scholarly journals Variational construction of positive entropy invariant measures of Lagrangian systems and Arnold diffusion

2018 ◽  
Vol 40 (3) ◽  
pp. 799-864
Author(s):  
SINIŠA SLIJEPČEVIĆ
Keyword(s):  
Degrees Of Freedom ◽  
Invariant Measures ◽  
Invariant Tori ◽  
A Priori ◽  
Lagrangian Systems ◽  
Positive Entropy ◽  
Stable And Unstable Manifolds ◽  
Closed Set ◽  
Peierls Barrier ◽  
Topological Obstruction

We develop a variational method for constructing positive entropy invariant measures of Lagrangian systems without assuming transversal intersections of stable and unstable manifolds, and without restrictions to the size of non-integrable perturbations. We apply it to a family of $2\frac{1}{2}$ degrees of freedom a priori unstable Lagrangians, and show that if we assume that there is no topological obstruction to diffusion (precisely formulated in terms of topological non-degeneracy of minima of the Peierls barrier), then there exists a vast family of ‘horseshoes’, such as ‘shadowing’ ergodic positive entropy measures having precisely any closed set of invariant tori in its support. Furthermore, we give bounds on the topological entropy and the ‘drift acceleration’ in any part of a region of instability in terms of a certain extremal value of the Fréchet derivative of the action functional, generalizing the angle of splitting of separatrices. The method of construction is new, and relies on study of formally gradient dynamics of the action (coupled parabolic semilinear partial differential equations on unbounded domains). We apply recently developed techniques of precise control of the local evolution of energy (in this case the Lagrangian action), energy dissipation and flux.

The Influence of Loading Position in A Priori High Stress Detection using Mode Superposition

2020 ◽  
Vol 1 (1) ◽  
pp. 93-102
Author(s):  
Carsten Strzalka ◽  
◽  
Manfred Zehn ◽  
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
A Priori ◽  
High Stress ◽  
Von Mises Stress ◽  
Detailed Knowledge ◽  
Variable Loading ◽  
Numerical Effort ◽  
Von Mises

For the analysis of structural components, the finite element method (FEM) has become the most widely applied tool for numerical stress- and subsequent durability analyses. In industrial application advanced FE-models result in high numbers of degrees of freedom, making dynamic analyses time-consuming and expensive. As detailed finite element models are necessary for accurate stress results, the resulting data and connected numerical effort from dynamic stress analysis can be high. For the reduction of that effort, sophisticated methods have been developed to limit numerical calculations and processing of data to only small fractions of the global model. Therefore, detailed knowledge of the position of a component’s highly stressed areas is of great advantage for any present or subsequent analysis steps. In this paper an efficient method for the a priori detection of highly stressed areas of force-excited components is presented, based on modal stress superposition. As the component’s dynamic response and corresponding stress is always a function of its excitation, special attention is paid to the influence of the loading position. Based on the frequency domain solution of the modally decoupled equations of motion, a coefficient for a priori weighted superposition of modal von Mises stress fields is developed and validated on a simply supported cantilever beam structure with variable loading positions. The proposed approach is then applied to a simplified industrial model of a twist beam rear axle.

scholarly journals A Traveler’s Guide to the Multiverse: Promises, Pitfalls, and a Framework for the Evaluation of Analytic Decisions

2021 ◽  
Vol 4 (1) ◽  
pp. 251524592095492
Author(s):  
Marco Del Giudice ◽  
Steven W. Gangestad
Keyword(s):  
Degrees Of Freedom ◽  
A Priori ◽  
Simulated Data ◽  
Style Analysis ◽  
Data Set ◽  
Scant Attention ◽  
Equivalence Type ◽  
The Impact ◽  
Biased Estimates

