Properties of Linear and Nonlinear Operators

2012 ◽  
pp. 55-107
Author(s):  
Pavel Drábek ◽  
Jaroslav Milota
Keyword(s):  
Nonlinear Operators ◽  
Linear And Nonlinear

Assimilation of Nonlinear Observations with the Maximum Likelihood Ensemble Filter

2021 ◽  
Author(s):  
Saori Nakashita ◽  
Takeshi Enomoto
Keyword(s):  
Maximum Likelihood ◽  
Data Assimilation ◽  
Cost Function ◽  
Satellite Observations ◽  
Observation Error ◽  
Nonlinear Operators ◽  
Linear And Nonlinear ◽  
Nonlinear Observation ◽  
The Cost ◽  
Maximum Likelihood Ensemble Filter

<p>Satellite observations have been a growing source for data assimilation in the operational numerical weather prediction. Remotely sensed observations require a nonlinear observation operator.  Most ensemble-based data assimilation methods are formulated for tangent linear observation operators, which are often substituted by nonlinear observation operators. By contrast, the Maximum Likelihood Ensemble Filter (MLEF), which has features of both variational and ensemble approaches, is formulated for linear and nonlinear operators in an identical form and can use non-differentiable observation operators.<span> </span></p><p>In this study, we investigate the performance of MLEF and Ensemble Transform Kalman Filter (ETKF) with the tangent linear and nonlinear observation operators in assimilation experiments of nonlinear observations with a one-dimensional Burgers model.</p><p>The ETKF analysis with the nonlinear operator diverges when the observation error is small due to unrealistically large increments associated with the high order observation terms. The filter divergence can be avoided by localization of the extent of observation influence, but the analysis error is still larger than that of MLEF. In contrast, MLEF is found to be more stable and accurate without localization owing to the minimization of the cost function. Notably, MLEF can make an accurate analysis solution even without covariance inflation, eliminating the labor of parameter adjustment. In addition, the smaller observation error is, or the stronger observation nonlinearity is, MLEF with the nonlinear operators can assimilate observations more effectively than MLEF with the tangent linear operators. This result indicates that MLEF can incorporate nonlinear effects and evaluate the observation term in the cost function appropriately. These encouraging results imply that MLEF is suitable for assimilation of satellite observations with high nonlinearity.</p>

scholarly journals Variational Methods for NLEV Approximation Near a Bifurcation Point

2012 ◽  
Vol 2012 ◽  
pp. 1-32 ◽  
Author(s):  
Raffaele Chiappinelli
Keyword(s):  
The Other ◽  
Bounded Domains ◽  
Nonlinear Operators ◽  
Finite Dimensional ◽  
Semilinear Elliptic ◽  
Nonlinear Eigenvalues ◽  
Linear And Nonlinear ◽  
Bifurcation Problems ◽  
Lusternik Schnirelmann ◽  
Adjoint Operators

We review some more and less recent results concerning bounds on nonlinear eigenvalues (NLEV) for gradient operators. In particular, we discuss the asymptotic behaviour of NLEV (as the norm of the eigenvector tends to zero) in bifurcation problems from the line of trivial solutions, considering perturbations of linear self-adjoint operators in a Hilbert space. The proofs are based on the Lusternik-Schnirelmann theory of critical points on one side and on the Lyapounov-Schmidt reduction to the relevant finite-dimensional kernel on the other side. The results are applied to some semilinear elliptic operators in bounded domains ofℝN. A section reviewing some general facts about eigenvalues of linear and nonlinear operators is included.

scholarly journals Qualitative Analysis for Controllable Dynamical Systems: Stability with Control Lyapunov Functions

2021 ◽  
Author(s):  
Adela Ionescu
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Main Idea ◽  
Excitable Media ◽  
Technological Systems ◽  
Control Lyapunov Function ◽  
Nonlinear Operators ◽  
Rich Information ◽  
Linear And Nonlinear ◽  
Controllable Systems

The present chapter focuses on some recent work on the qualitatively analysis of dynamical systems, namely stability, a powerful tool with multiple connected appliances. Among them, feedback is a powerful idea which is used extensively in natural and technological systems. In engineering, feedback has been rediscovered and patented many times in many different contexts. Stabilizing a dynamical system could be often easier if we approach controllable systems. When the dynamical system is in a controllable form, we can place bounds on its behavior by analyzing the improvement of the linear and nonlinear operators that describe the system. In this chapter it is analyzed how a control in a simple form, could influence the possibility to construct the so-called Control Lyapunov Function (CLF) in order to stabilize the dynamical system in study. The main idea is to test multiple cases, in order to get a rich information panel and to make easier the problem of finding a CLF, which is generally a difficult task. As applications, models from excitable media are chosen.

Preface of the “Symposium on Linear and Nonlinear Operators and Applications”

2011 ◽  
Author(s):  
Gheorghe Morosanu ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras ◽  
Zacharias Anastassi
Keyword(s):  
Nonlinear Operators ◽  
Linear And Nonlinear
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