scholarly journals Approximate factorization of positive matrices by using methods of tropical optimization

2020 ◽  
Vol 16 (4) ◽  
pp. 357-374
Author(s):  
Nikolai K. Krivulin ◽  
◽  
Elizaveta Yu. Romanova ◽  
Keyword(s):  
Approximation Problem ◽  
Logarithmic Scale ◽  
Box Constraints ◽  
Approximate Factorization ◽  
Tropical Mathematics ◽  
Tropical Optimization ◽  
The Matrix ◽  
Rank One ◽  
Positive Matrices ◽  
Factor Decomposition

The problem of rank-one factorization of positive matrices with missing (unspecified) entries is considered where a matrix is approximated by a product of column and row vectors that are subject to box constraints. The problem is reduced to the constrained approximation of the matrix, using the Chebyshev metric in logarithmic scale, by a matrix of unit rank. Furthermore, the approximation problem is formulated in terms of tropical mathematics that deals with the theory and applications of algebraic systems with idempotent addition. By using methods of tropical optimization, direct analytical solutions to the problem are derived for the case of an arbitrary positive matrix and for the case when the matrix does not contain columns (rows) with all entries missing. The results obtained allow one to find the vectors of the factor decomposition by using expressions in a parametric form which is ready for further analysis and immediate calculation. In conclusion, an example of approximate rank-one factorization of a matrix with missing entries is provided.

Velocity estimates derived from three‐dimensional seismic data

Geophysics ◽  
1983 ◽  
Vol 48 (11) ◽  
pp. 1486-1497 ◽  
Author(s):  
Kwame Owusu ◽  
G. H. F. Gardner ◽  
Wulf F. Massell
Keyword(s):  
Seismic Data ◽  
Constant Velocity ◽  
Three Dimensional ◽  
Logarithmic Scale ◽  
Model Data ◽  
Reconstruction Process ◽  
Input Field ◽  
Scattering Center ◽  
Straight Line ◽  
The Matrix

A new computer algorithm is described by which velocity estimates can be derived from three‐dimensional (3-D) multifold seismic data. The velocity estimate, referred to as “imaging velocity,” is that which best describes the diffraction hyperboloid due to a scatterer. The scattering center is best imaged when this velocity is used in the reconstruction process. The method is based on the 3-D Kirchhoff summation migration before stack. The implementation consists of two basic phases: (1) differentiating the input field traces and resampling them to a logarithmic time scale, and (2) shifting, weighting, and summing each resampled trace to a range of depth levels also chosen on a logarithmic scale. Peak amplitudes in the resulting image matrix give a time T and depth Z from which velocity is obtained using the relation [Formula: see text] The locus of constant velocity is a slanted straight line in the coordinate system of the matrix. In the usual application of migration for velocity analysis, each input trace of N samples is migrated for each of M constant velocity functions requiring [Formula: see text] moveout shift calculations. In the new method presented here, a constant shift is calculated for a given resampled trace, for each depth into which it is summed. This reduces the number of calculations per trace to about N, resulting in a significant improvement in computing efficiency. The operation of the algorithm is illustrated using synthetic and physical model data.

scholarly journals A note on adaptivity in factorized approximate inverse preconditioning

2020 ◽  
Vol 28 (2) ◽  
pp. 149-159
Author(s):  
Jiří Kopal ◽  
Miroslav Rozložník ◽  
Miroslav Tůma
Keyword(s):  
Large Scale ◽  
Condition Numbers ◽  
Algebraic Equations ◽  
Approximate Factorization ◽  
Approximate Inverse ◽  
Linear Algebraic Equations ◽  
Wide Range ◽  
The Matrix ◽  
Preconditioned Iterative ◽  
Principal Submatrices

AbstractThe problem of solving large-scale systems of linear algebraic equations arises in a wide range of applications. In many cases the preconditioned iterative method is a method of choice. This paper deals with the approximate inverse preconditioning AINV/SAINV based on the incomplete generalized Gram–Schmidt process. This type of the approximate inverse preconditioning has been repeatedly used for matrix diagonalization in computation of electronic structures but approximating inverses is of an interest in parallel computations in general. Our approach uses adaptive dropping of the matrix entries with the control based on the computed intermediate quantities. Strategy has been introduced as a way to solve di cult application problems and it is motivated by recent theoretical results on the loss of orthogonality in the generalized Gram– Schmidt process. Nevertheless, there are more aspects of the approach that need to be better understood. The diagonal pivoting based on a rough estimation of condition numbers of leading principal submatrices can sometimes provide inefficient preconditioners. This short study proposes another type of pivoting, namely the pivoting that exploits incremental condition estimation based on monitoring both direct and inverse factors of the approximate factorization. Such pivoting remains rather cheap and it can provide in many cases more reliable preconditioner. Numerical examples from real-world problems, small enough to enable a full analysis, are used to illustrate the potential gains of the new approach.

scholarly journals The least-squares solutions of the matrix equation $ A^{\ast}XB+B^{\ast}X^{\ast}A = D $ and its optimal approximation

2021 ◽  
Vol 7 (3) ◽  
pp. 3680-3691
Author(s):  
Huiting Zhang ◽  
◽  
Yuying Yuan ◽  
Sisi Li ◽  
Yongxin Yuan ◽  
...  
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Canonical Correlation ◽  
Approximation Problem ◽  
Optimal Approximation ◽  
Linear Matrix ◽  
General Representation ◽  
Linear Matrix Equation ◽  
The Matrix

