Optimal Eigenvalues Research Articles

Spectral optimization for weighted anisotropic problems with Robin conditions

We study a weighted eigenvalue problem with anisotropic diffusion in bounded Lipschitz domains Ω⊂RN, N≥1, under Robin boundary conditions, proving the existence of two positive principal eigenvalues λ± respectively associated with a positive and a negative eigenfunction. Next, we analyze the minimization of λ± with respect to the sign-changing weight, showing that the optimal eigenvalues Λ± are equal if the domain has a center of symmetry and the optimal weights are of bang-bang type, namely piece-wise constant functions, each one taking only two values. As a consequence, the problem is equivalent to the minimization with respect to the subsets of Ω satisfying a volume constraint. Then, we completely solve the optimization problem in one dimension, in the case of homogeneous Dirichlet or Neumann conditions, showing new phenomena induced by the presence of the anisotropic diffusion. The optimization problem for λ+ naturally arises in the study of the optimal spatial arrangement of resources for a species to survive in an heterogeneous habitat.

Lyapunov control of finite-dimensional quantum systems based on bi-objective quantum-behaved particle swarm optimization algorithm

This paper presents a Lyapunov control scheme to drive finite-dimensional closed and Markovian open quantum systems into any target pure state with as high fidelity and as little time as possible. The control law is established by a Lyapunov function with an unknown Hermitian operator, where the eigenvectors of the operator are constructed based on the condition that the LaSalle invariant set contains the desired target state and a set of optimal eigenvalues of the Hermitian operator is searched by the quantum-behaved particle swarm optimization (QPSO) algorithm. In particular, for open systems, when the denominator in the control law is quite small, we propose an improved QPSO algorithm with constraints to enhance the flexibility in implementation. Finally, numerical simulations are carried out on a five-level and a three-qubit closed quantum systems, and a five-level and a ten-level open quantum systems. Simulation results demonstrate that the proposed control scheme can achieve high-fidelity transfer to desired target states for high-dimensional closed and open quantum systems.

Differential diagnosis of pulmonary tuberculosis and talsromycosis marneffei by computed tomography-derived radiomics in patients with acquired immunodeficiency syndrome

OBJECTIVE: To investigate the value of computed tomography (CT)-derived radiomics features in the differential diagnosis of pulmonary tuberculosis (PTB) and talaromycosis marneffei (TSM) in patients with acquired immunodeficiency syndrome (AIDS). MATERIALS AND METHODS: The venous phase images for 166 patients with AIDS (PTB, n = 66; TSM, n = 99) were retrospectively analyzed, and the radiomics features of lung lesions and mediastinal lymph nodes were extracted. The samples were divided into a training set and a test set in a ratio of 8:2. The optimal eigenvalues were used to establish four prediction models: radiomics model 1 (PTB group and TSM lung lesions), radiomics model 2 (PTB group and TSM lung lesions), radiomics model 3 (pulmonary lesions without lymph node enhancement), and radiomics model 4 (pulmonary lesions with lymph node enhancement). The working characteristic curve was used to evaluate the predictive performance of the model. RESULTS: The accuracy, sensitivity, specificity, and area under the curve values were 0.67, 0.78, 0.78, and 0.735, respectively, for the radiomics model 1 test set; 0.67, 0.62, 0.67, and 0.654, respectively, for radiomics model 2; 0.89, 0.76, 0.80, and 0.833, respectively, for radiomics model 3; and 0.76, 0.80, 0.88, and 0.886, respectively, for radiomics model 4. CONCLUSION: The prediction model based on CT-derived radiomics features has value for the identification of PTB and TSM. The radiomics model based on the optimal eigenvalues of lung lesions combined with lymph node plain scan images is compared with the establishment of a single lung. The focal omics feature model has better predictive power.

