Towards a priori estimators of the accuracy of determining parameters by the least-squares method

1965 ◽  
Vol 13 (4) ◽  
pp. 359
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
A Priori

scholarly journals An Optimum Deconvolution Method

1979 ◽  
Vol 49 ◽  
pp. 287-290
Author(s):  
C.R. Subrahmanya
Keyword(s):  
Least Squares ◽  
Prior Knowledge ◽  
Least Squares Method ◽  
A Priori ◽  
Statistical Treatment ◽  
Stable Solution ◽  
Brightness Distribution ◽  
Deconvolution Method ◽  
Explicit Recognition ◽  
Optimum Solution

An optimum solution to a deconvolution problem has to fulfil three general criteria: (a) an explicit recognition of the smoothing nature of convolution; (b) a statistical treatment of noise, e.g., using the least-squares criterion; and (c) requiring the solution to conform to all our prior knowledge about it. In the usual least-squares method, one minimises a variance of ‘residuals’, or the departures of the observed data from the values expected according to the recovered solution. However, this condition does not lead to a stable solution in the case of deconvolution, since the only stable solutions are those conforming to a criterion of ‘regularisation’ or smoothness (see, e.g., Tikhonov and Arsenin 1977). In our method, the stability is achieved by minimising the variance of the second-differences of the solution simultaneously with the fulfilment of the least-squares criterion. Such a procedure was first used by Phillips(1962). However, the solution thus obtained is still unsatisfactory since it usually does not conform to oura prioriinformation. When we seek the brightness distribution of an object, the most frequent violation of our prior knowledge is that of positiveness. This motivated us to develop an Optimum Deconvolution Method (ODM) which constrains the solution to satisfy prior knowledge while retaining the features of least-squares and smoothness criteria.

A least-squares method for estimating the correlated error of GRACE models

2020 ◽  
Vol 221 (3) ◽  
pp. 1736-1749
Author(s):  
John W Crowley ◽  
Jianliang Huang
Keyword(s):  
Least Squares ◽  
Spherical Harmonic ◽  
Climate Model ◽  
Least Squares Method ◽  
A Priori ◽  
Temporal Trends ◽  
Standard Form ◽  
New Method ◽  
Correlated Errors ◽  
Global Trends

SUMMARY A new least-squares method is developed for estimating and removing the correlated errors (stripes) from the Gravity Recovery and Climate Experiment (GRACE) and GRACE Follow-On (GRACE-FO) mission data. This method is based on a joint parametric model of the correlated errors and temporal trends in the spherical harmonic coefficients of GRACE models. Three sets of simulation data are created from the Global Land Data Assimilation System (GLDAS), the Regional Atmospheric Climate Model 2.3 (RACMO2.3) and GRACE models and used to test it. The results show that the new method improves the decorrelation method by Swenson & Wahr significantly. Its application to the release 5 (RL05) and new release 6 (RL06) spherical harmonic solutions from the Center for Space Research (CSR) at The University of Texas at Austin demonstrates its effectiveness and provides a relative assessment of the two releases. A comparison to the Swenson & Wahr and Kusche et al. methods highlights the deficiencies in past destriping methods and shows how the inclusion and decoupling of temporal trends helps to overcome them. A comparison to the CSR mascon and JPL mascon solutions demonstrates that the new method yields global trends that have greater amplitude than those produced by the CSR RL05 mascon solution and are of comparable quality to the JPL RL06 mascon solution. Furthermore, these results are obtained without the need for a priori information, scale factors or complex regularization methods and the solutions remain in the standard form of spherical harmonics rather than discrete mascons. The latter could introduce additional discretization error when converting to the spherical harmonic model, upon which many post-processing methods and applications are built.

A Study on the Galerkin Least-Squares Method for the Oldroyd-B Model

2018 ◽  
Vol 18 (2) ◽  
pp. 181-198
Author(s):  
Tsu-Fen Chen ◽  
Hyesuk Lee ◽  
Chia-Chen Liu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Error Estimate ◽  
Least Squares ◽  
Viscoelastic Fluid ◽  
Least Squares Method ◽  
A Priori ◽  
Fluid Flows ◽  
Planar Channel ◽  
A Priori Error Estimate

AbstractWe consider a reduced Galerkin least-squares finite element method for the Oldroyd-B model of viscoelastic fluid flows. Model problems considered are the flow past a planar channel and a 4-to-1 contraction problems. An a priori error estimate for the reduced Galerkin least-squares method is derived and numerical results supporting the estimate are presented.

Inversion of detrital zircon data to constrain spatially varying erosion rates

2020 ◽  
Author(s):  
Fien De Doncker ◽  
Frédéric Herman ◽  
Matthew Fox
Keyword(s):  
Least Squares ◽  
Detrital Zircon ◽  
Erosion Rate ◽  
Least Squares Method ◽  
A Priori ◽  
Age Distribution ◽  
Inversion Method ◽  
Provenance Analysis ◽  
Erosion Rates ◽  
Geological Unit

