Eigenvalue Optimization Problem Research Articles

Existence of harmonic maps and eigenvalue optimization in higher dimensions

AbstractWe prove the existence of nonconstant harmonic maps of optimal regularity from an arbitrary closed manifold $(M^{n},g)$ ( M n , g ) of dimension $n>2$ n > 2 to any closed, non-aspherical manifold $N$ N containing no stable minimal two-spheres. In particular, this gives the first general existence result for harmonic maps from higher-dimensional manifolds to a large class of positively curved targets. In the special case of the round spheres $N=\mathbb{S}^{k}$ N = S k , $k\geqslant 3$ k ⩾ 3 , we obtain a distinguished family of nonconstant harmonic maps $M\to \mathbb{S}^{k}$ M → S k of index at most $k+1$ k + 1 , with singular set of codimension at least 7 for $k$ k sufficiently large. Furthermore, if $3\leqslant n\leqslant 5$ 3 ⩽ n ⩽ 5 , we show that these smooth harmonic maps stabilize as $k$ k becomes large, and correspond to the solutions of an eigenvalue optimization problem on $M$ M , generalizing the conformal maximization of the first Laplace eigenvalue on surfaces.

Finite element method for an eigenvalue optimization problem of the Schrödinger operator

<abstract><p>In this paper, we study the optimization algorithm to compute the smallest eigenvalue of the Schrödinger operator with volume constraint. A finite element discretization of this problem is established. We provide the error estimate for the numerical solution. The optimal solution can be approximated by a fixed point iteration scheme. Then a monotonic decreasing algorithm is presented to solve the eigenvalue optimization problem. Numerical simulations demonstrate the efficiency of the method.</p></abstract>

Delay-Dependent Sliding Mode Variable Structure Control of Vehicle Magneto-Rheological Semi-Active Suspension

The vehicle semi-active suspension with Magneto-Rheological Damper (MRD) has been a hot research topic of this decade, featuring the challenging task of the robust control with actuator time delay considerations. In this study, a delay dependent sliding mode variable structure control, based on the Linear Matrix Inequality (LMI), is proposed to suppress the vibration of the Magneto-Rheological Semi-Active Suspension (MRSS) control system. In accordance with the nonlinear characteristics of MRD, a dynamic model of automotive semi-active suspension system, considering time delay, is established. By defining a parameter-dependent Lyapunov switching functional, the conditions for asymptotic stability of closed-loop time delay system are derived, while the sliding mode variable structure control with reduced conservatism is designed. According to the method of LMI, the asymptotic stability problem of sliding mode is transformed into a feasibility problem, which can be solved by the solver ‘feasp’ in LMI toolbox. In addition, the calculation of the critical time delay of MRSS is expressed as a generalized eigenvalue optimization problem. For comparison purposes, three representative controllers, including a conventional sliding mode controller, a delay dependent controller, and a smith compensation, are studied. Simulation and real vehicle testing on bump and random road responses show that, the designed delay dependent controller can ensure the stability of the suspension system, weaken the influence of time delay on the control performance and effectively improve the ride comfort of the vehicle.

Minimum Variance Pole Placement in Uncertain Linear Control Systems

The pole-placement issue for linear multi-input multi-output (MIMO) dynamic systems with uncertain parameters has been addressed in this article. A static feedback matrix has been designed for minimizing variances of closed-loop poles (CLPs) and for assigning poles to the nominal system at the desired places. It is assumed that the joint probability density function (PDF) of uncertain parameters is known and the system has more than one input. A new unknown vector is used like an eigenvector for a stochastic closed-loop system matrix to state the problem. The variances of poles are considered as cost functions, and the means of poles are termed constraints. This form of the problem statement has helped us to simply find a solution. In the first step, the optimization problem with constraints was handled by solving the equality constraint, and then, the problem was converted to a classic extended eigenvalue optimization problem. Later, the eigenvalue optimization problem was solved by the Rayleigh quotient and the feedback matrix was accomplished. Finally, this approach was simulated and validated using the MATLAB simulations, and the results were compared with a robust pole-placement method, which MATLAB control toolbox uses. The Monte Carlo simulations showed lower covariance for CLPs around the mean poles as compared to the robust pole-placement method.

Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems

Recently Fraser and Schoen showed that the solution of a certain extremal Steklov eigenvalue problem on a compact surface with boundary can be used to generate a free boundary minimal surface, i.e., a surface contained in the ball that has (i) zero mean curvature and (ii) meets the boundary of the ball orthogonally (doi:10.1007/s00222-015-0604-x). In this paper, we develop numerical methods that use this connection to realize free boundary minimal surfaces. Namely, on a compact surface, Σ, with genus γ and b boundary components, we maximize σj(Σ, g) L(∂Σ, g) over a class of smooth metrics, g, where σj(Σ, g) is the jth nonzero Steklov eigenvalue and L(∂Σ, g) is the length of ∂Σ. Our numerical method involves (i) using conformal uniformization of multiply connected domains to avoid explicit parameterization for the class of metrics, (ii) accurately solving a boundary-weighted Steklov eigenvalue problem in multi-connected domains, and (iii) developing gradient-based optimization methods for this non-smooth eigenvalue optimization problem. For genus γ = 0 and b = 2, …, 9, 12, 15, 20 boundary components, we numerically solve the extremal Steklov problem for the first eigenvalue. The corresponding eigenfunctions generate a free boundary minimal surface, which we display in striking images. For higher eigenvalues, numerical evidence suggests that the maximizers are degenerate, but we compute local maximizers for the second and third eigenvalues with b = 2 boundary components and for the third and fifth eigenvalues with b = 3 boundary components.

Symmetry and rigidity results for composite membranes and plates

The composite membrane problem is an eigenvalue optimization problem deeply studied from the beginning of the '00's. In this note we survey most of the results proved by several authors over the last twenty years, up to the recent paper [14] written in collaboration with Giovanni Cupini.We finally introduce an eigenvalue optimization problem for a fourth order operator, called composite plate problem and we present the symmetry and rigidity results obtained in this framework. These last mentioned results are part of the papers [12,13], written in collaboration with Francesca Colasuonno.

Nonsmooth algorithms for minimizing the largest eigenvalue with applications to inner numerical radius

Abstract Nonsmoothness at optimal points is a common phenomenon in many eigenvalue optimization problems. We consider two recent algorithms to minimize the largest eigenvalue of a Hermitian matrix dependent on one parameter, both proven to be globally convergent unaffected by nonsmoothness. One of these algorithms models the eigenvalue function with a piece-wise quadratic function and is effective in dealing with nonconvex problems. The other algorithm projects the Hermitian matrix into subspaces formed of eigenvectors and is effective in dealing with large-scale problems. We generalize the latter slightly to cope with nonsmoothness. For both algorithms we analyze the rate of convergence in the nonsmooth setting, when the largest eigenvalue is multiple at the minimizer and zero is strictly in the interior of the generalized Clarke derivative, and prove that both algorithms converge rapidly. The algorithms are applied to, and the deduced results are illustrated on the computation of the inner numerical radius, the modulus of the point on the boundary of the field of values closest to the origin, which carries significance for instance for the numerical solution of a symmetric definite generalized eigenvalue problem and the iterative solution of a saddle point linear system.

Optimization of Cascading Processes in Arbitrary Networks with Stochastic Interactions

We consider the problem of optimal propagation of cascades in a network, where the propagation process depends on the values that a decision-maker assigns to the network and independent influence rates. Assuming that there are costs associated with changing the values of these rates, we investigate the question of resource allocation that minimizes the time by which the cascading process reaches all nodes. To this end, we propose a model that represents the cascading process in the network as a continuous time Markov chain (CTMC). We show how the propagation problem can be framed as an eigenvalue optimization problem on the generator of the CTMC. In turn, this problem is successively transformed into a maximum minimum cut problem and into a generalized maximum flow problem on a linearly-sized auxiliary graph. Therefore, the optimization problem can be solved in strongly polynomial time in terms of the size of the network. In addition, whenever there are no preexisting influence rates, the problem can also be transformed into the problem of finding a minimum spanning arborescence on a similar auxiliary graph. In this setting the optimal solution provides a hierarchy that classifies the nodes in terms of their importance to the cascade propagation.

Optimization of Communication Network Topology in Distributed Control Systems Subject to Prescribed Decay Rate.

In this paper, we propose a simple cohesive framework to find an optimal directed control network topology with minimum number of links while a prescribed decay rate is satisfied in the transient response of a distributed control system. In order to guarantee the system's decay rate to be faster than a prespecified value, a constraint on the dominant eigenvalue of the system is required to be considered. This results in a nonconvex optimization problem as eigenvalue of a parametric nonsymmetric matrix is a nonconvex, nonsmooth, and even non-Lipschitz function. Here, we present a convex equivalent optimization problem whose minimizer also solves this eigenvalue optimization problem. This optimization problem proposes a state-feedback matrix which results in a decay rate faster than a given value while input signal costs are considered. The equivalent optimization problem in combination with sparsity-promoting optimal control constitutes a combinatorial optimization problem. Using alternating direction method of multipliers, the problem is decomposed into a chain of analytically solvable subproblems which are differentiable and separable. The proposed optimization framework includes relative preference between the topology of the control network and the decay rate of the system. The simulation results show the effectiveness of the proposed framework.

