The Experts below are selected from a list of 29403 Experts worldwide ranked by ideXlab platform
Thomas F Eibert - One of the best experts on this subject based on the ideXlab platform.
-
fast integral equation solution by multilevel green s function Interpolation combined with multilevel fast multipole method
IEEE Transactions on Antennas and Propagation, 2012Co-Authors: Dennis T Schobert, Thomas F EibertAbstract:A fast wideband integral equation (IE) solver combining the multilevel interpolatory fast Fourier transform accelerated approach (MLIPFFT) with the multilevel fast multipole method (MLFMM) is discussed. On electrically fine levels within an oct-tree multilevel structure, coupling computations are performed by MLIPFFT. This method is based on a 3D Lagrange factorization of the pertinent Green's functions with a smooth approximation error in space and it does not suffer a low frequency breakdown as known from MLFMM. For high frequency integral equation problems, MLIPFFT has decreased computational efficiency as the Nyquist theorem requires increasing numbers of samples in 3 dimensions. Due to a transition from the Interpolation Point based MLIPFFT source/receive formulation towards an appropriate k-space representation at a certain level within the oct-tree, the high frequency efficient MLFMM can be employed for coarse levels. The hybrid algorithm is hence well suited for fast wideband integral equation solutions. Both, mixed-potential and direct-field formulations are considered. Furthermore, a method for MLIPFFT extrapolation error reduction based on fine level Interpolation domain spreading is introduced. In several numerical examples, the performance of the proposed algorithm is demonstrated.
-
a wideband fast integral equation solver combining multilevel fast multipole and multilevel green s function Interpolation method with fast fourier transform acceleration
URSI General Assembly and Scientific Symposium, 2011Co-Authors: Dennis T Schobert, Thomas F EibertAbstract:A wideband fast integral solver employing a fast Fourier transform accelerated multilevel Green's function Interpolation method (MLIPFFT) combined with the multilevel fast multipole method (MLFMM) is presented. On fine levels of the employed oct-tree structure, the low frequency stable MLIPFFT is utilized. At a certain wavelength dependent threshold for the box size, the Interpolation Point based representation of the MLIPFFT is converted into its k-space representation suitable for an MLFMM. On the coarser levels, MLFMM translations are used then, where the MLIPFFT becomes less efficient. The functionality of this hybrid algorithm is demonstrated in an example.
Dennis T Schobert - One of the best experts on this subject based on the ideXlab platform.
-
fast integral equation solution by multilevel green s function Interpolation combined with multilevel fast multipole method
IEEE Transactions on Antennas and Propagation, 2012Co-Authors: Dennis T Schobert, Thomas F EibertAbstract:A fast wideband integral equation (IE) solver combining the multilevel interpolatory fast Fourier transform accelerated approach (MLIPFFT) with the multilevel fast multipole method (MLFMM) is discussed. On electrically fine levels within an oct-tree multilevel structure, coupling computations are performed by MLIPFFT. This method is based on a 3D Lagrange factorization of the pertinent Green's functions with a smooth approximation error in space and it does not suffer a low frequency breakdown as known from MLFMM. For high frequency integral equation problems, MLIPFFT has decreased computational efficiency as the Nyquist theorem requires increasing numbers of samples in 3 dimensions. Due to a transition from the Interpolation Point based MLIPFFT source/receive formulation towards an appropriate k-space representation at a certain level within the oct-tree, the high frequency efficient MLFMM can be employed for coarse levels. The hybrid algorithm is hence well suited for fast wideband integral equation solutions. Both, mixed-potential and direct-field formulations are considered. Furthermore, a method for MLIPFFT extrapolation error reduction based on fine level Interpolation domain spreading is introduced. In several numerical examples, the performance of the proposed algorithm is demonstrated.
-
a wideband fast integral equation solver combining multilevel fast multipole and multilevel green s function Interpolation method with fast fourier transform acceleration
URSI General Assembly and Scientific Symposium, 2011Co-Authors: Dennis T Schobert, Thomas F EibertAbstract:A wideband fast integral solver employing a fast Fourier transform accelerated multilevel Green's function Interpolation method (MLIPFFT) combined with the multilevel fast multipole method (MLFMM) is presented. On fine levels of the employed oct-tree structure, the low frequency stable MLIPFFT is utilized. At a certain wavelength dependent threshold for the box size, the Interpolation Point based representation of the MLIPFFT is converted into its k-space representation suitable for an MLFMM. On the coarser levels, MLFMM translations are used then, where the MLIPFFT becomes less efficient. The functionality of this hybrid algorithm is demonstrated in an example.