<abstract><p>In this paper, the least-squares solutions to the linear matrix equation $ A^{\ast}XB+B^{\ast}X^{\ast}A = D $ are discussed. By using the canonical correlation decomposition (CCD) of a pair of matrices, the general representation of the least-squares solutions to the matrix equation is derived. Moreover, the expression of the solution to the corresponding weighted optimal approximation problem is obtained.</p></abstract>

scholarly journals REGULARITY RADIUS: PROPERTIES, APPROXIMATION AND A NOT A PRIORI EXPONENTIAL ALGORITHM

2018 ◽  
Vol 33 ◽  
pp. 122-136 ◽  
Author(s):  
David Hartman ◽  
Milan Hladik
Keyword(s):  
Approximation Algorithms ◽  
A Priori ◽  
Interval Matrix ◽  
Original Matrix ◽  
Totally Positive ◽  
Classes Of Matrices ◽  
The Matrix ◽  
Rank One ◽  
Radius Of Regularity ◽  
Special Classes

The radius of regularity, sometimes spelled as the radius of nonsingularity, is a measure providing the distance of a given matrix to the nearest singular one. Despite its possible application strength this measure is still far from being handled in an efficient way also due to findings of Poljak and Rohn providing proof that checking this property is NP-hard for a general matrix. There are basically two approaches to handle this situation. Firstly, approximation algorithms are applied and secondly, tighter bounds for radius of regularity are considered. Improvements of both approaches have been recently shown by Hartman and Hlad\'{i}k (doi:10.1007/978-3-319-31769-4\_9) utilizing relaxation of the radius computation to semidefinite programming. An estimation of the regularity radius using any of the above mentioned approaches is usually applied to general matrices considering none or just weak assumptions about the original matrix. Surprisingly less explored area is represented by utilization of properties of special classes of matrices as well as utilization of classical algorithms extended to be used to compute the considered radius. This work explores a process of regularity radius analysis and identifies useful properties enabling easier estimation of the corresponding radius values. At first, checking finiteness of this characteristic is shown to be a polynomial problem along with determining a sharp upper bound on the number of nonzero elements of the matrix to obtain infinite radius. Further, relationship between maximum (Chebyshev) norm and spectral norm is used to construct new bounds for the radius of regularity. Considering situations where the known bounds are not tight enough, a new method based on Jansson-Rohn algorithm for testing regularity of an interval matrix is presented which is not a priory exponential along with numerical experiments. For a situation where an input matrix has a special form, several corresponding results are provided such as exact formulas for several special classes of matrices, e.g., for totally positive and inverse non-negative, or approximation algorithms, e.g., rank-one radius matrices. For tridiagonal matrices, an algorithm by Bar-On, Codenotti and Leoncini is utilized to design a polynomial algorithm to compute the radius of regularity.

scholarly journals Petrophysical Analysis of an Iraqi Gas Field (Mansuriya Gas Field)

2020 ◽  
Vol 26 (3) ◽  
pp. 100-116
Author(s):  
Hasan Saleh Azeez ◽  
Dr. Abdul Aali Al-Dabaj ◽  
Dr.Samaher Lazim
Keyword(s):  
Gamma Ray ◽  
Water Saturation ◽  
Total Porosity ◽  
Logarithmic Scale ◽  
Formation Factor ◽  
Gas Field ◽  
The Matrix ◽  
Rock Parameters ◽  
Potassium 40

Mansuriya Gas field is an elongated anticlinal structure aligned from NW to SE, about 25 km long and 5-6 km wide. Jeribe formation is considered the main reservoir where it contains condensate fluid and has a uniform thickness of about 60 m. The reservoir is significantly over-pressured, (TPOC, 2014). This research is about well logs analysis, which involves the determination of Archie petrophysical parameters, water saturation, porosity, permeability and lithology. The interpretations and cross plots are done using Interactive Petrophysics (IP) V3.5 software. The rock parameters (a, m and n) values are important in determining the water saturation where (m) can be calculated by plotting the porosity from core and the formation factor from core on logarithmic scale for both and the slope which represent (m) then Pickett plot method is used to determine the other parameters after calculating Rw from water analysis . The Matrix Identification (MID), M-N and Density-Neutron crossplots indicates that the lithology of Jeribe Formation consists of dolomite, limestone with some anhydrite also gas-trend is clear in the Jeribe Formation. The main reservoir, Jeribe Formation carbonate, is subdivided into 8 zones namely  J1 to J8, based mainly on porosity log (RHOB and NPHI) trend, DT trend and saturation trend.  Jeribe formation was considered to be clean in terms of shale content .The higher gamma ray because of the uranium component which is often associated with dolomitisationl and when it is removed and only comprises the thorium and potassium-40 contributions, showed the gamma response to be low compared to the total gamma ray response that also contains the uranium   contribution.While the Jeribe formation is considered to be clean in terms of shale content so the total porosity is equal to the effective porosity.No porosity cut off is found if cutoff permeability 0.01 md is applied while the porosity cut off approximately equal to 0.1 only for unit J6 & J8 if cutoff permeability 0.1 md is applied . It can be concluded that no saturation cutoff for the units of Jeribe formation is found after a cross plot between water saturation and log porosity for the reservoir units of Jeribe formation and applied the calculated cut off porosity. The permeability has been predicted using two methods: FZI and Classical, the two methods yield approximately the same results for all wells.

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