Underwater image dehazing using global color features

Accurate underwater imaging is a challenging task. Unlike regular photography, visibility underwater is drastically hazed by water molecules and suspended particles distorting light rays and causing differential absorption of different wavelengths of color. Existing research produces well-adjusted but not perfect results. In this work, we present a novel method that dehazed underwater images by estimating global background light based on the optimal eigenvalues of a matrix constructed from three components: (1) gradient of the pairwise wavelength of color channels (blue–red (br), blue–green (bg), and green–red (gr)) (2) gradient of the wavelength of color channels, and (3) the color channels themselves. We estimate transmission maps via scene depth by exploiting the difference in absorption of the different color channel wavelengths. The proposed technique is executed by augmenting UWCNN (Under-Water CNN) with graph-cut theory. The resultant dehazed images successfully show greatly improved color. The proposed results outperform the existing methods in terms of entropy, UIQMnorm, UICM, UISM, and UCIQE. These improvements will provide stronger imaging tools to domains like submarine search and rescue, navigation, oceanography, and mapping. There is room for future research because our process slightly darkens images which leads to marginally lower UIConM values. Future study can also further evaluate the effect of color channels while assuming pixels with the highest intensity are used as the global ambient light.

Active triggering control of pneumatic rehabilitation gloves based on surface electromyography sensors.

The portable and inexpensive hand rehabilitation robot has become a practical rehabilitation device for patients with hand dysfunction. A pneumatic rehabilitation glove with an active trigger control system is proposed, which is based on surface electromyography (sEMG) signals. It can trigger the hand movement based on the patient’s hand movement trend, which may improve the enthusiasm and efficiency of patient training. Firstly, analysis of sEMG sensor installation position on human’s arm and signal acquisition process were carried out. Then, according to the statistical law, three optimal eigenvalues of sEMG signals were selected as the follow-up neural network classification input. Using the back propagation (BP) neural network, the classifier of hand movement is established. Moreover, the mapping relationship between hand sEMG signals and hand actions is built by training and testing. Different patients choose the same optimal eigenvalues, and the calculation formula of eigenvalues’ amplitude is unique. Due to the differences among individuals, the weights and thresholds of each node in the BP neural network model corresponding to different patients are not the same. Therefore, the BP neural network model library is established, and the corresponding network is called for operation when different patients are trained. Finally, based on sEMG signal trigger, the pneumatic glove training control algorithm was proposed. The combination of the trigger signal waveform and the motion signal waveform indicates that the pneumatic rehabilitation glove is triggered to drive the patient’s hand movement. Preliminary tests have confirmed that the accuracy rate of trend recognition for hand movement is about 90%. In the future, clinical trials of patients will be conducted to prove the effectiveness of this system.

Robust Observer Based on Fixed-Time Sliding Mode Control of Position/Velocity for a T-S Fuzzy MEMS Gyroscope

This study focused on a control system of the nonlinear micro-electro-mechanical systems (MEMS) gyroscope. First, sector nonlinearity was used to model a MEMS gyroscope in the Takagi-Sugeno (T-S) fuzzy system. Second, a state observer was designed based on linear matrix inequality (LMI) to identify the optimal eigenvalues of the state tracking error function. Then, full-state fixed-time sliding mode control (FTSMC) was constructed to control the system. Third, a case study of a harmonic disturbance observer was used to address the unknown disturbance of the system. A disturbance observer (DOB) was simply designed based on the error signals of the system outputs and observer outputs. The output signals precisely converged to the predefined trajectories in a very short time, with no overshoots and small of steady-state errors. Moreover, the estimated output states were precisely tracked by the system outputs. These important factors were used to confirm that the control of the T-S fuzzy MEMS was effective and easy to achieve. The study used MATLAB simulation to archive the verification. The maximum of tracking error was ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> ∈ [-4.657:5.565]×10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-11</sup> , and the maximum settling time was T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e3</sub> ~ 0.144 for the error of the ẏ- axis and the settling time of the ẋ- axis, respectively.