M J D Powell - One of the best experts on this subject based on the ideXlab platform.
-
the newuoa software for unconstrained optimization without derivatives
2006Co-Authors: M J D PowellAbstract:The NEWUOA software seeks the least value of a function F(x), x∈Rn, when F(x) can be calculated for any vector of variables x. The algorithm is iterative, a quadratic model Q≈F being required at the beginning of each iteration, which is used in a trust region procedure for adjusting the variables. When Q is revised, the new Q interpolates F at m Points, the value m = 2n + 1 being recommended. The remaining freedom in the new Q is taken up by minimizing the Frobenius norm of the change to ∇2Q. Only one Interpolation Point is altered on each iteration. Thus, except for occasional origin shifts, the amount of work per iteration is only of order (m+n)2, which allows n to be quite large. Many questions were addressed during the development of NEWUOA, for the achievement of good accuracy and robustness. They include the choice of the initial quadratic model, the need to maintain enough linear independence in the Interpolation conditions in the presence of computer rounding errors, and the stability of the updating of certain matrices that allow the fast revision of Q. Details are given of the techniques that answer all the questions that occurred. The software was tried on several test problems. Numerical results for nine of them are reported and discussed, in order to demonstrate the performance of the software for up to 160 variables.
-
uobyqa unconstrained optimization by quadratic approximation
Mathematical Programming, 2002Co-Authors: M J D PowellAbstract:UOBYQA is a new algorithm for general unconstrained optimization calculations, that takes account of the curvature of the objective function, F say, by forming quadratic models by Interpolation. Therefore, because no first derivatives are required, each model is defined by ½(n+1)(n+2) values of F, where n is the number of variables, and the Interpolation Points must have the property that no nonzero quadratic polynomial vanishes at all of them. A typical iteration of the algorithm generates a new vector of variables, \(\widetilde{\underline{x}}\)t say, either by minimizing the quadratic model subject to a trust region bound, or by a procedure that should improve the accuracy of the model. Then usually F(\(\widetilde{\underline{x}}\)t) is obtained, and one of the Interpolation Points is replaced by \(\widetilde{\underline{x}}\)t. Therefore the paper addresses the initial positions of the Interpolation Points, the adjustment of trust region radii, the calculation of \(\widetilde{\underline{x}}\)t in the two cases that have been mentioned, and the selection of the Point to be replaced. Further, UOBYQA works with the Lagrange functions of the Interpolation equations explicitly, so their coefficients are updated when an Interpolation Point is moved. The Lagrange functions assist the procedure that improves the model, and also they provide an estimate of the error of the quadratic approximation to F, which allows the algorithm to achieve a fast rate of convergence. These features are discussed and a summary of the algorithm is given. Finally, a Fortran implementation of UOBYQA is applied to several choices of F, in order to investigate accuracy, robustness in the presence of rounding errors, the effects of first derivative discontinuities, and the amount of work. The numerical results are very promising for n≤20, but larger values are problematical, because the routine work of an iteration is of fourth order in the number of variables.
-
on the lagrange functions of quadratic models that are defined by Interpolation
Optimization Methods & Software, 2001Co-Authors: M J D PowellAbstract:Quadratic models are of fundamental importance to the efficiency of many optimization algorithms when second derivatives of the objective function influence the required values of the variables. They may be constructed by Interpolation to function values for suitable choices of the Interpolation Points. We consider the Lagrange functions of this technique, because they have some highly useful properties. In particular, they show whether a change to an Interpolation Point preserves nonsingularity of the Interpolation equations, and they provide a bound on the error of the quadratic model. Further, they can be updated efficiently when an Interpolation Point is moved. These features are explained. Then it is shown that the error bound can control the adjustment of a trust region radius in a way that gives excellent convergence properties in an algorithm for unconstrained minimization calculations. Finally, a convenient procedure for generating the initial Interpolation Points is described.
