Sparse Moment Problem
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| Author | : Saroj Aryal |
| Publisher | : |
| Total Pages | : 73 |
| Release | : 2013 |
| Genre | : Moment problems (Mathematics) |
| ISBN | : 9781303423789 |
The well-known theorems of Stieltjes, Hamburger and Hausdorff establish conditions on infinite sequences of real numbers to be moment sequences. Further works by Carathéodory, Schur and Nevanlinna connect moment problems to problems in function theory and functions belonging to various spaces. In many problems associated with realization of a signal or an image, data may be corrupted or missing. Reconstruction of a function from moment sequences with missing terms is an interesting problem leading to advances in image and/or signal reconstruction. It is easy to show that a subsequence of a moment sequence may not be a moment sequence. Conditions are obtained to show how rigid the space of sub-moment sequences is and necessary and sufficient conditions for a sequence to be a sub-moment sequence are established. A deep connection between the sub-moment measures and the moment measures is derived and the determinacy of the moment and sub-moment problems are related. This problem is further related to completion of positive Hankel matrices. Data obtained from physical experiments and natural processes are often moment multisequences. Conditions are established for a subsequence of a moment multisequence to be positive. Furthermore, subsequences of a moment multisequence that are also moment multisequences are fully characterized.
| Author | : Victor Magron |
| Publisher | : World Scientific |
| Total Pages | : 223 |
| Release | : 2023-04-25 |
| Genre | : Mathematics |
| ISBN | : 1800612966 |
Many applications, including computer vision, computer arithmetic, deep learning, entanglement in quantum information, graph theory and energy networks, can be successfully tackled within the framework of polynomial optimization, an emerging field with growing research efforts in the last two decades. One key advantage of these techniques is their ability to model a wide range of problems using optimization formulations. Polynomial optimization heavily relies on the moment-sums of squares (moment-SOS) approach proposed by Lasserre, which provides certificates for positive polynomials. On the practical side, however, there is 'no free lunch' and such optimization methods usually encompass severe scalability issues. Fortunately, for many applications, including the ones formerly mentioned, we can look at the problem in the eyes and exploit the inherent data structure arising from the cost and constraints describing the problem.This book presents several research efforts to resolve this scientific challenge with important computational implications. It provides the development of alternative optimization schemes that scale well in terms of computational complexity, at least in some identified class of problems. It also features a unified modeling framework to handle a wide range of applications involving both commutative and noncommutative variables, and to solve concretely large-scale instances. Readers will find a practical section dedicated to the use of available open-source software libraries.This interdisciplinary monograph is essential reading for students, researchers and professionals interested in solving optimization problems with polynomial input data.
| Author | : Didier Henrion |
| Publisher | : World Scientific |
| Total Pages | : 248 |
| Release | : 2020-11-04 |
| Genre | : Mathematics |
| ISBN | : 1786348551 |
The Moment-SOS hierarchy is a powerful methodology that is used to solve the Generalized Moment Problem (GMP) where the list of applications in various areas of Science and Engineering is almost endless. Initially designed for solving polynomial optimization problems (the simplest example of the GMP), it applies to solving any instance of the GMP whose description only involves semi-algebraic functions and sets. It consists of solving a sequence (a hierarchy) of convex relaxations of the initial problem, and each convex relaxation is a semidefinite program whose size increases in the hierarchy.The goal of this book is to describe in a unified and detailed manner how this methodology applies to solving various problems in different areas ranging from Optimization, Probability, Statistics, Signal Processing, Computational Geometry, Control, Optimal Control and Analysis of a certain class of nonlinear PDEs. For each application, this unconventional methodology differs from traditional approaches and provides an unusual viewpoint. Each chapter is devoted to a particular application, where the methodology is thoroughly described and illustrated on some appropriate examples.The exposition is kept at an appropriate level of detail to aid the different levels of readers not necessarily familiar with these tools, to better know and understand this methodology.
| Author | : Fernanda Botelho |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 250 |
| Release | : 2017-04-18 |
| Genre | : Mathematics |
| ISBN | : 1470427729 |
This volume contains the proceedings of the Workshop on Problems and Recent Methods in Operator Theory, held at the University of Memphis, Memphis, TN, from October 15–16, 2015 and the AMS Special Session on Advances in Operator Theory and Applications, in Memory of James Jamison, held at the University of Memphis, Memphis, TN, from October 17–18, 2015. Operator theory is at the root of several branches of mathematics and offers a broad range of challenging and interesting research problems. It also provides powerful tools for the development of other areas of science including quantum theory, physics and mechanics. Isometries have applications in solid-state physics. Hermitian operators play an integral role in quantum mechanics very much due to their “nice” spectral properties. These powerful connections demonstrate the impact of operator theory in various branches of science. The articles in this volume address recent problems and research advances in operator theory. Highlighted topics include spectral, structural and geometric properties of special types of operators on Banach spaces, with emphasis on isometries, weighted composition operators, multi-circular projections on function spaces, as well as vector valued function spaces and spaces of analytic functions. This volume gives a succinct overview of state-of-the-art techniques from operator theory as well as applications to classical problems and long-standing open questions.