Breaking Waves and Spectral Analysis of the Two‐Dimensional KdV–Bogoyavlenskii Equation

AbstractWe study here the initial value problem for a two‐dimensional Korteweg–de Vries (KdV) equation, first derived by Calogero and Bogoyavlenskii, by means of the inverse scattering transform. The dynamics of the discrete spectrum of an associated Schrödinger operator is far richer than that of KdV equation. Even for optimal eigenvalues, generic smooth solutions may develop shocks with multiple branches and/or cusp singularities in finite time. However, evolution may move poles of the transmission coefficient off the imaginary axis, destroy or even create them. We characterize conditions to prevent these pathologies before explosion time and describe ample classes of solutions, corresponding to both continuous and discrete spectrum. We also find that in certain conditions new eigenvalues might be created; in these cases a minimal set of initial spectral data must incorporate additionally the transmission coefficient on the entire plane. The previous results are applied to describe the Cauchy problem corresponding to initial data combinations of delta terms and derivatives and show that for long time the delta singularity may persist or be smoothed to a cusp‐discontinuity. Finally, we give conditions under which the evolution is reduced to the classical KdV.

Asymptotic Behaviour of Cuboids Optimising Laplacian Eigenvalues

We prove that in dimension $$n \ge 2$$ , within the collection of unit-measure cuboids in $$\mathbb {R}^n$$ (i.e. domains of the form $$\prod _{i=1}^{n}(0, a_n)$$ ), any sequence of minimising domains $$R_k^\mathcal {D}$$ for the Dirichlet eigenvalues $$\lambda _k$$ converges to the unit cube as $$k \rightarrow \infty $$ . Correspondingly we also prove that any sequence of maximising domains $$R_k^\mathcal {N}$$ for the Neumann eigenvalues $$\mu _k$$ within the same collection of domains converges to the unit cube as $$k\rightarrow \infty $$ . For $$n=2$$ this result was obtained by Antunes and Freitas in the case of Dirichlet eigenvalues and van den Berg, Bucur and Gittins for the Neumann eigenvalues. The Dirichlet case for $$n=3$$ was recently treated by van den Berg and Gittins. In addition we obtain stability results for the optimal eigenvalues as $$k \rightarrow \infty $$ . We also obtain corresponding shape optimisation results for the Riesz means of eigenvalues in the same collection of cuboids. For the Dirichlet case this allows us to address the shape optimisation of the average of the first k eigenvalues.

Optimisation of Eigenvalues of the Dirichlet Laplacian with a Surface Area Restriction

We perform a numerical optimisation of the low frequencies of the Dirichlet Laplacian with perimeter and surface area restrictions, in two and 3-dimensions, respectively. In the former case, we handle the first 50 eigenvalues and measure the rate at which the corresponding optimisers approach the disk, while in the latter we optimise the first twenty eigenvalues. We derive theoretical compatibility conditions which must be satisfied by a sequence of optimisers and test our numerical results against these. We also consider the cases of rectangles with a fixed perimeter and parallelepipeds with a surface restriction for which we compute the first $$10^7$$107 and $$10^6$$106 optimal eigenvalues, respectively. In this context, we prove convergence to the cube in any dimensions and compare the numerical results with our theoretical estimates for the rate of convergence.

Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin Laplacian

We consider the problem of minimising the n th-eigenvalue of the Robin Laplacian in RN . Although for n = 1,2 and a positive boundary parameter α it is known that the minimisers do not depend on α , we demonstrate numerically that this will not always be the case and illustrate how the optimiser will depend on α . We derive a Wolf–Keller type result for this problem and show that optimal eigenvalues grow at most with n 1/N , which is in sharp contrast with the Weyl asymptotics for a fixed domain. We further show that the gap between consecutive eigenvalues does go to zero as n goes to infinity. Numerical results then support the conjecture that for each n there exists a positive value of α n such that the n th eigenvalue is minimised by n disks for all 0 n and, combined with analytic estimates, that this value is expected to grow with n 1/N .

Optimization of Input Covariance Matrix for Multi-Antenna Correlated Channels

Optimizing the input covariance matrix of a multiple-antenna transmit system with partial channelstructure feedback is an important issue to fully exploit the channel capacity. Efficient design of the optimal input covariance matrix, however, remains unavailable although its eigenvector structure was clearly revealed in a recent publication. In this paper, we obtain an explicit derivative function forming a solid basis for optimizing the optimal input covariance matrix. This new derivative expression enables us to further develop an efficient iterative algorithm for determining the optimal eigenvalues. The technique is illustrated through numerical examples.