<p>Landscapes evolve through surface processes that are often transient in space and time. To understand the underlying geomorphic processes, one must assess how erosion rates vary spatially. This can be done using provenance analysis. Here, we introduce a formal inversion method to derive erosion patterns using detrital zircon age data as fingerprints. Zircons are omnipresent in Earth’s crust and contain information about the time since (re)crystallization in their U/Th-Pb ratio. For each geological unit having undergone a specific tectonic or magmatic history, one can find a unique age-frequency signature. Hence, erosion and sedimentation of grains originating from diverse source areas lead to a mix of the varying age-frequency signatures in sediments found at the outlet of a catchment. Considering that the age signal is not altered during erosion-transportation-deposition events, and given that recent technological advances enable precise dating of large amounts of grains, U/Th-Pb zircon ages provide an appropriate fingerprinting tool. Our inversion approach relies on the least-squares method with a priori information and model covariance to deal with non-uniqueness. We show with synthetic and natural examples that we are able to retrieve erosion rate patterns of a catchment when the age distribution for each geological unit is well known. Furthermore, relying on the nested form of catchments and their subcatchments, we demonstrate that adding samples taken at the outlet of subcatchments improves the estimation of erosion rate patterns. We conclude that the least squares inverse model applied on detrital zircon data has great potential for investigating erosion rates.</p>

Recent Improvements of the Algorithm of Mode Isolation

2003 ◽  
Author(s):  
Jerry H. Ginsberg ◽  
Matthew S. Allen
Keyword(s):  
Frequency Response ◽  
Least Squares ◽  
Natural Frequencies ◽  
Least Squares Method ◽  
A Priori ◽  
Single Mode ◽  
Computational Effort ◽  
Complex Frequency ◽  
Linear Least Squares ◽  
Mode Isolation

The Algorithm of Mode Isolation (AMI) identifies the natural frequencies, modal damping ratios, and mode vectors of a system by proceesing complex frequency response data. It uses an iterative procedure based on the fact that a general frequency response function is a superposition of modal contributions. The iterations focus successively on a singel mode. The mode that is in focus is isolated by subtracting the other modal contributions using prior estimates of their modal properties. This process leads to a self-contained identification of the number of modes that participate in any frequency band, whereas other techniques require a priori guesses. This paper describes modifications intended to improve AMI’s accuracy and reduce its computational effort. These involve the use of a new linear least squares method for identifying the natural frequency and dmaping ratio of a single mode, a linear least squares global fit of the data in order to identify mode vectors. Results are presented for a model of a cantilever beam with suspended spring-mass-dashpot system. This system was used by Drexel, Ginsberg, and Zaki [Journal of Vibration and Acoustics, 2003 (forthcoming)] to assess the prior version’s ability to identify weakly excited modes and modes having close natural frequencies in the presence of high noise levels. Application of the modeified version of AMI to the same system is shown to lead to significantly more accurate damping ratios are mode vectors, with equally good natural frequencies.

scholarly journals A least-squares Galerkin approach to gradient and Hessian recovery for nondivergence-form elliptic equations

2021 ◽  
Author(s):  
Omar Lakkis ◽  
Amireh Mousavi
Keyword(s):  
Least Squares ◽  
Elliptic Equations ◽  
Adaptive Mesh Refinement ◽  
Least Squares Method ◽  
A Priori ◽  
Adaptive Mesh ◽  
A Posteriori ◽  
Nondivergence Form ◽  
Convergence Results ◽  
Discrete Spaces

Abstract We propose a least-squares method involving the recovery of the gradient and possibly the Hessian for elliptic equation in nondivergence form. As our approach is based on the Lax–Milgram theorem with the curl-free constraint built into the target (or cost) functional, the discrete spaces require no inf-sup stabilization. We show that standard conforming finite elements can be used yielding a priori and a posteriori convergence results. We illustrate our findings with numerical experiments with uniform or adaptive mesh refinement.

‘Least squares’ method—theorem or law?

1980 ◽  
Vol 59 (9) ◽  
pp. 8
Author(s):  
D.E. Turnbull
Keyword(s):  
Least Squares ◽  
Least Squares Method

An Augmented Iteratively Re-weighted and Refined Least Squares Method for lp Minimization Problem in Inverse Modeling of Potential Field Magnetic Data

2020 ◽  
Vol 1 (3) ◽  
Author(s):  
Maysam Abedi
Keyword(s):  
Magnetic Field ◽  
Magnetic Susceptibility ◽  
Least Squares ◽  
Minimization Problem ◽  
Potential Field ◽  
Field Data ◽  
Least Squares Method ◽  
Porphyry Copper ◽  
Magnetic Data ◽  
Magnetic Field Data

The presented work examines application of an Augmented Iteratively Re-weighted and Refined Least Squares method (AIRRLS) to construct a 3D magnetic susceptibility property from potential field magnetic anomalies. This algorithm replaces an lp minimization problem by a sequence of weighted linear systems in which the retrieved magnetic susceptibility model is successively converged to an optimum solution, while the regularization parameter is the stopping iteration numbers. To avoid the natural tendency of causative magnetic sources to concentrate at shallow depth, a prior depth weighting function is incorporated in the original formulation of the objective function. The speed of lp minimization problem is increased by inserting a pre-conditioner conjugate gradient method (PCCG) to solve the central system of equation in cases of large scale magnetic field data. It is assumed that there is no remanent magnetization since this study focuses on inversion of a geological structure with low magnetic susceptibility property. The method is applied on a multi-source noise-corrupted synthetic magnetic field data to demonstrate its suitability for 3D inversion, and then is applied to a real data pertaining to a geologically plausible porphyry copper unit.  The real case study located in  Semnan province of  Iran  consists  of  an arc-shaped  porphyry  andesite  covered  by  sedimentary  units  which  may  have  potential  of  mineral  occurrences, especially  porphyry copper. It is demonstrated that such structure extends down at depth, and consequently exploratory drilling is highly recommended for acquiring more pieces of information about its potential for ore-bearing mineralization.

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