The direct geodesic problem and an approximate analytical solution in Cartesian coordinates on a triaxial ellipsoid

2020 ◽  
Vol 14 (2) ◽  
pp. 205-213
Author(s):  
G. Panou ◽  
R. Korakitis
Keyword(s):  
Numerical Method ◽  
Analytical Method ◽  
Taylor Series Expansion ◽  
Oblate Spheroid ◽  
Accurate Solution ◽  
Triaxial Ellipsoid ◽  
Data Set ◽  
Cartesian Coordinates ◽  
Similar Accuracy ◽  
Geodesic Problem

AbstractIn this work, the direct geodesic problem in Cartesian coordinates on a triaxial ellipsoid is solved by an approximate analytical method. The parametric coordinates are used and the parametric to Cartesian coordinates conversion and vice versa are presented. The geodesic equations on a triaxial ellipsoid in Cartesian coordinates are solved using a Taylor series expansion. The solution provides the Cartesian coordinates and the angle between the line of constant v and the geodesic at the end point. An extensive data set of geodesics, previously studied with a numerical method, is used in order to validate the presented analytical method in terms of stability, accuracy and execution time. We conclude that the presented method is suitable for a triaxial ellipsoid with small eccentricities and an accurate solution is obtained. At a similar accuracy level, this method is about thirty times faster than the corresponding numerical method. Finally, the presented method can also be applied in the degenerate case of an oblate spheroid, which is extensively used in geodesy.

scholarly journals Geodesic equations and their numerical solution in Cartesian coordinates on a triaxial ellipsoid

2019 ◽  
Vol 9 (1) ◽  
pp. 1-12 ◽  
Author(s):  
G. Panou ◽  
R. Korakitis
Keyword(s):  
Numerical Solution ◽  
Numerical Method ◽  
Oblate Spheroid ◽  
Triaxial Ellipsoid ◽  
Data Set ◽  
Cartesian Coordinates ◽  
Geodesic Equations ◽  
Ellipsoidal Coordinates ◽  
Extensive Test ◽  
Geodesic Problem

Abstract In this work, the geodesic equations and their numerical solution in Cartesian coordinates on an oblate spheroid, presented by Panou and Korakitis (2017), are generalized on a triaxial ellipsoid. A new exact analytical method and a new numerical method of converting Cartesian to ellipsoidal coordinates of a point on a triaxial ellipsoid are presented. An extensive test set for the coordinate conversion is used, in order to evaluate the performance of the two methods. The direct geodesic problem on a triaxial ellipsoid is described as an initial value problem and is solved numerically in Cartesian coordinates. The solution provides the Cartesian coordinates and the angle between the line of constant λ and the geodesic, at any point along the geodesic. Also, the Liouville constant is computed at any point along the geodesic, allowing to check the precision of the method. An extensive data set of geodesics is used, in order to demonstrate the validity of the numerical method for the geodesic problem. We conclude that a complete, stable and precise solution of the problem is accomplished.

scholarly journals Research Article. Geodesic equations and their numerical solutions in geodetic and Cartesian coordinates on an oblate spheroid

2017 ◽  
Vol 7 (1) ◽  
Author(s):  
G. Panou ◽  
R. Korakitis
Keyword(s):  
Initial Value Problem ◽  
Numerical Solutions ◽  
Oblate Spheroid ◽  
Initial Value ◽  
Data Set ◽  
Cartesian Coordinates ◽  
Geodesic Equations ◽  
Research Article ◽  
Fast Solution ◽  
Geodesic Problem

AbstractThe direct geodesic problem on an oblate spheroid is described as an initial value problem and is solved numerically using both geodetic and Cartesian coordinates. The geodesic equations are formulated by means of the theory of differential geometry. The initial value problem under consideration is reduced to a system of first-order ordinary differential equations, which is solved using a numerical method. The solution provides the coordinates and the azimuths at any point along the geodesic. The Clairaut constant is not used for the solution but it is computed, allowing to check the precision of the method. An extensive data set of geodesics is used, in order to evaluate the performance of the method in each coordinate system. The results for the direct geodesic problem are validated by comparison to Karney’s method. We conclude that a complete, stable, precise, accurate and fast solution of the problem in Cartesian coordinates is accomplished.

Performance of a solution of the direct geodetic problem by Taylor series of Cartesian coordinates

2021 ◽  
Vol 11 (1) ◽  
pp. 122-130
Author(s):  
C. Marx
Keyword(s):  
Test Data ◽  
Taylor Series ◽  
Solution Method ◽  
Mathematical Formulation ◽  
Current Method ◽  
Series Expansions ◽  
Triaxial Ellipsoid ◽  
Present Contribution ◽  
Data Set ◽  
Cartesian Coordinates