Face recognition using bacterial foraging strategy

This article presents an efficient algorithm for LDA-based face recognition with the selection of optimal principal components using E-coli Bacterial Foraging Optimization Technique. Different methods were suggested in the literature to select the largest eigenvalues and their corresponding eigenvectors for linear discriminant analysis (LDA). Some researchers have suggested eliminating the three largest eigenvalues to avoid the effect under varying illumination conditions. But, there is no unified approach for selecting optimal eigenvalues to enhance the performance of an algorithm. In this context, a GA–PCA algorithm has been proposed to select optimal eigenvalues and their corresponding eigenvectors in LDA. They proposed a fitness function to find the optimal eigenvectors using the Genetic Algorithm (GA). However, the crossover method used results in differences in offspring, and mutation never allowed them for a physical dispersal of the child in a chosen area. This prevents us in selecting optimal eigenvectors for improvising accuracy of the face recognition algorithm. This has motivated the authors to develop a new algorithm called BFO-Fisher which uses a nutrient concentration function (cost function) for optimization. In this work, the cost function is maximized through hill climbing via a type of biased random walk which is not possible in GA. Here the proposed BFO-Fisher algorithm offers us two distinct additional advantages—(i) the proposed algorithm can supplement the features of GA, and (ii) the random bias incorporated into the BFO algorithm guides us to move in the direction of increasingly favorable environment, which is desirable. In this experiment, both Yale and UMIST Databases are used for the performance evaluation. Experimental results presented in this article reveal the fact that about 3% (Rank 1) improvement can be achieved as compared to the GA-Fisher algorithm.

Capacity Approximations for a Deterministic MIMO Channel

Abstract : In this paper, we derive closed form approximations for the capacity of a point-to-point, deterministic Gaussian MIMO communication channel. We focus on the behavior of the inverse eigenvalues of the Gram matrix associated with the gain matrix of the MIMO channel, by considering small variance and large power assumptions. We revisit the concept of deterministic MIMO capacity by pointing out that, under transmitter power constraint, the optimal transmit covariance matrix is not necessarily diagonal. We discuss the water filling algorithm for obtaining the optimal eigenvalues of the transmitter covariance matrix, and the water fill level in conjunction with the Karush-Kuhn-Tucker optimality conditions. We revise the Telatar conjecture for the capacity of a non-ergodic channel. We also provide deterministic examples and numerical simulations of the capacity, which are discussed in terms of our mathematical framework.

Optimal mass distribution in vibrating systems

The problem of determining the distribution of mass in vibrating systems that optimize the extreme eigenvalue is considered. We give explicit solutions for the fundamental discrete models of an axially vibrating rod and a Bernoulli–Euler beam. The problems are first recast as affine sum of matrices with linear total mass constraint. This formulation allows determination of the eigenvectors corresponding to the optimal eigenvalues by solving a set of quadratic equations recursively. The optimal mass distribution is then determined by the recovered eigenvectors.

Model Order Reduction of 1D Diffusion Systems Via Residue Grouping

A model order reduction method is developed and applied to 1D diffusion systems with negative real eigenvalues. Spatially distributed residues are found either analytically (from a transcendental transfer function) or numerically (from a finite element or finite difference state space model), and residues with similar eigenvalues are grouped together to reduce the model order. Two examples are presented from a model of a lithium ion electrochemical cell. Reduced order grouped models are compared to full order models and models of the same order in which optimal eigenvalues and residues are found numerically. The grouped models give near-optimal performance with roughly 1∕20 the computation time of the full order models and require 1000–5000 times less CPU time for numerical identification compared to the optimization procedure.

