Splines In Space Research Articles

Inverse gravimetry: background material and multiscale mollifier approaches

This paper represents an extended version of the publications Freeden (in: Freeden, Nashed, Sonar (eds) Handbook of Geomathematics, 2nd edn, vol 1, Springer, New York, pp 3–78, 2015) and Freeden and Nashed (in: Freeden, Nashed (eds) Handbook of Mathematical Geodesy, Geosystems Mathematics, Birkhauser, Basel, pp 641–685, 2018c). It deals with the ill-posed problem of transferring input gravitational potential information in the form of Newtonian volume integral values to geological output characteristics of the density contrast function. Some essential properties of the Newton volume integral are recapitulated. Different methodologies of the resolution of the inverse gravimetry problem and their numerical implementations are examined including their dependence on the data source. Three types of input information may be distinguished, namely internal (borehole), terrestrial (surface), and/or external (spaceborne) gravitational data sets. Singular integral theory based inversion of the Newtonian integral equation such as a Haar-type solution is handled in a multiscale framework to decorrelate specific geological signal signatures with respect to inherently given features. Reproducing kernel Hilbert space regularization techniques are studied (together with their transition to certain mollified variants) to provide geological contrast density distributions by “downward continuation” from terrestrial and/or spaceborne data. Numerically, reproducing kernel Hilbert space spline solutions are formulated in terms of Gaussian approximating sums for use of gravimeter data systems.

On the nonlinear dynamics of the traveling-wave solutions of the Serre system

We numerically study nonlinear phenomena related to the dynamics of traveling wave solutions of the Serre equations including the stability, the persistence, the interactions and the breaking of solitary waves. The numerical method utilizes a high-order finite-element method with smooth, periodic splines in space and explicit Runge–Kutta methods in time. Other forms of solutions such as cnoidal waves and dispersive shock waves are also considered. The differences between solutions of the Serre equations and the Euler equations are also studied.

Splines in the Space of Shells

AbstractCubic splines in Euclidean space minimize the mean squared acceleration among all curves interpolating a given set of data points. We extend this observation to the Riemannian manifold of discrete shells in which the associated metric measures both bending and membrane distortion. Our generalization replaces the acceleration with the covariant derivative of the velocity. We introduce an effective time‐discretization for this novel paradigm for navigating shell space. Further transferring this concept to the space of triangular surface descriptors—edge lengths, dihedral angles, and triangle areas—results in a simplified interpolation method with high computational efficiency.

Three-dimensional state space spline finite strip analysis of angle-plied laminates

In this paper, a combined spline finite strip and state space approach is introduced to obtain three-dimensional solutions of laminated composite plates with general boundary conditions. Spline and linear polynomial functions are used, respectively, as the shape functions in the longitudinal and transverse directions of the strips. The variations of the displacements and stresses through the thickness of a strip are calculated through solving differential equations in the form of the state space equations. This method allows the displacements and stresses to be computed simultaneously at all nodes along the nodal lines. In comparison with three dimensional finite element solutions, the current method guarantees the continuity of all the displacements and through-thickness stresses across all interfaces of material layer and, therefore, provides more reliable and accurate results. It can also reduce the computational effort when compared with the traditional three-dimensional finite element and finite strip methods.

On the Galerkin/Finite-Element Method for the Serre Equations

A highly accurate numerical scheme is presented for the Serre system of partial differential equations, which models the propagation of dispersive shallow water waves in the fully-nonlinear regime. The fully-discrete scheme utilizes the Galerkin / finite-element method based on smooth periodic splines in space, and an explicit fourth-order Runge-Kutta method in time. Computations compared with exact solitary and cnoidal wave solutions show that the scheme achieves the optimal orders of accuracy in space and time. These computations also show that the stability of this scheme does not impose restrictive conditions on the temporal step size. In addition, solitary, cnoidal, and dispersive shock waves are studied in detail using this numerical scheme for the Serre system and compared with the 'classical' Boussinesq system for small-amplitude shallow water waves. The results show that the interaction of solitary waves in the Serre system is more inelastic. The efficacy of the numerical scheme for modeling dispersive shocks is shown by comparison with asymptotic results. These results have application to the modeling of shallow water waves of intermediate or large amplitude, such as occurs in the nearshore zone.