Abstract The direct geodetic problem is regarded on the biaxial and triaxial ellipsoid. A known solution method suitable for low eccentricities, which uses differential equations in Cartesian coordinates and Taylor series expansions of these coordinates, is advanced in view of its practical application. According to previous works, this approach has the advantages that no singularities occur in the determination of the coordinates, its mathematical formulation is simple and it is not computationally intensive. The formulas of the solution method are simplified in the present contribution. A test of this method using an extensive test data set on a biaxial earth ellipsoid shows its accuracy and practicability for distances of any length. Based on the convergence behavior of the series of the test data set, a truncation criterion for the series expansions is compiled taking into account accuracy requirements of the coordinates. Furthermore, a procedure is shown which controls the truncation of the series expansions by accuracy requirements of the direction to be determined in the direct problem. The conducted tests demonstrate the correct functioning of the methods for the series truncation. However, the considered solution method turns out to be significantly slower than another current method for biaxial ellipsoids, which makes it more relevant for triaxial ellipsoids.

scholarly journals Analytical and numerical methods of converting Cartesian to ellipsoidal coordinates

2021 ◽  
Vol 11 (1) ◽  
pp. 111-121
Author(s):  
G. Panou ◽  
R. Korakitis
Keyword(s):  
Numerical Methods ◽  
Analytical Method ◽  
Oblate Spheroid ◽  
Comparative Assessment ◽  
Triaxial Ellipsoid ◽  
The Moon ◽  
The Earth ◽  
Ellipsoidal Coordinates ◽  
Set Of Points ◽  
Exact Analytical Method

Abstract In this work, two analytical and two numerical methods of converting Cartesian to ellipsoidal coordinates of a point in space are presented. After slightly modifying a well-known exact analytical method, a new exact analytical method is developed. Also, two well-known numerical methods, which were developed for points exactly on the surface of a triaxial ellipsoid, are generalized for points in space. The four methods are validated with numerical experiments using an extensive set of points for the case of the Earth. Then, a theoretical and a numerical comparative assessment of the four methods is made. Furthermore, the new exact analytical method is applied for an almost oblate spheroid and for the case of the Moon and the results are compared. We conclude that, the generalized Panou and Korakitis’ numerical method, starting with approximate values from the new exact analytical method, is the best choice in terms of accuracy of the resulting ellipsoidal coordinates.

Possible application of approximate models for calculation of selectivity of consecutive reactions under the effect of mass transfer

1982 ◽  
Vol 47 (5) ◽  
pp. 1301-1309 ◽  
Author(s):  
František Kaštánek ◽  
Marie Fialová
Keyword(s):  
Mass Transfer ◽  
Numerical Method ◽  
Analytical Method ◽  
Approximate Analytical Method ◽  
Approximate Models ◽  
Consecutive Reactions ◽  
The Difference

The possibility of use of approximate models for calculation of selectivity of consecutive reactions is critically analysed. Simple empirical criteria are proposed which enable safer application of approximate analytical reactions. A more universal modification has been formulated by use of which the difference of selectivity calculated by the exact numerical method and by the approximate analytical method is at maximum 12%.

scholarly journals Associated equations and their corresponding resonance curve

1999 ◽  
Vol 21 (3) ◽  
pp. 147-155
Author(s):  
Nguyen Van Dinh
Keyword(s):  
Numerical Method ◽  
Analytical Method ◽  
Nonlinear Oscillations ◽  
Resonance Curve ◽  
Practical Case ◽  
High Harmonics

In the theory of nonlinear oscillations, in order to identify the resonance curve we usually try to eliminate the diphase Ѳ in the equations of stationary oscillations. We obtain thus a certain frequency-amplitude relationship. In simple cases when the mentioned equations contain only and linearly the first harmonics (sin Ѳ, cos Ѳ) the elimination of Ѳ is elementary, by using the trigono-metrical identity sin2 Ѳ+ cos2 Ѳ = 1. In general, high harmonics (sin2 Ѳ, cos2 Ѳ, etc.) are present. Consequently the expressions of sin Ѳ, cos Ѳ are cumbersome or do not exist and the analytical elimination of Ѳ is quite inconvenient or impossible. For this reason, to identify the resonance curve of complicated systems, we use the numerical method. Below, intending to develop the analytical method, we shall propose a procedure enabling us to transform the "original" complicated equations of stationary oscillations into the so-called associated ones, only and linearly containing sin Ѳ, cos Ѳ. The equivalence of the original and associated equations will be treated and the associated resonance 'curve-that is determined by the associated equations-will be analyzed The discussion will be restricted to a simple practical case in which, beside sin Ѳ and cos Ѳ, only sin2 Ѳ and cos2 Ѳ are present. Nevertheless, the method proposed and the results obtained can be generalized.

scholarly journals The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?

