and a completely novel potential theoretic approach, assigning energy to point configurations and greedily minimizing the total potential arising from pair-wise point interactions. Such energy minimization questions are certainly not new, in the static setting--physicist Thomson posed the question of how to minimize the potential of electrons on a sphere as far back as 1904. However, a greedy approach to uniform distribution via energy minimization is novel, particularly through the lens of Wasserstein, and yields provably Wasserstein-optimal point sequences using the Green\u27s function of the Laplacian as our energy function on manifolds of dimension at least 3 (with dimension 2 losing at most a square root log factor from the optimal bound). We connect this to known results from Graham, Pausinger, and Proinov regarding best possible uniform bounds on the Wasserstein 2-distance of point sequences in the unit interval. We also present many open questions and conjectures on the optimal asymptotic bounds for total energy of point configurations and the growth of the total energy function as points are added, motivated by numerical investigations that display remarkably well-behaved qualities in the dynamical system induced by greedy minimization. In Chapter 5, we consider specific point sequences and bounds on the transport distance from the point measure they generate to the uniform measure. We provide provably optimal rates for the van der Corput sequence, the Kronecker sequence, regular grids and the measures induced by quadratic residues in a field of prime order. We also prove an upper bound for higher degree monomial residues in fields of prime order, and conjecture this to be optimal. In Chapter 6, we consider numerical integration error bounds over Lipschitz functions, asking how closely we can estimate the integral of a function by averaging its values at finitely many points. This is a rather classical question that was answered completely by Bakhalov in 1959 and has since become a standard example (`the easiest case which is perfectly understood\u27). Somewhat surprisingly perhaps, we show that the result is not sharp and improve it in two ways: by refining the function space and by proving that these results can be true uniformly along a subsequence. These bounds refine existing results that were widely considered to be optimal, and we show the intimate connection between transport distance and integration error. Our results are new even for the classical discrete grid. In Chapter 7, we study the case of finite graphs--we show that the fundamental question underlying this thesis can also be meaningfully posed on finite graphs where it leads to a fascinating combinatorial problem. We show that the philosophy introduced in Chapter 4 can be meaningfully adapted and obtain a potential-theoretic algorithm that produces such a sequence on graphs. We show that, using spectral techniques, we are able to obtain empirically strong bounds on the 1-Wasserstein distance between measures on subsets of vertices and the uniform measure, which for graphs of large diameter are much stronger than the trivial diameter bound