35,473 research outputs found
Output frequency response function-based analysis for nonlinear Volterra systems
Analysis of nonlinear systems has been studied extensively. Based on some recently developed results, a new systematic approach to the analysis of nonlinear Volterra systems in
the frequency domain is proposed in this paper, which provides a novel insight into the frequency domain analysis and design of nonlinear systems subject to a general input instead of only specific harmonic inputs using input-output experimental data. A general procedure to conduct an output frequency response function (OFRF) based analysis is given, and some fundamental results and techniques are established for this purpose. A case study for the analysis of a circuit system is provided to illustrate this new frequency domain method
On the convergence of maximum variance unfolding
Maximum Variance Unfolding is one of the main methods for (nonlinear)
dimensionality reduction. We study its large sample limit, providing specific
rates of convergence under standard assumptions. We find that it is consistent
when the underlying submanifold is isometric to a convex subset, and we provide
some simple examples where it fails to be consistent
Reciprocity-driven Sparse Network Formation
A resource exchange network is considered, where exchanges among nodes are
based on reciprocity. Peers receive from the network an amount of resources
commensurate with their contribution. We assume the network is fully connected,
and impose sparsity constraints on peer interactions. Finding the sparsest
exchanges that achieve a desired level of reciprocity is in general NP-hard. To
capture near-optimal allocations, we introduce variants of the Eisenberg-Gale
convex program with sparsity penalties. We derive decentralized algorithms,
whereby peers approximately compute the sparsest allocations, by reweighted l1
minimization. The algorithms implement new proportional-response dynamics, with
nonlinear pricing. The trade-off between sparsity and reciprocity and the
properties of graphs induced by sparse exchanges are examined.Comment: 19 page
Spin Glasses and Nonlinear Constraints in Portfolio Optimization
We discuss the portfolio optimization problem with the obligatory deposits
constraint. Recently it has been shown that as a consequence of this nonlinear
constraint, the solution consists of an exponentially large number of optimal
portfolios, completely different from each other, and extremely sensitive to
any changes in the input parameters of the problem, making the concept of
rational decision making questionable. Here we reformulate the problem using a
quadratic obligatory deposits constraint, and we show that from the physics
point of view, finding an optimal portfolio amounts to calculating the
mean-field magnetizations of a random Ising model with the constraint of a
constant magnetization norm. We show that the model reduces to an eigenproblem,
with 2N solutions, where N is the number of assets defining the portfolio.
Also, in order to illustrate our results, we present a detailed numerical
example of a portfolio of several risky common stocks traded on the Nasdaq
Market.Comment: 10 pages, 4 figure
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Adaptive grid semidefinite programming for finding optimal designs
We find optimal designs for linear models using anovel algorithm that iteratively combines a semidefinite programming(SDP) approach with adaptive grid techniques.The proposed algorithm is also adapted to find locally optimaldesigns for nonlinear models. The search space is firstdiscretized, and SDP is applied to find the optimal designbased on the initial grid. The points in the next grid set arepoints that maximize the dispersion function of the SDPgeneratedoptimal design using nonlinear programming. Theprocedure is repeated until a user-specified stopping rule isreached. The proposed algorithm is broadly applicable, andwe demonstrate its flexibility using (i) models with one ormore variables and (ii) differentiable design criteria, suchas A-, D-optimality, and non-differentiable criterion like Eoptimality,including the mathematically more challengingcasewhen theminimum eigenvalue of the informationmatrixof the optimal design has geometric multiplicity larger than 1. Our algorithm is computationally efficient because it isbased on mathematical programming tools and so optimalityis assured at each stage; it also exploits the convexity of theproblems whenever possible. Using several linear and nonlinearmodelswith one or more factors, we showthe proposedalgorithm can efficiently find optimal designs
Numerical Analysis
Acknowledgements: This article will appear in the forthcoming Princeton Companion to Mathematics, edited by Timothy Gowers with June Barrow-Green, to be published by Princeton University Press.\ud
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In preparing this essay I have benefitted from the advice of many colleagues who corrected a number of errors of fact and emphasis. I have not always followed their advice, however, preferring as one friend put it, to "put my head above the parapet". So I must take full responsibility for errors and omissions here.\ud
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With thanks to: Aurelio Arranz, Alexander Barnett, Carl de Boor, David Bindel, Jean-Marc Blanc, Mike Bochev, Folkmar Bornemann, Richard Brent, Martin Campbell-Kelly, Sam Clark, Tim Davis, Iain Duff, Stan Eisenstat, Don Estep, Janice Giudice, Gene Golub, Nick Gould, Tim Gowers, Anne Greenbaum, Leslie Greengard, Martin Gutknecht, Raphael Hauser, Des Higham, Nick Higham, Ilse Ipsen, Arieh Iserles, David Kincaid, Louis Komzsik, David Knezevic, Dirk Laurie, Randy LeVeque, Bill Morton, John C Nash, Michael Overton, Yoshio Oyanagi, Beresford Parlett, Linda Petzold, Bill Phillips, Mike Powell, Alex Prideaux, Siegfried Rump, Thomas Schmelzer, Thomas Sonar, Hans Stetter, Gil Strang, Endre Süli, Defeng Sun, Mike Sussman, Daniel Szyld, Garry Tee, Dmitry Vasilyev, Andy Wathen, Margaret Wright and Steve Wright
Stability results for logarithmic Sobolev and Gagliardo-Nirenberg inequalities
This paper is devoted to improvements of functional inequalities based on
scalings and written in terms of relative entropies. When scales are taken into
account and second moments fixed accordingly, deficit functionals provide
explicit stability measurements, i.e., bound with explicit constants distances
to the manifold of optimal functions. Various results are obtained for the
Gaussian logarithmic Sobolev inequality and its Euclidean counterpart, for the
Gaussian generalized Poincar{\'e} inequalities and for the Gagliardo-Nirenberg
inequalities. As a consequence, faster convergence rates in diffusion equations
(fast diffusion, Ornstein-Uhlenbeck and porous medium equations) are obtained
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