Decisions made by researchers while analyzing data (e.g., how to measure variables, how to handle outliers) are sometimes arbitrary, without an objective justification for choosing one alternative over another. Multiverse-style methods (e.g., specification curve, vibration of effects) estimate an effect across an entire set of possible specifications to expose the impact of hidden degrees of freedom and/or obtain robust, less biased estimates of the effect of interest. However, if specifications are not truly arbitrary, multiverse-style analyses can produce misleading results, potentially hiding meaningful effects within a mass of poorly justified alternatives. So far, a key question has received scant attention: How does one decide whether alternatives are arbitrary? We offer a framework and conceptual tools for doing so. We discuss three kinds of a priori nonequivalence among alternatives—measurement nonequivalence, effect nonequivalence, and power/precision nonequivalence. The criteria we review lead to three decision scenarios: Type E decisions (principled equivalence), Type N decisions (principled nonequivalence), and Type U decisions (uncertainty). In uncertain scenarios, multiverse-style analysis should be conducted in a deliberately exploratory fashion. The framework is discussed with reference to published examples and illustrated with the help of a simulated data set. Our framework will help researchers reap the benefits of multiverse-style methods while avoiding their pitfalls.

scholarly journals Adaptive modelling of variably saturated seepage problems

2021 ◽  
Author(s):  
B Ashby ◽  
C Bortolozo ◽  
A Lukyanov ◽  
T Pryer
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Degrees Of Freedom ◽  
A Priori ◽  
Water Flux ◽  
Posteriori Error ◽  
Adaptive Finite Element Method ◽  
Posteriori Error Estimate ◽  
Adaptive Finite Element ◽  
Element Method

Summary In this article, we present a goal-oriented adaptive finite element method for a class of subsurface flow problems in porous media, which exhibit seepage faces. We focus on a representative case of the steady state flows governed by a nonlinear Darcy–Buckingham law with physical constraints on subsurface-atmosphere boundaries. This leads to the formulation of the problem as a variational inequality. The solutions to this problem are investigated using an adaptive finite element method based on a dual-weighted a posteriori error estimate, derived with the aim of reducing error in a specific target quantity. The quantity of interest is chosen as volumetric water flux across the seepage face, and therefore depends on an a priori unknown free boundary. We apply our method to challenging numerical examples as well as specific case studies, from which this research originates, illustrating the major difficulties that arise in practical situations. We summarise extensive numerical results that clearly demonstrate the designed method produces rapid error reduction measured against the number of degrees of freedom.

scholarly journals IASI-METOP and MIPAS-ENVISAT data fusion

2010 ◽  
Vol 10 (1) ◽  
pp. 183-211 ◽  
Author(s):  
S. Ceccherini ◽  
U. Cortesi ◽  
S. Del Bianco ◽  
P. Raspollini ◽  
B. Carli
Keyword(s):  
Data Fusion ◽  
Degrees Of Freedom ◽  
Solution Method ◽  
Information Gain ◽  
A Priori ◽  
A Priori Information ◽  
Priori Information ◽  
Single Instrument ◽  
Fusion Products ◽  
Temporal Coverage

Abstract. The combination of data obtained with different sensors (data fusion) is a powerful technique that can provide target products of the best quality in terms of precision and accuracy, as well as spatial and temporal coverage and resolution. In this paper the results are presented of the data fusion of measurements of ozone vertical profile performed by two space-borne interferometers (IASI on METOP and MIPAS on ENVISAT) using the new measurement-space-solution method. With this method both the loss of information due to interpolation and the propagation of possible biases (caused by a priori information) are avoided. The data fusion products are characterized by means of retrieval errors, information gain, averaging kernels and number of degrees of freedom. The analysis is performed both on simulated and real measurements and the results demonstrate and quantify the improvement of data fusion products with respect to measurements of a single instrument.

Construction of invariant measures supported within the gaps of Aubry–Mather sets

1996 ◽  
Vol 16 (1) ◽  
pp. 51-86 ◽  
Author(s):  
Giovanni Forni
Keyword(s):  
Variational Approach ◽  
Invariant Measures ◽  
Positive Entropy ◽  
Almost Periodic ◽  
Invariant Curves ◽  
Mather Theory ◽  
Entropy Measures ◽  
Twist Maps ◽  
Minimization Procedure ◽  
Area Preserving

AbstractThis paper represents a contribution to the variational approach to the understanding of the dynamics of exact area-preserving monotone twist maps of the annulus, currently known as the Aubry–Mather theory. The method introduced by Mather to construct invariant measures of Denjoy type is extended to produce almost-periodic measures, having arbitrary rationally independent frequencies, and positive entropy measures, supported within the gaps of Aubry–Mather sets which do not lie on invariant curves. This extension is based on a generalized version of the Percival's Lagrangian and on a new minimization procedure, which also gives a simplified proof of the basic existence theorem for the Aubry–Mather sets.

