Error Estimations in Linear Inverse Problems With a Priori Information

2011 ◽  
Author(s):  
Anatoly G. Yagola ◽  
Yury M. Korolev
Keyword(s):  
Exact Solution ◽  
A Priori ◽  
Posteriori Error ◽  
Posteriori Error Estimate ◽  
Compact Set ◽  
A Posteriori ◽  
A Priori Information ◽  
A Posteriori Error ◽  
Error Estimations ◽  
Priori Information

We consider an inverse problem for an operator equation Az = u. The exact operator A and the exact right-hand side u are unknown. Only their upper and lower estimations are available. We provide techniques of calculating upper and lower estimations for the exact solution belonging to a compact set in this case, as well as a posteriori error estimations. We obtain approximate solutions with an optimal a posteriori error estimate. We also make use of a priori information about the exact solution, e.g. its monotonicity and convexity. The developed software package was applied to solving practical ill-posed problems.

scholarly journals Ozone profile smoothness as a priori information in the inversion of limb measurements

2004 ◽  
Vol 22 (10) ◽  
pp. 3411-3420 ◽  
Author(s):  
V. F. Sofieva ◽  
J. Tamminen ◽  
H. Haario ◽  
E. Kyrölä ◽  
M. Lehtinen
Keyword(s):  
A Priori ◽  
Optimal Estimation ◽  
Maximum A Posteriori ◽  
A Posteriori ◽  
A Priori Information ◽  
Ozone Profile ◽  
Stellar Occultation ◽  
Atmospheric Profiles ◽  
Priori Information ◽  
Typical Measurement

Abstract. In this work we discuss inclusion of a priori information about the smoothness of atmospheric profiles in inversion algorithms. The smoothness requirement can be formulated in the form of Tikhonov-type regularization, where the smoothness of atmospheric profiles is considered as a constraint or in the form of Bayesian optimal estimation (maximum a posteriori method, MAP), where the smoothness of profiles can be included as a priori information. We develop further two recently proposed retrieval methods. One of them - Tikhonov-type regularization according to the target resolution - develops the classical Tikhonov regularization. The second method - maximum a posteriori method with smoothness a priori - effectively combines the ideas of the classical MAP method and Tikhonov-type regularization. We discuss a grid-independent formulation for the proposed inversion methods, thus isolating the choice of calculation grid from the question of how strong the smoothing should be. The discussed approaches are applied to the problem of ozone profile retrieval from stellar occultation measurements by the GOMOS instrument on board the Envisat satellite. Realistic simulations for the typical measurement conditions with smoothness a priori information created from 10-years analysis of ozone sounding at Sodankylä and analysis of the total retrieval error illustrate the advantages of the proposed methods. The proposed methods are equally applicable to other profile retrieval problems from remote sensing measurements.

scholarly journals divand-1.0: <i>n</i>-dimensional variational data analysis for ocean observations

2014 ◽  
Vol 7 (1) ◽  
pp. 225-241 ◽  
Author(s):  
A. Barth ◽  
J.-M. Beckers ◽  
C. Troupin ◽  
A. Alvera-Azcárate ◽  
L. Vandenbulcke
Keyword(s):  
Cost Function ◽  
Dimensional Analysis ◽  
Computational Cost ◽  
Posteriori Error ◽  
Cholesky Factorization ◽  
Ocean Current ◽  
Posteriori Error Estimate ◽  
Two Dimensional ◽  
A Posteriori ◽  
A Posteriori Error

Abstract. A tool for multidimensional variational analysis (divand) is presented. It allows the interpolation and analysis of observations on curvilinear orthogonal grids in an arbitrary high dimensional space by minimizing a cost function. This cost function penalizes the deviation from the observations, the deviation from a first guess and abruptly varying fields based on a given correlation length (potentially varying in space and time). Additional constraints can be added to this cost function such as an advection constraint which forces the analysed field to align with the ocean current. The method decouples naturally disconnected areas based on topography and topology. This is useful in oceanography where disconnected water masses often have different physical properties. Individual elements of the a priori and a posteriori error covariance matrix can also be computed, in particular expected error variances of the analysis. A multidimensional approach (as opposed to stacking two-dimensional analysis) has the benefit of providing a smooth analysis in all dimensions, although the computational cost is increased. Primal (problem solved in the grid space) and dual formulations (problem solved in the observational space) are implemented using either direct solvers (based on Cholesky factorization) or iterative solvers (conjugate gradient method). In most applications the primal formulation with the direct solver is the fastest, especially if an a posteriori error estimate is needed. However, for correlated observation errors the dual formulation with an iterative solver is more efficient. The method is tested by using pseudo-observations from a global model. The distribution of the observations is based on the position of the Argo floats. The benefit of the three-dimensional analysis (longitude, latitude and time) compared to two-dimensional analysis (longitude and latitude) and the role of the advection constraint are highlighted. The tool divand is free software, and is distributed under the terms of the General Public Licence (GPL) (http://modb.oce.ulg.ac.be/mediawiki/index.php/divand).

scholarly journals A Priori and A Posteriori Error Estimates for a Crank Nicolson Type Scheme of an Elliptic Problem with Dynamical Boundary Conditions

2016 ◽  
Vol 8 (2) ◽  
pp. 1
Author(s):  
Rola Ali Ahmad ◽  
Toufic El Arwadi ◽  
Houssam Chrayteh ◽  
Jean-Marc Sac-Epee
Keyword(s):  
Error Estimates ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Time Discretization ◽  
Posteriori Error ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Dynamical Boundary Conditions ◽  
Error Indicators ◽  
Crank Nicolson

In this article we claim that we are going to give a priori and a posteriori error estimates for a Crank Nicolson type scheme. The problem is discretized by the finite elements in space. The main result of this paper consists in establishing two types of error indicators, the first one linked to the time discretization and the second one to the space discretization.

scholarly journals A priori and a posteriori error analysis for a hybrid formulation of a prestressed shell model.

2021 ◽  
Author(s):  
Serge Nicaise ◽  
Ismail Merabet ◽  
Rayhana REZZAG BARA
Keyword(s):  
Shell Model ◽  
A Priori ◽  
Functional Space ◽  
Finite Element Approximation ◽  
Posteriori Error ◽  
Error Estimator ◽  
Element Approximation ◽  
A Posteriori ◽  
A Posteriori Error ◽  
A Priori Error Estimate

This work deals with the finite element approximation of a prestressed shell model using a new formulation where the unknowns (the displacement and the rotation of fibers normal to the midsurface) are described in Cartesian and local covariant basis respectively. Due to the constraint involved in the definition of the functional space, a penalized version is then considered. We obtain a non robust a priori error estimate of this penalized formulation, but a robust one is obtained for its mixed formulation. Moreover, we present a reliable and efficient a posteriori error estimator of the penalized formulation. Numerical tests are included that confirmthe efficiency of our residual a posteriori estimator.

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