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HermiteFit, a novel algorithm for fitting a protein structure into a low-resolution electron-density map, is presented. The algorithm accelerates the rotation of the Fourier image of the electron density by using three-dimensional orthogonal Hermite functions. As part of the new method, an algorithm for the rotation of the density in the Hermite basis and an algorithm for the conversion of the expansion coefficients into the Fourier basis are presented. HermiteFit was implemented using the cross-correlation or the Laplacian-filtered cross-correlation as the fitting criterion. It is demonstrated that in the Hermite basis the Laplacian filter has a particularly simple form. To assess the quality of density encoding in the Hermite basis, an analytical way of computing the crystallographic R factor is presented. Finally, the algorithm is validated using two examples and its efficiency is compared with two widely used fitting methods, ADP_EM and colores from the Situs package. HermiteFit will be made available at http://nano-d.inrialpes.fr/software/HermiteFit or upon request from the authors.

Purpose: Most commercially available treatment planning systems (TPSs) approximate the continuous delivery of volumetric modulated arc therapy (VMAT) plans with a series of discretized static beams for treatment planning, which can make VMAT dose computation extremely inefficient. In this study, we developed a polar‐coordinate‐based pencil beam (PB) algorithm for efficient VMAT dose computation with high‐resolution gantry angle sampling that can improve the computational efficiency and reduce the dose discrepancy due to the angular under‐sampling effect. Methods and Materials: 6 MV 1×1mm2$1 \times 1{\rm{\;m}}{{\rm{m}}^2}$ pencil beams were simulated on a uniform cylindrical phantom under an EGSnrc Monte Carlo (MC) environment. The MC‐generated PB kernels were collected in the polar coordinate system for each bixel on a 40×40cm2$40 \times 40{\rm{\;c}}{{\rm{m}}^2}$ fluence map and subsequently fitted via a series of Gaussians. The fluence was calculated using a detectors' eye view with off‐axis and MLC transmission factors corrected. Doses of VMAT arc on the phantom were computed by summing the convolution results between the corresponding PB kernels and fluence for each bixel in the polar coordinate system. The convolution was performed using fast Fourier transform to expedite the computing speed. The calculated doses were converted to the Cartesian coordinate system and compared with the reference dose computed by a collapsed cone convolution (CCC) algorithm of the TPS. A heterogeneous phantom was created to study the heterogeneity corrections using the proposed algorithm. Ten VMAT arcs were included to evaluate the algorithm performance. Gamma analysis and computation complexity theory were used to measure the dosimetric accuracy and computational efficiency, respectively. Results: The dosimetric comparisons on the homogeneous phantom between the proposed PB algorithm and the CCC algorithm for 10 VMAT arcs demonstrate that the proposed algorithm can achieve a dosimetric accuracy comparable to that of the CCC algorithm with average gamma passing rates of 96% (2%/2mm) and 98% (3%/3mm). In addition, the proposed algorithm can provide better computational efficiency for VMAT dose computation using a PC equipped with a 4‐core processor, compared to the CCC algorithm utilizing a dual 10‐core server. Moreover, the computation complexity theory reveals that the proposed algorithm has a great advantage with regard to computational efficiency for VMAT dose computation on homogeneous medium, especially when a fine angular sampling rate is applied. This can support a reduction in dose errors from the angular under‐sampling effect by using a finer angular sampling rate, while still preserving a practical computing speed. For dose calculation on the heterogeneous phantom, the proposed algorithm with heterogeneity corrections can still offer a reasonable dosimetric accuracy with comparable computational efficiency to that of the CCC algorithm. Conclusions: We proposed a novel polar‐coordinate‐based pencil beam algorithm for VMAT dose computation that enables a better computational efficiency while maintaining clinically acceptable dosimetric accuracy and reducing dose error caused by the angular under‐sampling effect. It also provides a flexible VMAT dose computation structure that allows adjustable sampling rates and direct dose computation in regions of interest, which makes the algorithm potentially useful for clinical applications such as independent dose verification for VMAT patient‐specific QA. [ABSTRACT FROM AUTHOR]

Distributed digital array radar (DDAR) can operate in multiple modes and perform multiple functions. Compared with traditional phased array radars, the performance advantages of DDAR are obvious in many aspects. One of them is that DDAR can form space mesh in a wide scope. This article proposes a new fast mesh forming method. Unlike traditional multiple beam forming, the authors' algorithm can form a mesh along range and azimuth. First, the range profile is obtained based on fast Fourier transform. Second, the mesh forming is divided into several steps that are accomplished in polar formatted plane. Finally, the polar formatted result is transformed into Cartesian coordinates. The authors tested the method using RADBASE data, which shows that the authors' algorithm has good performance. [ABSTRACT FROM AUTHOR]