Symmetry and rigidity for the hinged composite plate problem

The composite plate problem is an eigenvalue optimization problem related to the fourth order operator (−Δ)2. In this paper we continue the study started in [10], focusing on symmetry and rigidity issues in the case of the hinged composite plate problem, a specific situation that allows us to exploit classical techniques like the moving plane method.

Subspace Acceleration for the Crawford Number and Related Eigenvalue Optimization Problems

This paper is concerned with subspace acceleration techniques for computing the Crawford number, that is, the distance between zero and the numerical range of a matrix $A$. Our approach is based on an eigenvalue optimization characterization of the Crawford number. We establish local convergence of order $1+\sqrt{2}\approx 2.4$ for an existing subspace method applied to such and other eigenvalue optimization problems involving a Hermitian matrix that depends analytically on one parameter. For the particular case of the Crawford number, we show that the relevant part of the objective function is strongly concave. In turn, this enables us to develop a subspace method that only uses three-dimensional subspaces but still achieves global convergence and a local convergence that is at least quadratic. A number of numerical experiments confirm our theoretical results and reveal that the established convergence orders appear to be tight.

A Subspace Method for Large-Scale Eigenvalue Optimization

We consider the minimization or maximization of the $J$th largest eigenvalue of an analytic and Hermitian matrix-valued function, and build on Mengi, Yildirim, and Kilic [SIAM J. Matrix Anal. Appl., 35, pp. 699--724, 2014]. This work addresses the setting when the matrix-valued function involved is very large. We describe subspace procedures that convert the original problem into a small-scale one by means of orthogonal projections and restrictions to certain subspaces, and that gradually expand these subspaces based on the optimal solutions of small-scale problems. Global convergence and superlinear rate-of-convergence results with respect to the dimensions of the subspaces are presented in the infinite dimensional setting, where the matrix-valued function is replaced by a compact operator depending on parameters. In practice, it suffices to solve eigenvalue optimization problems involving matrices with sizes on the scale of tens, instead of the original problem involving matrices with sizes on the scale...

Kernel-based covariate functional balancing for observational studies

Covariate balance is often advocated for objective causal inference since it mimics randomization in observational data. Unlike methods that balance specific moments of covariates, our proposal attains uniform approximate balance for covariate functions in a reproducing-kernel Hilbert space. The corresponding infinite-dimensional optimization problem is shown to have a finite-dimensional representation in terms of an eigenvalue optimization problem. Large-sample results are studied, and numerical examples show that the proposed method achieves better balance with smaller sampling variability than existing methods.

Eigenvector sensitivity when tracking modes with repeated eigenvalues

While eigenvalue optimization problems have been widely studied during the last three decades, particularly for structural and topology optimization problems, there are very few examples of problems involving mode shapes either in the cost or the constraints. In this paper, we propose a general framework for computing eigenvector sensitivity whenever tracking specific mode shapes selected beforehand. Of course, the approach is valid for either non-repeated or repeated eigenvalues, but here the emphasis is placed on the multiple eigenvalues case. Both mathematical validity of the algorithm and numerical corroboration through several examples are included.

Frequency response as a surrogate eigenvalue problem in topology optimization

SummaryThis article discusses the use of frequency response surrogates for eigenvalue optimization problems in topology optimization that may be used to avoid solving the eigenvalue problem. The motivation is to avoid complications that arise from multiple eigenvalues and the computational complexity associated with computation of eigenvalues in very large problems. Copyright © 2017 John Wiley & Sons, Ltd.

