Least-squares gradient calculation from multi-point observations of scalar and vector fields: methodology and applications with Cluster in the plasmasphere

Abstract. This paper describes a general-purpose algorithm for computing the gradients in space and time of a scalar field, a vector field, or a divergence-free vector field, from in situ measurements by one or more spacecraft. The algorithm provides total error estimates on the computed gradient, including the effects of measurement errors, the errors due to a lack of spatio-temporal homogeneity, and errors due to small-scale fluctuations. It also has the ability to diagnose the conditioning of the problem. Optimal use is made of the data, in terms of exploiting the maximum amount of information relative to the uncertainty on the data, by solving the problem in a weighted least-squares sense. The method is illustrated using Cluster magnetic field and electron density data to compute various gradients during a traversal of the inner magnetosphere. In particular, Cluster is shown to cross azimuthal density structure, and the existence of field-aligned currents in the plasmasphere is demonstrated.

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Form Error Evaluation: An Iterative Reweighted Least Squares Algorithm*

Journal of Manufacturing Science and Engineering ◽

10.1115/1.1765144 ◽

2004 ◽

Vol 126 (3) ◽

pp. 535-541 ◽

Cited By ~ 21

Author(s):

Xiangyang Zhu ◽

Han Ding ◽

Michael Y. Wang

Keyword(s):

Least Squares ◽

Weighted Least Squares ◽

Chebyshev Approximation ◽

Approximation Problem ◽

General Purpose ◽

Geometric Features ◽

Error Evaluation ◽

Form Error ◽

Minimum Zone ◽

Effectiveness And Efficiency

This paper establishes the equivalence between the solution to a linear Chebyshev approximation problem and that of a weighted least squares (WLS) problem with the weighting parameters being appropriately defined. On this basis, we present an algorithm for form error evaluation of geometric features. The algorithm is implemented as an iterative procedure. At each iteration, a WLS problem is solved and the weighting parameters are updated. The proposed algorithm is of general-purpose, it can be used to evaluate the exact minimum zone error of various geometric features including flatness, circularity, sphericity, cylindericity and spatial straightness. Numerical examples are presented to show the effectiveness and efficiency of the algorithm.

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Vector fields satisfying the barycenter property

Open Mathematics ◽

10.1515/math-2018-0040 ◽

2018 ◽

Vol 16 (1) ◽

pp. 429-436 ◽

Cited By ~ 2

Author(s):

Manseob Lee

Keyword(s):

Vector Field ◽

Vector Fields ◽

Axiom A ◽

Divergence Free Vector ◽

Divergence Free ◽

Volume Preserving

AbstractWe show that if a vector fieldXhas theC1robustly barycenter property then it does not have singularities and it is Axiom A without cycles. Moreover, if a genericC1-vector field has the barycenter property then it does not have singularities and it is Axiom A without cycles. Moreover, we apply the results to the divergence free vector fields. It is an extension of the results of the barycenter property for generic diffeomorphisms and volume preserving diffeomorphisms [1].

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A pasting lemma and some applications for conservative systems

Ergodic Theory and Dynamical Systems ◽

10.1017/s014338570700017x ◽

2007 ◽

Vol 27 (5) ◽

pp. 1399-1417 ◽

Cited By ~ 43

Author(s):

ALEXANDER ARBIETO ◽

CARLOS MATHEUS

Keyword(s):

Vector Field ◽

Compact Manifold ◽

Vector Fields ◽

Three Dimensional ◽

Invariant Set ◽

Dominated Splitting ◽

Divergence Free Vector ◽

Conservative Systems ◽

Divergence Free ◽

Volume Preserving

AbstractWe prove that in a compact manifold of dimension n≥2, C1+α volume-preserving diffeomorphisms that are robustly transitive in the C1-topology have a dominated splitting. Also we prove that for three-dimensional compact manifolds, an isolated robustly transitive invariant set for a divergence-free vector field cannot have a singularity. In particular, we prove that robustly transitive divergence-free vector fields in three-dimensional manifolds are Anosov. For this, we prove a ‘pasting’ lemma, which allows us to make perturbations in conservative systems.