| Author | : Helmut Harbrecht |
| Publisher | : |
| Total Pages | : 31 |
| Release | : 2006 |
| Genre | : |
| ISBN | : |
| Author | : Yousef Saad |
| Publisher | : SIAM |
| Total Pages | : 537 |
| Release | : 2003-04-01 |
| Genre | : Mathematics |
| ISBN | : 0898715342 |
Mathematics of Computing -- General.
| Author | : Hayoung Choi |
| Publisher | : |
| Total Pages | : 56 |
| Release | : 2015 |
| Genre | : Moment problems (Mathematics) |
| ISBN | : 9781339054636 |
Moment problems arise naturally in many areas of Mathematics, Economics and Operations research. While moment problems have numerous applications to extremal problems, optimization and limit theorems in probability theory, they rely on a complete set of moments or truncated moment sequences. Due to missing moment entries or availability of truncated moment sequences, sometimes we need to work in the space of incomplete moment sequences. Moment problems with missing entries are closely related to Hankel matrix completion problems. In this dissertation we give solutions to Hamburger moment problems with missing entries. The problem of completing partial positive sequences is considered. The main result is a characterization of positive (semi)definite completable patterns, namely patterns that guarantee the existence of a Hamburger moment completion of a partial positive (semi)definite sequence. Moreover, several patterns which are not positive definite completable are given. Furthermore, we characterize the determinate case by certain subsequence of given moment sequence. For a positive sequence if its subsequence with the pattern of arithmetic progression is determinate, then the sequence is determinate. Also, if a sequence is indeterminate then its subsequences with pattern of arithmetic progression are indeterminate. In the last part of this thesis, we apply moment completion problems to reconstruct Radon transform with missing data. Radon transform is a well known tool for reconstructing data from its projections. Reconstruction of Radon transform with missing data is closely related to reconstruction of a function from moment sequences with missing terms. A new range theorem is established for the Radon transform based on the Hamburger moment problem in two variables, and the sparse moment problem is converted into the Radon transform with missing data and vice versa. A modified Radon transform is introduced and its inversion formula is established.
| Author | : Jean-Luc Starck |
| Publisher | : Cambridge University Press |
| Total Pages | : 351 |
| Release | : 2010-05-10 |
| Genre | : Computers |
| ISBN | : 0521119138 |
This book presents the state of the art in sparse and multiscale image and signal processing, covering linear multiscale transforms, such as wavelet, ridgelet, or curvelet transforms, and non-linear multiscale transforms based on the median and mathematical morphology operators. Matlab and IDL code accompany these methods and applications to reproduce the experiments and illustrate the reasoning and methodology of the research available for download at the associated Web site.
| Author | : Miguel F. Anjos |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 955 |
| Release | : 2011-11-19 |
| Genre | : Business & Economics |
| ISBN | : 1461407699 |
Semidefinite and conic optimization is a major and thriving research area within the optimization community. Although semidefinite optimization has been studied (under different names) since at least the 1940s, its importance grew immensely during the 1990s after polynomial-time interior-point methods for linear optimization were extended to solve semidefinite optimization problems. Since the beginning of the 21st century, not only has research into semidefinite and conic optimization continued unabated, but also a fruitful interaction has developed with algebraic geometry through the close connections between semidefinite matrices and polynomial optimization. This has brought about important new results and led to an even higher level of research activity. This Handbook on Semidefinite, Conic and Polynomial Optimization provides the reader with a snapshot of the state-of-the-art in the growing and mutually enriching areas of semidefinite optimization, conic optimization, and polynomial optimization. It contains a compendium of the recent research activity that has taken place in these thrilling areas, and will appeal to doctoral students, young graduates, and experienced researchers alike. The Handbook’s thirty-one chapters are organized into four parts: Theory, covering significant theoretical developments as well as the interactions between conic optimization and polynomial optimization; Algorithms, documenting the directions of current algorithmic development; Software, providing an overview of the state-of-the-art; Applications, dealing with the application areas where semidefinite and conic optimization has made a significant impact in recent years.
| Author | : Raúl E. Curto |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 74 |
| Release | : 1996-01-01 |
| Genre | : Mathematics |
| ISBN | : 9780821862919 |