The mechanics of locating earthquakes

1976 ◽  
Vol 66 (1) ◽  
pp. 173-187
Author(s):  
Ray Buland
Keyword(s):  
Degrees Of Freedom ◽  
A Priori Estimates ◽  
A Priori ◽  
Joint Probability ◽  
Step Length ◽  
Earth Model ◽  
Random Errors ◽  
Qr Algorithm ◽  
Location Parameters ◽  
Chi Squared

abstract A complete reexamination of Geiger's method in the light of modern numerical analysis indicates that numerical stability can be insured by use of the QR algorithm and the convergence domain considerably enlarged by the introduction of step-length damping. In order to make the maximum use of all data, the method is developed assuming a priori estimates of the statistics of the random errors at each station. Numerical experiments indicate that the bulk of the joint probability density of the location parameters is in the linear region allowing simple estimates of the standard errors of the parameters. The location parameters are found to be distributed as one minus chi squared with m degrees of freedom, where m is the number of parameters, allowing the simple construction of confidence levels. The use of the chi-squared test with n-m degrees of freedom, where n is the number of data, is introduced as a means of qualitatively evaluating the correctness of the earth model.

Information Processing Systems in UAV Based on Bayesian Filtering in Conditions of Uncertainty

2020 ◽  
Vol 12 (4) ◽  
pp. 42-59
Author(s):  
Rinat Galiautdinov
Keyword(s):  
Information Processing ◽  
Degrees Of Freedom ◽  
Adaptive Filters ◽  
A Priori ◽  
Practical Implementation ◽  
Bayesian Filtering ◽  
Math Model ◽  
Digital Implementation ◽  
Estimation Algorithms ◽  
A Priori Uncertainty

In this article, the author considers the possibility of applying modern IT technologies to implement information processing algorithms in UAV motion control system. Filtration of coordinates and motion parameters of objects under a priori uncertainty is carried out using nonlinear adaptive filters: Kalman and Bayesian filters. The author considers numerical methods for digital implementation of nonlinear filters based on the convolution of functions, the possibilities of neural networks and fuzzy logic for solving the problems of tracking UAV objects (or missiles), the math model of dynamics, the features of the practical implementation of state estimation algorithms in the frame of added additional degrees of freedom. The considered algorithms are oriented on solving the problems in real time using parallel and cloud computing.

ON THE OPTIMAL POLYNOMIAL APPROXIMATION OF STOCHASTIC PDES BY GALERKIN AND COLLOCATION METHODS

2012 ◽  
Vol 22 (09) ◽  
pp. 1250023 ◽  
Author(s):  
JOAKIM BECK ◽  
RAUL TEMPONE ◽  
FABIO NOBILE ◽  
LORENZO TAMELLINI
Keyword(s):  
Tensor Product ◽  
Degrees Of Freedom ◽  
Functional Dependence ◽  
A Priori ◽  
Spectral Expansion ◽  
Sparse Grids ◽  
Collocation Methods ◽  
Stochastic Pdes ◽  
Convergence Properties ◽  
Sharp Estimates

In this work we focus on the numerical approximation of the solution u of a linear elliptic PDE with stochastic coefficients. The problem is rewritten as a parametric PDE and the functional dependence of the solution on the parameters is approximated by multivariate polynomials. We first consider the stochastic Galerkin method, and rely on sharp estimates for the decay of the Fourier coefficients of the spectral expansion of u on an orthogonal polynomial basis to build a sequence of polynomial subspaces that features better convergence properties, in terms of error versus number of degrees of freedom, than standard choices such as Total Degree or Tensor Product subspaces. We consider then the Stochastic Collocation method, and use the previous estimates to introduce a new class of Sparse Grids, based on the idea of selecting a priori the most profitable hierarchical surpluses, that, again, features better convergence properties compared to standard Smolyak or tensor product grids. Numerical results show the effectiveness of the newly introduced polynomial spaces and sparse grids.

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