Two hierarchies of spline interpolations. Practical algorithms for multivariate higher order splines

A systematic construction of higher order splines using two hierarchies of polynomials is presented. Explicit instructions on how to implement one of these hierarchies are given. The results are limited to interpolations on regular, rectangular grids, but an approach to other types of grids is also discussed

A Fast Mellin and Scale Transform

A fast algorithm for the discrete-scale (and -Mellin) transform is proposed. It performs a discrete-time discrete-scale approximation of the continuous-time transform, with subquadratic asymptotic complexity. The algorithm is based on a well-known relation between the Mellin and Fourier transforms, and it is practical and accurate. The paper gives some theoretical background on the Mellin, -Mellin, and scale transforms. Then the algorithm is presented and analyzed in terms of computational complexity and precision. The effects of different interpolation procedures used in the algorithm are discussed

Uncertainty Quantification of a Nonlinear Aeroelastic System Using Polynomial Chaos Expansion With Constant Phase Interpolation

The present study focuses on the uncertainty quantification of an aeroelastic instability system. This is a classical dynamical system often used to model the flow induced oscillation of flexible structures such as turbine blades. It is relevant as a preliminary fluid-structure interaction model, successfully demonstrating the oscillation modes in blade rotor structures in attached flow conditions. The potential flow model used here is also significant because the modern turbine rotors are, in general, regulated in stall and pitch in order to avoid dynamic stall induced vibrations. Geometric nonlinearities are added to this model in order to consider the possibilities of large twisting of the blades. The resulting system shows Hopf and period-doubling bifurcations. Parametric uncertainties have been taken into account in order to consider modeling and measurement inaccuracies. A quadrature based spectral uncertainty tool called polynomial chaos expansion is used to quantify the propagation of uncertainty through the dynamical system of concern. The method is able to capture the bifurcations in the stochastic system with multiple uncertainties quite successfully. However, the periodic response realizations are prone to time degeneracy due to an increasing phase shifting between the realizations. In order to tackle the issue of degeneracy, a corrective algorithm using constant phase interpolation, which was developed earlier by one of the authors, is applied to the present aeroelastic problem. An interpolation of the oscillatory response is done at constant phases instead of constant time and that results in time independent accuracy levels

Derivation of new staggered compact schemes with application to navier-stokes equations

A method is proposed for the derivation of new classes of staggered compact derivative and interpolation operators. The algorithm has its roots in an implicit interpolation theory consistent with compact schemes and reduces to the computation of the required staggered derivatives and interpolated quantities as a combination of nodal values and collocated compact derivatives. The new approach is cost-effective, simplifies the imposition of boundary conditions, and has improved spectral resolution properties, on equal order of accuracy, with respect to classical schemes. The method is applied to incompressible Navier-Stokes equations through the implementation into a staggered flow solver with a fractional step procedure, and tested on classical benchmarks.Postprint (published version

Approximate inverse based multigrid solution of large sparse linear systems

In this thesis we study the approximate inverse based multigrid algorithm FAPIN for the solution of large sparse linear systems of equations. This algorithm, which is closely related to the well known multigrid V-cycle, has proven successful in the numerical solution of several second order boundary value problems. Here we are mainly concerned with its application to fourth order problems. In particular, we demonstrate good multigrid performance with discrete problems arising from the beam equation and the biharmonic (plate) equation. The work presented also represents new experience with FAPIN using cubic B-spline, bicubic B-spline and piecewise bicubic Hermite basis functions. We recast a convergence proof in matrix notation for the nonsingular case. Central to our development are the concepts of an approximate inverse and an approximate pseudo-inverse of a matrix. In particular, we use least squares approximate inverses (and related approximate pseudo-inverses) found by solving a Frobenius matrix norm minimization problem. These approximate inverses are used in the multigrid smoothers of our FAPIN algorithms

High-order adaptive methods for computing invariant manifolds of maps

The author presents efficient and accurate numerical methods for computing invariant manifolds of maps which arise in the study of dynamical systems. In order to decrease the number of points needed to compute a given curve/surface, he proposes using higher-order interpolation/approximation techniques from geometric modeling. He uses B´ezier curves/triangles, fundamental objects in curve/surface design, to create adaptive methods. The methods are based on tolerance conditions derived from properties of B´ezier curves/triangles. The author develops and tests the methods for an ordinary parametric curve; then he adapts these methods to invariant manifolds of planar maps. Next, he develops and tests the method for parametric surfaces and then he adapts this method to invariant manifolds of three-dimensional maps

Grid generation for the solution of partial differential equations

A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given

Fast Fourier Transform at Nonequispaced Nodes and Applications

The direct computation of the discrete Fourier transform at arbitrary nodes requires O(NM) arithmetical operations, too much for practical purposes. For equally spaced nodes the computation can be done by the well known fast Fourier transform (FFT) in only O(N log N) arithmetical operations. Recently, the fast Fourier transform for nonequispaced nodes (NFFT) was developed for the fast approximative computation of the above sums in only O(N log N + M log 1/e), where e denotes the required accuracy. The principal topics of this thesis are generalizations and applications of the NFFT. This includes the following subjects: - Algorithms for the fast approximative computation of the discrete cosine and sine transform at nonequispaced nodes are developed by applying fast trigonometric transforms instead of FFTs. - An algorithm for the fast Fourier transform on hyperbolic cross points with nonequispaced spatial nodes in 2 and 3 dimensions based on the NFFT and an appropriate partitioning of the hyperbolic cross is proposed. - A unified linear algebraic approach to recent methods for the fast computation of matrix-vector-products with special dense matrices, namely the fast multipole method, fast mosaic-skeleton approximation and H-matrix arithmetic, is given. Moreover, the NFFT-based summation algorithm by Potts and Steidl is further developed and simplified by using algebraic polynomials instead of trigonometric polynomials and the error estimates are improved. - A new algorithm for the characterization of engineering surface topographies with line singularities is proposed. It is based on hard thresholding complex ridgelet coefficients combined with total variation minimization. The discrete ridgelet transform is designed by first using a discrete Radon transform based on the NFFT and then applying a dual-tree complex wavelet transform. - A new robust local scattered data approximation method is introduced. It is an advancement of the moving least squares approximation (MLS) and generalizes an approach of van den Boomgard and van de Weijer to scattered data. In particular, the new method is space and data adaptive