Non-polynomial Spline Difference Schemes for Solving Second-order Hyperbolic Equations

In this paper, a class of improved methods based on non-polynomial cubic splines in space and finite difference in time direction are constructed for the second-order hyperbolic equations with initial boundary value problems. Truncation error and stability analysis of the methods have been carried out. It is shown that by suitably choosing the parameters, many known methods can be derived from ours. We also obtain a new high accuracy scheme of ( ) 4 4 h k + ο , which is conditionally stable for 1 ≤ r .Finally, a numerical experiment is tested and results are compared with other published numerical solutions.

Convergence of discrete and penalized least squares spherical splines

We study the convergence of discrete and penalized least squares spherical splines in spaces with stable local bases. We derive a bound for error in the approximation of a sufficiently smooth function by the discrete and penalized least squares splines. The error bound for the discrete least squares splines is explicitly dependent on the mesh size of the underlying triangulation. The error bound for the penalized least squares splines additionally depends on the penalty parameter.

Sextic spline solution of variable coefficient fourth-order parabolic equations

We report new three-level implicit methods of O(k 2+h 4) and O(k 4+h 4) for the numerical solution of fourth-order non-homogeneous parabolic partial differential equation with variable coefficients. We use sextic spline in space and finite difference in time directions. Sextic spline function relations are derived by using off-step points. The linear stability of the presented method is investigated. We solve test problems numerically to validate the derived methods. Numerical comparison with other existence methods shows the superiority of our presented scheme.

Quartic spline methods for solving one-dimensional telegraphic equations

In this paper, by using a quartic spline in space and generalized trapezoidal formula in time direction, one-dimensional telegraphic equations are solved. We present new classes of two-level schemes. The accuracy of these methods are O ( k 2 + h 4 ) . It has been shown that by suitably choosing parameter, a high accuracy scheme of O ( k 3 + h 4 ) can be derived from our methods. Numerical results demonstrate the superiority of the new scheme.

Numerical Methods Based on Non-Polynomial Sextic Spline for Solution of Variable Coefficient Fourth-Order Wave Equations

A new technique based on non-polynomial sextic spline functions connecting spline functions values at mid knots and their corresponding values of the fourth-order derivatives is developed. We derive various classes of three level implicit spline methods for solution of fourth-order non-homogeneous parabolic partial differential equation with variable coefficient. These new numerical methods are based on non-polynomial sextic spline in space and finite difference in time directions. The linear stability of the presented methods is investigated. We solve test problems numerically to validate the derived methods. Numerical comparison with other existing methods shows the superiority of our presented scheme.

Parameters spline methods for the solution of hyperbolic equations

In this paper, by using a parameters cubic spline in space and compact finite difference in time direction, we get a class of finite difference schemes for solving second-order hyperbolic equations with mixed boundary conditions. Stability analysis of the methods have been carried out. It has been shown that by suitable choosing the cubic spline parameters most of the previous known methods for homogeneous and non-homogeneous cases can be derived from our methods. We also obtain new high accuracy schemes of O ( h 2 + τ 2 h 2 ) and O ( h 4 + τ 2 h 4 ) . Numerical comparison with Rashidinia’s method [J. Rashidinia et al., Spline methods for the solutions of hyperbolic equations, Appl. Math. Comput. 190 (2007) 882–886] shows the superiority of our presented schemes.

Non‐polynomials spline methods for solving a fourth‐order parabolic equation

AbstractIn this paper a fourth‐order variable coefficient parabolic partial differential equation, that governs the behaviour of a vibrating beam, is solved by using a three level method based on non‐polynomial quintic spline in space and finite difference discretization in time. We also obtain two new high accuracy schemes of O (k4, h6) and O (k4, h8) and two new schemes which are analogues of Jain's formula for the non‐homogeneous case. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Spline methods for the solution of hyperbolic equation with variable coefficients

AbstractIn this study, we developed the methods based on nonpolynomial cubic spline for numerical solution of second‐order nonhomogeneous hyperbolic partial differential equation. Using nonpolynomial cubic spline in space and finite difference in time directions, we obtained the implicit three level methods of O(k2 + h2) and O(k2 + h4). The proposed methods are applicable to the problems having singularity at x = 0, too. Stability analysis of the presented methods have been carried out. The presented methods are applied to the nonhomogeneous examples of different types. Numerical comparison with Mohanty's method (Mohanty, Appl Math Comput, 165 (2005), 229–236) shows the superiority of our presented schemes. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

