736 research outputs found
Harmonic density interpolation methods for high-order evaluation of Laplace layer potentials in 2D and 3D
We present an effective harmonic density interpolation method for the
numerical evaluation of singular and nearly singular Laplace boundary integral
operators and layer potentials in two and three spatial dimensions. The method
relies on the use of Green's third identity and local Taylor-like
interpolations of density functions in terms of harmonic polynomials. The
proposed technique effectively regularizes the singularities present in
boundary integral operators and layer potentials, and recasts the latter in
terms of integrands that are bounded or even more regular, depending on the
order of the density interpolation. The resulting boundary integrals can then
be easily, accurately, and inexpensively evaluated by means of standard
quadrature rules. A variety of numerical examples demonstrate the effectiveness
of the technique when used in conjunction with the classical trapezoidal rule
(to integrate over smooth curves) in two-dimensions, and with a Chebyshev-type
quadrature rule (to integrate over surfaces given as unions of non-overlapping
quadrilateral patches) in three-dimensions
Efficient Spectral Domain MoM for the Design of Circularly Polarized Reflectarray Antennas Made of Split Rings
The method of moments (MoM) in the spectral domain is used for the analysis of the scattering of a plane wave by a multilayered periodic structure containing conducting concentric split rings in the unit cell. Basis functions accounting for edge singularities are used in the approximation of the current density on the split rings, which makes it possible a fast convergence of MoM with respect to the number of basis functions. Since the 2-D Fourier transforms of the basis functions cannot be obtained in closed-form, judicious tricks (controlled truncation of infinite summations, interpolations, etc.) are used for the efficient numerical determination of these Fourier transforms. The implemented spectral domain MoM software has been used in the design of a circularly polarized reflectarray antenna based on split rings under the local periodicity condition. The antenna has been analyzed with our spectral domain MoM software, with CST and with HFSS, and good agreement has been found among all sets of results. Our software has proven to be around 27 times faster than CST and HFSS
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Numerical Solution of Linear Ordinary Differential Equations and Differential-Algebraic Equations by Spectral Methods
This thesis involves the implementation of spectral methods, for numerical solution of linear Ordinary Differential Equations (ODEs) and linear Differential-Algebraic Equations (DAEs). First we consider ODEs with some ordinary problems, and then, focus on those problems in which the solution function or some coefficient functions have singularities. Then, by expressing weak and strong aspects of spectral methods to solve these kinds of problems, a modified pseudo-spectral method which is more efficient than other spectral methods is suggested and tested on some examples.
We extend the pseudo-spectral method to solve a system of linear ODEs and linear DAEs and compare this method with other methods such as Backward Difference Formulae (BDF), and implicit Runge-Kutta (RK) methods using some numerical examples. Furthermore, by using appropriate choice of Gauss-Chebyshev-Radau points, we will show that this method can be used to solve a linear DAE whenever some of coefficient functions have singularities by providing some examples. We also used some problems that have already been considered by some authors by finite difference methods, and compare their results with ours.
Finally, we present a short survey of properties and numerical methods for solving DAE problems and then we extend the pseudo-spectral method to solve DAE problems with variable coefficient functions. Our numerical experience shows that spectral and pseudo-spectral methods and their modified versions are very promising for linear ODE and linear DAE problems with solution or coefficient functions having singularities.
In section 3.2, a modified method for solving an ODE is introduced which is new work. Furthermore, an extension of this method for solving a DAE or system of ODEs which has been explained in section 4.6 of chapter four is also a new idea and has not been done by anyone previously.
In all chapters, wherever we talk about ODE or DAE we mean linear
The exponentially convergent trapezoidal rule
It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators
Studies in numerical quadrature
Various types of quadrature formulae for oscillatory integrals are
studied with a view to improving the accuracy of existing techniques.
Concentration is directed towards the production of practical algorithms
which facilitate the efficient evaluation of integrals of this type
arising in applications. [Continues.
Polynomial and rational measure modifications of orthogonal polynomials via infinite-dimensional banded matrix factorizations
We describe fast algorithms for approximating the connection coefficients
between a family of orthogonal polynomials and another family with a
polynomially or rationally modified measure. The connection coefficients are
computed via infinite-dimensional banded matrix factorizations and may be used
to compute the modified Jacobi matrices all in linear complexity with respect
to the truncation degree. A family of orthogonal polynomials with modified
classical weights is constructed that support banded differentiation matrices,
enabling sparse spectral methods with modified classical orthogonal
polynomials
Dynamical decoupling of a qubit with always-on control fields
We consider dynamical decoupling schemes in which the qubit is continuously
manipulated by a control field at all times. Building on the theory of the
Uhrig Dynamical Decoupling sequence (UDD) and its connections to Chebyshev
polynomials, we derive a method of always-on control by expressing the UDD
control field as a Fourier series. We then truncate this series and numerically
optimize the series coefficients for decoupling, constructing the CAFE
(Chebyshev and Fourier Expansion) sequence. This approach generates a bounded,
continuous control field. We simulate the decoupling effectiveness of our
sequence vs. a continuous version of UDD for a qubit coupled to fully-quantum
and semi-classical dephasing baths and find comparable performance. We derive
filter functions for continuous-control decoupling sequences, and we assess how
robust such sequences are to noise on control fields. The methods we employ
provide a variety of tools to analyze continuous-control dynamical decoupling
sequences.Comment: 22 pages, 10 figure
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