Maximization of Laplace−Beltrami eigenvalues on closed Riemannian surfaces

Let $(M,g)$ be a connected, closed, orientable Riemannian surface and denote by $\lambda_k(M,g)$ the $k$-th eigenvalue of the Laplace-Beltrami operator on $(M,g)$. In this paper, we consider the mapping $(M, g)\mapsto \lambda_k(M,g)$. We propose a computational method for finding the conformal spectrum $\Lambda^c_k(M,[g_0])$, which is defined by the eigenvalue optimization problem of maximizing $\lambda_k(M,g)$ for $k$ fixed as $g$ varies within a conformal class $[g_0]$ of fixed volume $textrm{vol}(M,g) = 1$. We also propose a computational method for the problem where $M$ is additionally allowed to vary over surfaces with fixed genus, $\gamma$. This is known as the topological spectrum for genus $\gamma$ and denoted by $\Lambda^t_k(\gamma)$. Our computations support a conjecture of N. Nadirashvili (2002) that $\Lambda^t_k(0) = 8 \pi k$, attained by a sequence of surfaces degenerating to a union of $k$ identical round spheres. Furthermore, based on our computations, we conjecture that $\Lambda^t_k(1) = \frac{8\pi^2}{\sqrt{3}} + 8\pi (k-1)$, attained by a sequence of surfaces degenerating into a union of an equilateral flat torus and $k-1$ identical round spheres. The values are compared to several surfaces where the Laplace-Beltrami eigenvalues are well-known, including spheres, flat tori, and embedded tori. In particular, we show that among flat tori of volume one, the $k$-th Laplace-Beltrami eigenvalue has a local maximum with value $\lambda_k = 4\pi^2 \left\lceil \frac{k}{2} \right\rceil^2 \left( \left\lceil \frac{k}{2} \right\rceil^2 - \frac{1}{4}\right)^{-\frac{1}{2}}$. Several properties are also studied computationally, including uniqueness, symmetry, and eigenvalue multiplicity.

A minimization problem for an elliptic eigenvalue problem with nonlinear dependence on the eigenparameter

In this paper we examine an eigenvalue optimization problem. Given two nonlinear functions α(λ) and β(λ), find a subset D of the unit ball of measure A for which the first Dirichlet eigenvalue of the operator −div((α(λ)χD+β(λ)χDc)∇u)=λu is as small as possible. This sort of nonlinear eigenvalue problems arises in the study of some quantum dots taking into account an electron effective mass. We establish the existence of a solution, and we propose a numerical algorithm to obtain an approximate description of the optimizer.

Combating Strong–Weak Spatial–Temporal Resonances in Time-Reversal Uplinks

In a time-reversal (TR) communication system, the signal-to-noise ratio (SNR) is boosted and the inter-user interference (IUI) is suppressed due to the spatial–temporal resonances, commonly known as the focusing effects, of the TR technique when implemented in a rich scattering environment. However, since the spatial–temporal resonances highly depend on the location-specific multipath profile, there exists a strong–weak spatial–temporal resonances effect. In the TR uplink system, different users at different locations enjoy different strengths of spatial–temporal resonances, i.e., the received signal-to-interference-noise ratios (SINRs) for different users vary, and the weak ones can be blocked from correct detection in the presence of strong ones. In this paper, we formulate the strong–weak spatial–temporal resonances in the multiuser TR uplink system as a max–min weighted SINR balancing problem by joint power control and signature design. Then, a novel two-stage adaptive algorithm that can guarantee the convergence is proposed. In stage I, the original nonconvex problem is relaxed into a Perron Frobenius eigenvalue optimization problem and an iterative algorithm is proposed to obtain the optimum efficiently. In stage II, the gradient search method is applied to update the relaxed feasible set until the global optimum for the original optimization problem is obtained. Numerical results show that our algorithm converges quickly, achieves a high energy-efficiency, and provides a performance guarantee to all users.

Low-Rank Spectral Optimization via Gauge Duality

Various applications in signal processing and machine learning give rise to highly structured spectral optimization problems characterized by low-rank solutions. Two important examples that motivate this work are optimization problems from phase retrieval and from blind deconvolution, which are designed to yield rank-1 solutions. An algorithm is described that is based on solving a certain constrained eigenvalue optimization problem that corresponds to the gauge dual which, unlike the more typical Lagrange dual, has an especially simple constraint. The dominant cost at each iteration is the computation of rightmost eigenpairs of a Hermitian operator. A range of numerical examples illustrate the scalability of the approach.

Observability for optimal sensor locations in data assimilation

In this paper we will discuss the application of observability to the planning of sensor configurations in numerical weather prediction (NWP). The dimensions used in NWP make conventional definitions of observability impractical. For this reason we will rely partial observability which is obtained using dynamic optimization to approximate the observability. Using this metric we will form an optimization problem to select sensor configurations that maximize the partial observability of the dynamical system. This leads to a max–min problem which using an empirical gramian matrix we reduce to an eigenvalue optimization problem. Atmospheric data assimilation is the process of combining prior knowledge with observations to form an estimate of the system state required to produce a forecast of future weather conditions. Optimal sensor configurations leading to improved forecast quality are of interest. Due to the potential size of our intended application we will focus on computational methods that are both efficient and scalable. We will also leverage existing tools used in data assimilation and introduce tools used in nonsmooth optimization.