Spline methods for the solutions of hyperbolic equations

Second-order hyperbolic equations with mixed boundary conditions are solved, by using a non-polynomial cubic spline in space and finite difference in time direction. We develop new classes of three level methods. Stability analysis of the methods have been carried out. It has been shown that by suitably choosing the cubic spline parameters most of the previous known methods for homogeneous and non-homogeneous cases can be derived from our methods. We also obtain new high accuracy schemes of O ( k 2 + h 2 ) and O ( k 2 + h 4 ) . Numerical example is given to illustrate the applicability and efficiency of the new methods.

Sextic spline solution for solving a fourth-order parabolic partial differential equation

A fourth-order parabolic partial differential equation in one space variable which arises in the study of transverse vibrations of a uniform flexible beam is solved. The numerical solution is obtained by using a new three-level method based on a sextic spline in space and finite-difference discretization in time. Stability analysis of the method has been carried out. It is shown that we obtain a scheme of O(k 2+h 4) and O(h 4+h 2 k 2). The method is tested on a problem which has appeared in physical applications. Comparison with some known methods shows the superiority of the present method.

Spline methods for the solution of fourth-order parabolic partial differential equations

In this paper a fourth-order non-homogeneous parabolic partial differential equation, that governs the behaviour of a vibrating beam, is solved by using a new three level method based on parametric quintic spline in space and finite difference discretization in time. Stability analysis of the method has been carried out. It has been shown that by suitably choosing the parameters most of the previous known methods for homogeneous and non-homogeneous cases can be derived from our method. We also obtain two new high accuracy schemes of O( k 4, h 6) and O( k 4, h 8) and two new schemes which are analogues of Jain’s formula for the non-homogeneous case. Comparison of our results with those of some known methods show the superiority of the present approach.

A phase space spline smoother for fitting trajectories.

This paper presents a phase space spline smoother, which is especially useful for finding a best-fit trajectory from multiple examples of a given physical motion. Unlike conventional spline smoothers, the phase space spline smoother can simultaneously fit position and velocity information. The use of velocity information is important for modeling the dynamic motion of physical systems because the state space of these systems typically includes both position and velocity variables. A detailed description of the computational procedure is presented, along with a discussion of computational expense and practical guidelines for variance estimation, preprocessing of the target dataset, smoothing of multidimensional datasets, and cross-validation for selection of smoothing weights. The smoother is demonstrated on a dataset of handwriting motions.

Accuracy and efficiency of alternative spline smoothing algorithms

Wahba (1978) showed that a smoothing spline could be obtained as the estimate of the signal in a signal plus noise model. By expressing Wahba's (1978) stochastic model in state space form we can obtain a spline smoothing algorithm which can also be used to estimate the smoothing parameter from the data by either generalized cross-validation or marginal likelihood, and to compute Bayesian confidence intervals for the unobserved function and its derivatives at all abscissae. In this paper we investigate the speed and accuracy of the state space spline smoothing algorithm and show that it compares favourably to the algorithms of Hutchinson and de Hogg (1985) and Woltring (1986). We also present a square root version of the algorithm which is numerically more accurate and is useful in the computation of higher order splines.

A Finite Element Projection Method for the Solution of Particle Transport Problems

A method for solving particle transport problems has been developed. In this method the particle flux is expressed as a linear and separable sum of odd and even components in the direction variables. Then a Bubnov-Galerkin projection technique and an equivalent variational Raleigh-Ritz solution are applied to the second-order transport equation. A dual finite element basis of polynomial splines in space and spherical harmonics in angle is used. The general theoretical and numerical problem formalism is carried out for a seven-dimensional problem with anisotropic scattering, time dependence, three spatial and two angular variables, and with a multigroup treatment of the energy dependence. The boundary conditions for most physical problems of interest are dealt with explicitly and rigorously by a classical minimization (variational) principle. Finally, the computational validation of the method is obtained by a computer solution to the monoenergetic steady-state air-over-ground problem in a cylindrical (r,z) geometry and with an exponentially varying atmosphere.