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Representation of vector fields

Global Journal of Mathematical Analysis ◽

10.14419/gjma.v3i2.4577 ◽

2015 ◽

Vol 3 (2) ◽

pp. 73

Author(s):

Alexander G. Ramm

Keyword(s):

Vector Field ◽

Explicit Formula ◽

Simple Proof ◽

Vector Fields ◽

Smooth Domain ◽

Gradient Field ◽

Smooth Vector ◽

Smooth Vector Field ◽

Divergence Free Vector ◽

Divergence Free

<p>A simple proof is given for the explicit formula which allows one to recover a \(C^2\) – smooth vector field \(A=A(x)\) in \(\mathbb{R}^3\), decaying at infinity, from the knowledge of its \(\nabla \times A\) and \(\nabla \cdot A\). The representation of \(A\) as a sum of the gradient field and a divergence-free vector fields is derived from this formula. Similar results are obtained for a vector field in a bounded \(C^2\) - smooth domain.</p>

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On the convexity condition for the Semi-Geostrophic system

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2021018 ◽

2021 ◽

Author(s):

Adrian Tudorascu

Keyword(s):

Vector Field ◽

Critical Points ◽

Vector Fields ◽

Energy Functional ◽

Convexity Condition ◽

Local Minimizers ◽

Flow Maps ◽

Divergence Free Vector ◽

Absolute Density ◽

The Absolute

We show that conservative distributional solutions to the Semi-Geostrophic systems in a rigid domain are in some well-defined sense critical points of a time-shifted energy functional involving measure-preserving rearrangements of the absolute density and momentum, which arise as one-parameter flow maps of continuously differentiable, compactly supported divergence free vector fields. We also show directly, with no recourse to Monge- Kantorovich theory, that the convexity requirement on the modified pressure potentials arises naturally if these critical points are local minimizers of said energy functional for any admis- sible vector field. The obligatory connection with the Monge-Kantorovich theory is addressed briefly.

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Identification of Small-scale Unmanned Helicopter Based on Least Squares and Adaptive Immune Genetic Algorithm

ROBOT ◽

10.3724/spj.1218.2012.00072 ◽

2012 ◽

Vol 34 (1) ◽

pp. 72 ◽

Cited By ~ 1

Author(s):

Yuhu DU ◽

Jiancheng FANG ◽

Wei SHENG ◽

Xusheng LEI

Keyword(s):

Genetic Algorithm ◽

Least Squares ◽

Small Scale ◽

Unmanned Helicopter ◽

Immune Genetic Algorithm ◽

Adaptive Immune

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Evaluation for estimating of the PDF and the CDF of Generalized Inverted Exponential Distribution with Application in Industry

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.9.1.39 ◽

2020 ◽

pp. 507-522

Author(s):

Parisa Torkaman

Keyword(s):

Least Squares ◽

Exponential Distribution ◽

Mean Squared Error ◽

Weighted Least Squares ◽

Real Data ◽

Minimum Variance ◽

Cumulative Distribution ◽

Estimation Methods ◽

Data Set ◽

Better Than

The generalized inverted exponential distribution is introduced as a lifetime model with good statistical properties. This paper, the estimation of the probability density function and the cumulative distribution function of with five different estimation methods: uniformly minimum variance unbiased(UMVU), maximum likelihood(ML), least squares(LS), weighted least squares (WLS) and percentile(PC) estimators are considered. The performance of these estimation procedures, based on the mean squared error (MSE) by numerical simulations are compared. Simulation studies express that the UMVU estimator performs better than others and when the sample size is large enough the ML and UMVU estimators are almost equivalent and efficient than LS, WLS and PC. Finally, the result using a real data set are analyzed.

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