Gravity interpretation with the aid of quadratic programming

Geophysics ◽  
1980 ◽  
Vol 45 (3) ◽  
pp. 403-419 ◽  
Author(s):  
N. J. Fisher ◽  
L. E. Howard
Keyword(s):  
Quadratic Programming ◽  
Least Squares ◽  
Theoretical Data ◽  
Linear Constraints ◽  
Upper And Lower Bounds ◽  
Programming Algorithm ◽  
Least Squares Problem ◽  
The Matrix ◽  
Gravity Interpretation ◽  
Value Decomposition

The inverse gravity problem is posed as a linear least‐squares problem with the variables being densities of two‐dimensional prisms. Upper and lower bounds on the densities are prescribed so that the problem becomes a linearly constrained least‐squares problem, which is solved using a quadratic programming algorithm designed for upper and lower bound‐type constraints. The solution to any problem is smoothed by damping, using the singular value decomposition of the matrix of gravitational attractions. If the solution is required to be monotonically increasing with depth, then this feature can be incorporated. The method is applied to both field and theoretical data. The results are plotted for (1) undamped, nonmonotonic, (2) damped, nonmonotonic, and (3) damped, monotonic solutions; these conditions illustrate the composite approach of interpretation where both damping techniques and linear constraints are used in refining a solution which at first is unacceptable on geologic grounds while fitting the observed data well.

scholarly journals A Modification of Quadratic Programming Algorithm

2020 ◽  
Vol 10 (1) ◽  
pp. 167-176
Keyword(s):  
Quadratic Programming ◽  
Objective Function ◽  
Simplex Method ◽  
Optimal Solution ◽  
Linear Constraints ◽  
Programming Algorithm ◽  
Basic Solution ◽  
Dual Simplex Method ◽  
Modified Method ◽  
Feasible Solutions

In the existing methods for solving Quadratic Programming Problems having linearly factorized objective function and linear constraints, all the linear factors of the objective function are supposed to be positive for all feasible solutions. Here, a modification of the existing methods is proposed and it has been proved that the modified method can be applied to find the optimal solution of the problem even if all the linear factors of the objective function are not necessarily positive for all feasible solutions. Moreover, the proposed method can be applied to find the optimal solution of the problem even if the basic solution at any stage is not feasible. If the initial basic solution is feasible, we use simplex method to find the optimal solution. If the basic solution at any stage is not feasible, we use dual simplex method to find the optimal solution. Numerical examples are given to illustrate the method and the results are compared with the results obtained by other methods.

scholarly journals A Purely Algebraic Justification of the Kabsch-Umeyama Algorithm

2019 ◽  
Vol 124 ◽  
Author(s):  
Jim Lawrence ◽  
Javier Bernal ◽  
Christoph Witzgall
Keyword(s):  
Euclidean Space ◽  
Least Squares ◽  
Rigid Motion ◽  
Singular Value ◽  
Dimensional Euclidean Space ◽  
Least Squares Problem ◽  
Orthogonal Procrustes Problem ◽  
Motion Problem ◽  
Procrustes Problem ◽  
Value Decomposition

The constrained orthogonal Procrustes problem is the least-squares problem that calls for a rotation matrix that optimally aligns two matrices of the same order. Over past decades, the algorithm of choice for solving this problem has been the Kabsch-Umeyama algorithm which is essentially no more than the computation of the singular value decomposition of a particular matrix. Its justification as presented separately by Kabsch and Umeyama is not totally algebraic as it is based on solving the minimization problem via Lagrange multipliers. In order to provide a more transparent alternative, it is the main purpose of this paper to present a purely algebraic justification of the algorithm through the exclusive use of simple concepts from linear algebra. For the sake of completeness, a proof is also included of the well-known and widely-used fact that the orientation-preserving rigid motion problem, i.e., the least-squares problem that calls for an orientation-preserving rigid motion that optimally aligns two corresponding sets of points in d-dimensional Euclidean space, reduces to the constrained orthogonal Procrustes problem.

Weighted total least squares problems with inequality constraints solved by standard least squares theory

2020 ◽  
Author(s):  
Xie Jian ◽  
Long Sichun
Keyword(s):  
Quadratic Programming ◽  
Least Squares ◽  
Target Function ◽  
Inequality Constraints ◽  
Point Clouds ◽  
Coefficient Matrix ◽  
Total Least Squares ◽  
Programming Algorithm ◽  
Model Parameters ◽  
Weighted Total Least Squares

<p>The errors-in-variables (EIV) model is applied to surveying and mapping fields such as empirical coordinate transformation, line/plane fitting and rigorous modelling of point clouds and so on as it takes the errors both in coefficient matrix and observation vector into account. In many cases, not all of the elements in coefficient matrix are random or some of the elements are functionally dependent. The partial EIV (PEIV) model is more suitable in dealing with such structured coefficient matrix. Furthermore, when some reliable prior information expressed by inequality constraints is considered, the adjustment result of inequality constrained PEIV (ICPEIV) model is expected to be improved. There are two kinds of algorithms to solve the ICPEIV model under the weighted total least squares (WTLS) criterion currently. On the one hand, one can linearize the PEIV model and transform it into a sequence of quadratic programming (QP) sub-problems. On the other hand, one can directly solve the nonlinear target function by common used programming algorithms.All the QP algorithms and nonlinear programming methods are complicated and not familiar to the geodesists, so the ICPEIV model is not widely used in geodesy.   </p><p>In this contribution, an algorithm based on standard least squares is proposed. First, the estimation of model parameters and random variables in coefficient matrix are separated according to the Karush-Kuhn-Tucker (KKT) conditions of the minimization problem. The model parameters are obtained by solving the QP sub-problems while the variables are determined by the functional relationship between them. Then the QP problem is transformed to a system of linear equations with nonnegative Lagrange multipliers which is solved by an improved Jacobi iterative algorithm. It is similar to the equality-constrained least squares problem. The algorithm is simple because the linearization process is not required and it has the same form of classical least squares adjustment. Finally, two empirical examples are presented. The linear approximation algorithm, the sequential quadratic programming algorithm and the standard least squares algorithm are used. The examples show that the new method is efficient in computation and easy to implement, so it is a beneficial extension of classical least squares theory.</p>

scholarly journals The Generalized Bisymmetric (Bi-Skew-Symmetric) Solutions of a Class of Matrix Equations and Its Least Squares Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Yifen Ke ◽  
Changfeng Ma
Keyword(s):  
Least Squares ◽  
General Expression ◽  
Numerical Experiments ◽  
Matrix Equations ◽  
Solvability Conditions ◽  
Least Squares Problem ◽  
The Matrix

The solvability conditions and the general expression of the generalized bisymmetric and bi-skew-symmetric solutions of a class of matrix equations(AX=B,XC=D)are established, respectively. If the solvability conditions are not satisfied, the generalized bisymmetric and bi-skew-symmetric least squares solutions of the matrix equations are considered. In addition, two algorithms are provided to compute the generalized bisymmetric and bi-skew-symmetric least squares solutions. Numerical experiments illustrate that the results are reasonable.

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