Optimal Transmission Strategies and Impact of Correlation in Multiantenna Systems with Different Types of Channel State Information

We study the optimal transmission strategy of a multiple-input single-output (MISO) wireless communication link. The receiver has perfect channel state information (CSI), while the transmitter has different types of CSI, i.e., either perfect CSI, or no CSI, or long-term knowledge of the channel covariance matrix. For the case in which the transmitter knows the channel covariance matrix, it was recently shown that the optimal eigenvectors of the transmit covariance matrix correspond with the eigenvectors of the channel covariance matrix. However, the optimal eigenvalues are difficult to compute. We derive a characterization of the optimum power allocation. Furthermore, we apply this result to provide an efficient algorithm which computes the optimum power allocation. In addition to this, we analyze the impact of correlation on the ergodic capacity of the MISO system with different CSI schemes. At first, we justify the belief that equal power allocation is optimal if the transmitter is uninformed and the transmit antennas are correlated. Next, we show that the ergodic capacity with perfect CSI and without CSI at the transmitter is Schur-concave, i.e., the more correlated the transmit antennas are, the less capacity is achievable. In addition, we show that the ergodic capacity with covariance knowledge at the transmitter is Schur-convex with respect to the correlation properties. These results completely characterize the impact of correlation on the ergodic capacity in MISO systems. Furthermore, the capacity loss or gain due to correlation is quantified. For no CSI and perfect CSI at the transmitter, the capacity loss due to correlation is bounded by some small constant, whereas the capacity gain due to correlation grows unbounded with the number of transmit antennas in the case in which transmitter knows the channel covariance matrix. Finally, we illustrate all theoretical results by numerical simulations.

Optimal Transmission with Imperfect Channel State Information at theTransmit Antenna Array

We study the optimal transmission strategy of a multiple-input single-output wireless communication link. The receiver has perfect channel state information while the transmitter has only long-term channel state information in terms of the channel covariance matrix. It was recently shown that the optimal eigenvectors of the transmit covariance matrix correspond with the eigenvalues of the channel covariance matrix. However, the optimal eigenvalues are difficult to compute. We study the properties of these optimal capacity achieving eigenvalues, and present a necessary and sufficient condition for the optimal eigenvalues of the transmit covariance matrix. Furthermore, we develop a necessary and sufficient condition for achieving capacity when transmitting in all directions. We compare the capacity gain of an optimal diversity system with a system which works with beamforming, and we derive an upper bound. We answer the main questions regarding the system design using the developed results. Additionally, we show in which way the multiplexing gain can be computed in case the channel covariance matrix is given. We compute the maximum number of required parallel data streams, and we define a multiplexing function in order to obtain a measure for the available multiplexing gain. Furthermore, we show that the capacity gain is small considering the additional complexity at the receiver. We illustrate all results by numerical simulations.

OPTIMAL DESIGN OF INTERNAL RING SUPPORT FOR RECTANGULAR PLATES AGAINST VIBRATION OR BUCKLING

This paper is concerned with the optimal design of an internal ring support for rectangular plates against vibration or buckling. Presented herein is an effective numerical method for solving the optimization problem. It involves the use of (1) the so-called pb-2 Ritz method for the eigenvalue analysis and (2) the Nelder and Mead simplex method for searching for the optimal radius and location of the ring support which maximize either the fundamental frequency or the buckling load of the plate. By considering plates with different boundary conditions, extensive optimal results have been obtained. The sensitivity of the optimal eigenvalues with respect to the design of the ring support is also investigated.

Two formulations of exponentially weighted enhanced LQ controllers

Properties of a class of multiinput multioutput digital controllers are examined. The controllers incorporate three features: (1) optimal gains associated with known exogenous inputs; (2) penalties imposed both on squares of first backward differences in control actions and on control actions squared; and (3) exponential weighting to ensure bounds on closed-loop eigenvalues. Two versions of exponential weighting are treated. Results are given for both the finite-horizon and infinite-horizon versions of the problems. The results include expressions for optimal feedback gains and for optimal preview gains. For the infinite-horizon versions, relationships are derived between the optimal closed-loop eigenvalues and the eigenvalues associated with the preview gains. Two examples are given to illustrate properties of three controllers. >

MINIMAX SOLUTIONS IN STOCHASTIC PROGRAMMING

Abstract Minimax and other game-theoretic solutions are analyzed under linear programming problems where the objective functions are stochastic due to output and price variations. By applying the complementarity principles of quadratic programming, it is shown that this class of models leads to generalized eigenvalue problems. A set of theorems is developed to characterize the optimal eigenvalues. Lines of potential applications are also indicated in brief.