Hermite Subdivision with Shape Constraints on a Rectangular Mesh

International audienceWe study a two parameter version of the Hermite subdivision scheme introduced in [7], wish gives $C^1$ interpolants on rectangular meshes. We prove $C^1$-convergence for a range of the two parameters. By introducing a control grid we can choose the parameters in the scheme so that the interpolant inherits positivity and/or directional monotonicity from the initial data. Several examples are given showing that a desired shape can be achieved even if we use only very crude estimates for the initial slopes

Numerical and Analytical Studies of Electromagnetic Waves: Hermite Methods, Supercontinuum Generation, and Multiple Poles in the SEM

The dissertation consists of three parts: Hermite methods, scattering from a lossless sphere, and analysis of supercontinuum generation. Hermite methods are a new class of arbitrary order algorithms to solve partial differential equations (PDE). In the first chapter, we discuss the fundamentals of Hermite methods in great detail. Hermite interpolation is discussed as well as the different time evolution schemes including Hermite-Taylor and Hermite-Runge-Kutta schemes. Further, an order adaptive Hermite method for initial value problems is described. Analytical studies and numerical simulations in both 1D and 2D are presented. To handle geometry, a hybrid Hermite discontinuous Galerkin method is introduced. A discontinuous Galerkin method is used next to the boundaries to handle the geometry and the boundary conditions, while a Hermite method is used in the interior of the computation domain to enhance the performance. Numerical simulations of 1D wave propagation and the solutions to 2D Maxwell\u27s TM equations are presented along with performance and accuracy data. In the second chapter, we study the scattering problem concerning the scattering poles from a lossless sphere for both acoustic and electromagnetic waves. We show that in certain cases there exist only first order scattering poles, but in some other cases, arbitrary order scattering poles can be found by imposing certain lossless impedance boundary conditions on the spherical scatterer. A method to construct arbitrary order scattering poles is discussed. The impedance loading function is required to satisfy Foster\u27s theorem so that the scattering problem is lossless. In the last chapter, we analyse the generation of supercontinua in photonic crystal fibers. We depart from the commonly used approach where a Taylor series expansion of the propagation constant is used to model the dispersive properties in a generalized nonlinear Schrodinger equation (gNLSE). Instead, we develop a mathematical model starting from numerically calculated group velocity dispersion (GVD) curves. Then, we construct a certain function over a broad frequency window and integrate the gNLSE in a way so that the spectral dependence of the propagation constant is preserved. We found that the generation of broadband supercontinua in air-silica microstructured fibers results from a delicate balance of dispersion and nonlinearity. Numerical simulations show that if the nonlinear self-steepening is strong enough, the model produces a shock that is not arrested by dispersion, whereas for weaker nonlinearity the pulse propagates the full extent of the fiber with the generation of a supercontinuum

Multiresolution strategies for the numerical solution of optimal control problems

Optimal control problems are often characterized by discontinuities or switchings in the control variables. One way of accurately capturing the irregularities in the solution is to use a high resolution (dense) uniform grid. This requires a large amount of computational resources both in terms of CPU time and memory. Hence, in order to accurately capture any irregularities in the solution using a few computational resources, one can refine the mesh locally in the region close to an irregularity instead of refining the mesh uniformly over the whole domain. Therefore, a novel multiresolution scheme for data compression has been designed which is shown to outperform similar data compression schemes. Specifically, we have shown that the proposed approach results in fewer grid points in the grid compared to a common multiresolution data compression scheme. The validity of the proposed mesh refinement algorithm has been verified by solving several challenging initial-boundary value problems for evolution equations in 1D. The examples have demonstrated the stability and robustness of the proposed algorithm. Next, a direct multiresolution-based approach for solving trajectory optimization problems is developed. The original optimal control problem is transcribed into a nonlinear programming (NLP) problem that is solved using standard NLP codes. The novelty of the proposed approach hinges on the automatic calculation of a suitable, nonuniform grid over which the NLP problem is solved, which tends to increase numerical efficiency and robustness. Control and/or state constraints are handled with ease, and without any additional computational complexity. The proposed algorithm is based on a simple and intuitive method to balance several conflicting objectives, such as accuracy of the solution, convergence, and speed of the computations. The benefits of the proposed algorithm over uniform grid implementations are demonstrated with the help of several nontrivial examples. Furthermore, two sequential multiresolution trajectory optimization algorithms for solving problems with moving targets and/or dynamically changing environments have been developed.Ph.D.Committee Chair: Tsiotras, Panagiotis; Committee Member: Calise, Anthony J.; Committee Member: Egerstedt, Magnus; Committee Member: Prasad, J. V. R.; Committee Member: Russell, Ryan P.; Committee Member: Zhou, Hao-Mi

Convergence analysis of a high-order Nyström integral-equation method for surface scattering problems

In this paper we present a convergence analysis for the Nyström method proposed in [J Comput Phys 169 (1):80–110, 2001] for the solution of the combined boundary integral equation formulations of sound-soft acoustic scattering problems in three-dimensional space. This fast and efficient scheme combines FFT techniques and a polar change of variables that cancels out the kernel singularity. We establish the stability of the algorithms in the L^2 norm and we derive convergence estimates in both the L^2 and L^∞ norms. In particular, our analysis establishes theoretically the previously observed super-algebraic convergence of the method in cases in which the right-hand side is smooth

From Hermite to stationary subdivision schemes in one and several variables

International audienceVector and Hermite subdivision schemes both act on vector data, but since the latter one interprets the vectors as function values and consecutive derivatives they differ by the "renormalization" of the Hermite scheme in any step. In this paper we give an algebraic factorization method in one and several variables to relate any Hermite subdivision scheme that satisfies the so-called spectral condition to a vector subdivision scheme. These factorizations are natural extensions of the "zero at π" condition known for the masks of refinable functions. Moreover, we show how this factorization can be used to investigate different forms of convergence of the Hermite scheme and why the multivariate situation is conceptionally more intricate than the univariate one. Finally, we give some examples of such factorizations

A sparse-grid isogeometric solver

Isogeometric Analysis (IGA) typically adopts tensor-product splines and NURBS as a basis for the approximation of the solution of PDEs. In this work, we investigate to which extent IGA solvers can benefit from the so-called sparse-grids construction in its combination technique form, which was first introduced in the early 90s in the context of the approximation of high-dimensional PDEs. The tests that we report show that, in accordance to the literature, a sparse-grid construction can indeed be useful if the solution of the PDE at hand is sufficiently smooth. Sparse grids can also be useful in the case of non-smooth solutions when some a-priori knowledge on the location of the singularities of the solution can be exploited to devise suitable non-equispaced meshes. Finally, we remark that sparse grids can be seen as a simple way to parallelize pre-existing serial IGA solvers in a straightforward fashion, which can be beneficial in many practical situations.Comment: updated version after revie

Scalar and Hermite subdivision schemes

AbstractA criterion of convergence for stationary nonuniform subdivision schemes is provided. For periodic subdivision schemes, this criterion is optimal and can be applied to Hermite subdivision schemes which are not necessarily interpolatory. For the Merrien family of Hermite subdivision schemes which involve two parameters, we are able to describe explicitly the values of the parameters for which the Hermite subdivision scheme is convergent