Controlling Stochastic Sensitivity by Feedback Regulators in Nonlinear Dynamical Systems with Incomplete Information

The problem of synthesis of stochastic sensitivity for equilibrium modes in nonlinear randomly forced dynamical systems with incomplete information is considered. We construct a feedback regulator that uses noisy data on some system state coordinates. For parameters of the regulator providing assigned stochastic sensitivity, a quadratic matrix equation is derived. Attainability of the assigned stochastic sensitivity is reduced to the solvability of this equation. We suggest a constructive algorithm for solving this quadratic matrix equation. These general theoretical results are used to solve the problem of stabilizing equilibrium modes of nonlinear stochastic oscillators under conditions of incomplete information. Details of our approach are illustrated on the example of a van der Pol oscillator.

About a fixed-point-type transformation to solve quadratic matrix equations using the Krasnoselskij method.

In this paper, we study the simplest quadratic matrix equation: $\mathcal{Q}(X)=X^2+BX+C=0$. We transform this equation into an equivalent fixed-point equation and based on it we construct the Krasnoselskij method. From this transformation, we can obtain iterative schemes more accurate than successive approximation method. Moreover, under suitable conditions, we establish different results for the existence and localization of a solution for this equation with the Krasnoselskij method. Finally, we see numerically that the predictor-corrector iterative scheme with the Krasnoselskij method as a predictor and the Newton method as corrector method, can improves the numerical application of the Newton method when approximating a solution of the quadratic matrix equation.

Inexact Inverse Subspace Iteration with Preconditioning Applied to Quadratic Matrix Polynomials

An inexact variant of inverse subspace iteration is used to find a small invariant pair of a large quadratic matrix polynomial. It is shown that linear convergence is preserved provided the inner iteration is performed with increasing accuracy. A preconditioned block GMRES solver is employed as inner iteration. The preconditioner uses the strategy of “tuning” which prevents the inner iteration from increasing and therefore results in a substantial saving in costs. The accuracy of the computed invariant pair can be improved by the addition of a post-processing step involving very few iterations of Newton’s method. The effectiveness of the proposed approach is demonstrated by numerical experiments.

Hyperspectral Unmixing Based on Constrained Bilinear or Linear-Quadratic Matrix Factorization

Unsupervised hyperspectral unmixing methods aim to extract endmember spectra and infer the proportion of each of these spectra in each observed pixel when considering linear mixtures. However, the interaction between sunlight and the Earth’s surface is often very complex, so that observed spectra are then composed of nonlinear mixing terms. This nonlinearity is generally bilinear or linear quadratic. In this work, unsupervised hyperspectral unmixing methods, designed for the bilinear and linear-quadratic mixing models, are proposed. These methods are based on bilinear or linear-quadratic matrix factorization with non-negativity constraints. Two types of algorithms are considered. The first ones only use the projection of the gradient, and are therefore linked to an optimal manual choice of their learning rates, which remains the limitation of these algorithms. The second developed algorithms, which overcome the above drawback, employ multiplicative projective update rules with automatically chosen learning rates. In addition, the endmember proportions estimation, with three alternative approaches, constitutes another contribution of this work. Besides, the reduction of the number of manipulated variables in the optimization processes is also an originality of the proposed methods. Experiments based on realistic synthetic hyperspectral data, generated according to the two considered nonlinear mixing models, and also on two real hyperspectral images, are carried out to evaluate the performance of the proposed approaches. The obtained results show that the best proposed approaches yield a much better performance than various tested literature methods.

A numerically exact non-reflecting boundary condition applied to the acoustic Helmholtz equation

When modeling wave propagation, truncation of the computational domain to a manageable size requires non-reflecting boundaries. To construct such a boundary condition on one side of a rectangular domain for a finite-difference discretization of the acoustic wave equation in the frequency domain, the domain is extended on that one side to infinity. Constant extrapolation in the direction perpendicular to the boundary provides the material properties in the exterior. Domain decomposition can split the enlarged domain into the original one and its exterior. Because the boundary-value problem for the latter is translation-invariant, the boundary Green functions obey a quadratic matrix equation. Selection of the solvent that corresponds to the outgoing waves provides the input for the remaining problem in the interior. The result is a numerically exact non-reflecting boundary condition on one side of the domain. When two non-reflecting sides have a common corner, translation invariance is lost. Treating each side independently in combination with a classic absorbing condition in the other direction restores translation invariance and enables application of the method at the expense of numerical exactness. Solving the quadratic matrix equation with Newton's method turns out to be more costly than solving the Helmholtz equation and may select unwanted incoming waves. A proposed direct method has a much lower cost and selects the correct branch. A test on a 2-D acoustic marine seismic problem with a free surface at the top, a classic second-order Higdon condition at the bottom and numerically exact boundaries at the two lateral sides demonstrates the capability of the method. Numerically exact boundaries on each side, each computed independently with a free-surface or Higdon condition, provide even better results.