-
the uniform convergence of thin plate spline Interpolation in two dimensions
Numerische Mathematik, 1994Co-Authors: M J D PowellAbstract:Let \(f\) be a function from \({\cal R}^2\) to \({\cal R}\) that has square integrable second derivatives and let \(s\) be the thin plate spline interpolant to \(f\) at the Points \(\{ \underline v_i : i \!=\! 1,2,\ldots,n \}\) in\({\cal R}^2\) . We seek bounds on the error \(| f(\underline x)-s(\underline x) |\) when \(\underline x\) is in the convex hull of the Interpolation Points or when \(\underline x\) is close to at least one of the Interpolation Points but need not be in the convex hull. We find, for example, that, if \(\underline x\) is inside a triangle whose vertices are any three of the Interpolation Points, then \(| f(\underline x)-s(\underline x) |\) is bounded above by a multiple of \(h\), where \(h\) is the length of the longest side of the triangle and where the multiplier is independent of the Interpolation Points. Further, if\({\cal D}\) is any bounded set in \({\cal R}^2\) that is not a subset of a single straight line, then we prove that a sequence of thin plate spline interpolants converges to\(f\) uniformly on \({\cal D}\). Specifically, we require \(h \!\rightarrow\! 0\), where \(h\) is now the least upper bound on the numbers \(\{ d( \underline x, {\cal V} ) : \underline x \!\in\! {\cal D} \}\) and where \(d( \underline x, {\cal V} )\),\(\underline x \!\in\! {\cal R}^2\) , is the least Euclidean distance from \(\underline x\) to an Interpolation Point. Our method of analysis applies integration by parts and the Cauchy--Schwarz inequality to the scalar product between second derivatives that occurs in the variational calculation of thin plate spline Interpolation.
Sohaib Tahir - One of the best experts on this subject based on the ideXlab platform.
-
Current Differential Protection for Distributed Transmission Lines using Low Sampling Frequency
International Journal of Engineering, 2015Co-Authors: M. Arshad Shehzad Hassan, Zaibin Jiao, Chenqing Wang, Xiaoning Kang, Guobing Song, Sohaib TahirAbstract:Current differential protection has been affected negatively by the distributed capacitive current of transmission lines. In order to solve the problem of distributed capacitance current of transmission line, the current differential protection in this paper is based on distributed parameters model of the transmission line. The current formula along with the transmission line is derived under this distributed parameter line model. The differential criterion is constructed with the current calculated from both ends to the set Point. In order to improve the practicality of the criterion, the implementation of the differential protection is given under low sampling frequency. By adding a cubic spline data Interpolation Point at each sampling interval, the calculation of the set Point distributed current under low sampling frequency is achieved. In order to improve the operating speed of current differential protection, the Point is set at the midPoint of the line, and the magnitude of the current is calculated with a half data window absolute value integrals. The results show that the proposed novel current differential principle is not affected by the distributed capacitance current. It has obvious advantages compared with traditional current differential protection principle for the low sampling frequency requirement, fast action speed and the small amount of computation.
Colin P Williams - One of the best experts on this subject based on the ideXlab platform.
-
random matrix model of adiabatic quantum computing
Physical Review A, 2005Co-Authors: David R Mitchell, Christoph Adami, Waynn Lue, Colin P WilliamsAbstract:We present an analysis of the quantum adiabatic algorithm for solving hard instances of 3-SAT san NPcomplete problemd in terms of random matrix theory sRMTd. We determine the global regularity of the spectral fluctuations of the instantaneous Hamiltonians encountered during the Interpolation between the starting Hamiltonians and the ones whose ground states encode the solutions to the computational problems of interest. At each Interpolation Point, we quantify the degree of regularity of the average spectral distribution via its Brody parameter, a measure that distinguishes regular si.e., Poissoniand from chaotic si.e., Wigner-typed distributions of normalized nearest-neighbor spacings. We find that for hard problem instances—i.e., those having a critical ratio of clauses to variables—the spectral fluctuations typically become irregular across a contiguous region of the Interpolation parameter, while the spectrum is regular for easy instances. Within the hard region, RMT may be applied to obtain a mathematical model of the probability of avoided level crossings and concomitant failure rate of the adiabatic algorithm due to nonadiabatic Landau-Zener-type transitions. Our model predicts that if the Interpolation is performed at a uniform rate, the average failure rate of the quantum adiabatic algorithm, when averaged over hard problem instances, scales exponentially with increasing problem size.