2021 ◽  
Vol 4 ◽  
Author(s):  
Stefano Markidis
Keyword(s):  
Neural Networks ◽  
Deep Learning ◽  
High Performance ◽  
Critical Role ◽  
Low Frequency ◽  
Accurate Solution ◽  
Limiting Factor ◽  
Data Set ◽  
Linear Solvers ◽  
Computational Performance

Physics-Informed Neural Networks (PINN) are neural networks encoding the problem governing equations, such as Partial Differential Equations (PDE), as a part of the neural network. PINNs have emerged as a new essential tool to solve various challenging problems, including computing linear systems arising from PDEs, a task for which several traditional methods exist. In this work, we focus first on evaluating the potential of PINNs as linear solvers in the case of the Poisson equation, an omnipresent equation in scientific computing. We characterize PINN linear solvers in terms of accuracy and performance under different network configurations (depth, activation functions, input data set distribution). We highlight the critical role of transfer learning. Our results show that low-frequency components of the solution converge quickly as an effect of the F-principle. In contrast, an accurate solution of the high frequencies requires an exceedingly long time. To address this limitation, we propose integrating PINNs into traditional linear solvers. We show that this integration leads to the development of new solvers whose performance is on par with other high-performance solvers, such as PETSc conjugate gradient linear solvers, in terms of performance and accuracy. Overall, while the accuracy and computational performance are still a limiting factor for the direct use of PINN linear solvers, hybrid strategies combining old traditional linear solver approaches with new emerging deep-learning techniques are among the most promising methods for developing a new class of linear solvers.

scholarly journals By using a new iterative method to the generalized system Zakharov-Kuznetsov and estimate the best parameters via applied the pso algorithm

2020 ◽  
Vol 19 (2) ◽  
pp. 1055 ◽  
Author(s):  
Abeer Aldabagh
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Iterative Method ◽  
Numerical Method ◽  
Pso Algorithm ◽  
Accurate Solution ◽  
Generalized System ◽  
A New Technique ◽  
Best Parameters ◽  
New Iterative Method

In this paper, a new iterative method was applied to the Zakharov-Kuznetsov system to obtain the approximate solution and the results were close to the exact solution, A new technique has been proposed to reach the lowest possible error, and the closest accurate solution to the numerical method is to link the numerical method with the pso algorithm which is denoted by the symbol (NIM-PSO). The results of the proposed Technique showed that they are highly efficient and very close to the exact solution, and they are also of excellent effectiveness for treating partial differential equation systems.

scholarly journals Predictive modelling using pathway scores: robustness and significance of pathway collections

2019 ◽  
Vol 20 (1) ◽  
Author(s):  
Marcelo P. Segura-Lepe ◽  
Hector C. Keun ◽  
Timothy M. D. Ebbels
Keyword(s):  
Gene Expression ◽  
Prediction Models ◽  
Disease Status ◽  
Predictive Modelling ◽  
Data Set ◽  
Prediction Rules ◽  
Similar Accuracy ◽  
Robust To Noise ◽  
Pathway Databases ◽  
Gene Expression Levels

Abstract Background Transcriptomic data is often used to build statistical models which are predictive of a given phenotype, such as disease status. Genes work together in pathways and it is widely thought that pathway representations will be more robust to noise in the gene expression levels. We aimed to test this hypothesis by constructing models based on either genes alone, or based on sample specific scores for each pathway, thus transforming the data to a ‘pathway space’. We progressively degraded the raw data by addition of noise and examined the ability of the models to maintain predictivity. Results Models in the pathway space indeed had higher predictive robustness than models in the gene space. This result was independent of the workflow, parameters, classifier and data set used. Surprisingly, randomised pathway mappings produced models of similar accuracy and robustness to true mappings, suggesting that the success of pathway space models is not conferred by the specific definitions of the pathway. Instead, predictive models built on the true pathway mappings led to prediction rules with fewer influential pathways than those built on randomised pathways. The extent of this effect was used to differentiate pathway collections coming from a variety of widely used pathway databases. Conclusions Prediction models based on pathway scores are more robust to degradation of gene expression information than the equivalent models based on ungrouped genes. While models based on true pathway scores are not more robust or accurate than those based on randomised pathways, true pathways produced simpler prediction rules, emphasizing a smaller number of pathways.

Random Collocation Method (RCM)

2010 ◽  
Author(s):  
Hiroshi Isshiki
Keyword(s):  
Numerical Method ◽  
Collocation Method ◽  
Analytical Method ◽  
Traditional Method ◽  
Analytical Approach ◽  
Numerical Procedure ◽  
Computational Method ◽  
Weak Point ◽  
The Past ◽  
Innovative Idea

Recently, young people’s concern on theory is becoming very poor. If there is a numerical procedure that is friendlier with theory, the distance between theory and calculation would be decreased much, and the interaction between them would become more active. When the geometry of the domain is simple, the traditional analytical method using function expansion is very convenient in many numerical problems. In many problems, it has given very useful solutions for various problems. However, its effectiveness is usually limited to simple geometries of the domain. In the past, a fusion of the analytical approach and computational one has not been pursued sufficiently. If it becomes possible, it may give a different paradigm for obtaining the numerical solution. In the present paper, an innovative idea named Random Collocation Method (RCM) is discussed on how to overcome the weak point of the traditional method by combining it with computational method. It is the purpose of the present paper to develop the simplest numerical method and to make the distance between the theory and numerical method as small as possible.

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