In this article, a fast‐iterative method and a fast‐direct method is proposed for solving one‐dimensional and two‐dimensional tempered fractional diffusion equations with constant coefficients. The proposed iterative method is accelerated by circulant preconditioning which is shown to converge superlinearly while the proposed direct method is based on circulant and skew‐circulant representation for Toeplitz matrix inversion. In one‐dimensional case, the operation cost of the proposed methods are both shown to be O ( N log N ) with O ( N ) memory requirement in each time step, where N is the number of spatial nodes. With the alternating direction implicit method, it is proven that the proposed fast solution algorithms can be extended to handle two‐dimensional tempered fractional diffusion equations with O ( N 2 log N ) operation cost and O ( N 2 ) memory requirement in each time step, where the number of spatial nodes in x ‐direction and y ‐direction both equal to N . Numerical examples are provided to illustrate the effectiveness and efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

We provide a fast algorithm for calculating the fractional difference of a time series. In standard implementations, the calculation speed (number of arithmetic operations) is of order T2, where T is the length of the time series. Our algorithm allows calculation speed of order Tlog T. For moderate and large sample sizes, the difference in computation time is substantial. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Since its discovery, the direct-methods origin-free modulus sum function S, originally denoted ZR by Rius [ Acta Cryst. (1993), A 49, 406–409], has been used for solving some relatively complex crystal structures from powder X-ray diffraction data. In these applications, phase refinement was normally carried out by maximizing the function S with a modified tangent formula. However, further progress in the powder diffraction field was hampered by the complexity of combining the tangent formula refinement that makes explicit use of triple-phase sums, with the introduction of constraints in real space necessary to counterbalance the information loss produced by peak overlap in powder diffraction. Recently, a considerably simpler and completely general phasing procedure (S-FFT) has been developed that maximizes S by means of the FFT algorithm and considers the triple-phase sums implicitly [Rius, Crespi & Torrelles (2007), Acta Cryst. A 63, 131–134]. As pointed out by the authors, this procedure could represent a source of progress in powder diffraction. Here, an algorithm combining the S-FFT procedure with real-space constraints according to the Shake-and-Bake philosophy [Weeks, de Titta, Miller & Hauptman (1993), Acta Cryst. D 49, 179–181] is given and its capability to treat powder diffraction data is proved with some realistic test structures. [ABSTRACT FROM AUTHOR]

A novel method combining the conjugate-gradient and nonuniform fast Fourier transform (CG–NUFFT) algorithms for solving integral equations in electromagnetics is presented. The CG–NUFFT method has been applied to one- and two-dimensional inhomogeneous media. Numerical results show the advantages of the CG–NUFFT method over the CG–FFT method for problems with both electrically large and small regions. © 2000 John Wiley & Sons, Inc. Microwave Opt Technol Lett 24: 385–389, 2000. [ABSTRACT FROM AUTHOR]

In this paper, both the fast Fourier transform (FFT) and the preconditioned CG technique are first introduced into the method lines (MOL) to further enhance the computational efficiency of this semianalytic method. Electromagnetic wave scattering by an infinite-plane metallic grating is used as examples to describe its implementation. For an arbitrary incident wave, the Helmholtz equation and boundary condition are first transformed into new ones so that the impedance matrix elements are calculated by the FFT technique. As a result, this Topelitz impedance matrix only requires O(N) memory storage for the conjugate gradient FFT method to solve the current distribution involving the computational complexity O(N log N). The banded diagonal impedance matrix is selected as a preconditioner to speed up the convergence rate of the CG algorithm. Our numerical results show that the PCG–FFT method converges to an accurate solution in a much smaller CPU time. © 2000 John Wiley & Sons, Inc. Microwave Opt Technol Lett 24: 170–175, 2000. [ABSTRACT FROM AUTHOR]

A new efficient algorithm based on the decomposition of strong and weak interactions among scatterers is proposed. The weak interactions, which account for the majority of the required CPU time and memory, are calculated using a canonical grid with a translation addition theorem. This facilitates the use of FFT and results in an N log N-type efficiency for CPU and O(N) for memory. [ABSTRACT FROM AUTHOR]

A generalized k-space (GkS) method is developed for transient electromagnetic modeling. This method consists of the solution of two equations: (1) the scattering equation in the spectral-time (k - 1) domain, and (2) the constitutive equation in the spatial-time (r - t) domain. Both are derived as local algebraic equations, and therefore can be solved with O(N) operations. The connection between the r - t domain and k - t domain is obtained by the spatial FFT algorithm. Therefore, in each time step, the number of complex multiply-add operations is O N log2, N, and the storage requirement is O(N). Because it treats the spatial derivatives by Fourier transform, the k-space method, compared to the finite-difference method, provides a high-order accuracy for the same discretization, it is shown that with the same accuracy requirement, the GkS method requires much fewer unknowns than the conventional finite-difference method. [ABSTRACT FROM AUTHOR]