This paper proposes an efficient method for computing selected generalized eigenpairs of a sparse Hermitian definite matrix pencil (A, B). Based on Zolotarev's best rational function approximations of the signum function and conformal mapping techniques, we construct the best rational function approximation of a rectangular function supported on an arbitrary interval. This new best rational function approximation is applied to construct spectrum filters of (A, B). Combining fast direct solvers and the shift-invariant GMRES, a hybrid fast algorithm is proposed to apply spectral filters efficiently. Compared to the state-of-the-art algorithm FEAST, the proposed rational function approximation is proved to be optimal among a larger function class, and the numerical implementation of the proposed method is also faster. The efficiency and stability of the proposed method are demonstrated by numerical examples from computational chemistry.
Many rational filters in the literature were constructed by discretizing the contour integral on the complex plane, with an appropriate quadrature rule (e.g., the Gauss-Legendre quadrature rule @cite_24 , the trapezoidal quadrature rule @cite_40 @cite_6 , and the Zolotarev quadrature rule @cite_3 ). Here @math is a closed contour on the complex plane intersecting the real axis at @math and @math with all desired eigenvalues inside @math and other eigenvalues outside (See Figure (left) for an example). Suppose @math and @math are the quadrature points and weights in the discretization of the contour @math , respectively, the contour integral is discretized as a rational function where @math , and @math for @math . Some other methods advanced with conformal maps @cite_23 @cite_19 and optimization @cite_13 @cite_21 can also provide good rational filters.
@cite_21 @cite_3 @cite_6 @cite_24 @cite_19 @cite_40 @cite_23 @cite_13
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Abstract Solving (nonlinear) eigenvalue problems by contour integration, requires an effective discretization for the corresponding contour integrals. In this paper it is shown that good rational filter functions can be computed using (nonlinear least squares) optimization techniques as opposed to designing those functions based on a thorough understanding of complex analysis. The conditions that such an effective filter function should satisfy, are derived and translated in a nonlinear least squares optimization problem solved by optimization algorithms from Tensorlab. Numerical experiments illustrate the validity of this approach. The limitations of commonly used separable extensions of one-dimensional transforms, such as the Fourier and wavelet transforms, in capturing the geometry of image edges are well known. In this paper, we pursue a "true" two-dimensional transform that can capture the intrinsic geometrical structure that is key in visual information. The main challenge in exploring geometry in images comes from the discrete nature of the data. Thus, unlike other approaches, such as curvelets, that first develop a transform in the continuous domain and then discretize for sampled data, our approach starts with a discrete-domain construction and then studies its convergence to an expansion in the continuous domain. Specifically, we construct a discrete-domain multiresolution and multidirection expansion using nonseparable filter banks, in much the same way that wavelets were derived from filter banks. This construction results in a flexible multiresolution, local, and directional image expansion using contour segments, and, thus, it is named the contourlet transform. The discrete contourlet transform has a fast iterated filter bank algorithm that requires an order N operations for N-pixel images. Furthermore, we establish a precise link between the developed filter bank and the associated continuous-domain contourlet expansion via a directional multiresolution analysis framework. We show that with parabolic scaling and sufficient directional vanishing moments, contourlets achieve the optimal approximation rate for piecewise smooth functions with discontinuities along twice continuously differentiable curves. Finally, we show some numerical experiments demonstrating the potential of contourlets in several image processing applications. We introduce a general purpose algorithm for rapidly computing certain types of oscillatory integrals which frequently arise in problems connected to wave propagation, general hyperbolic equations, and curvilinear tomography. The problem is to numerically evaluate a so-called Fourier integral operator (FIO) of the form @math at points given on a Cartesian grid. Here, @math is a frequency variable, @math is the Fourier transform of the input @math , @math is an amplitude, and @math is a phase function, which is typically as large as @math ; hence the integral is highly oscillatory. Because a FIO is a dense matrix, a naive matrix vector product with an input given on a Cartesian grid of size @math by @math would require @math operations. This paper develops a new numerical algorithm which requires @math operations and as low as @math in storage space (the constants in front of these estimates are small). It operates by localizing the integral over polar wedges with small angular aperture in the frequency plane. On each wedge, the algorithm factorizes the kernel @math into two components: (1) a diffeomorphism which is handled by means of a nonuniform FFT and (2) a residual factor which is handled by numerical separation of the spatial and frequency variables. The key to the complexity and accuracy estimates is the fact that the separation rank of the residual kernel is provably independent of the problem size. Several numerical examples demonstrate the numerical accuracy and low computational complexity of the proposed methodology. We also discuss the potential of our ideas for various applications such as reflection seismology. This paper introduces two efficient algorithms that compute the Contour Tree of a 3D scalar field F and its augmented version with the Betti numbers of each isosurface. The Contour Tree is a fundamental data structure in scientific visualization that is used to preprocess the domain mesh to allow optimal computation of isosurfaces with minimal overhead storage. The Contour Tree can also be used to build user interfaces reporting the complete topological characterization of a scalar field, as shown in Figure 1.The first part of the paper presents a new scheme that augments the Contour Tree with the Betti numbers of each isocontour in linear time. We show how to extend the scheme introduced in [3] with the Betti number computation without increasing its complexity. Thus, we improve on the time complexity from our previous approach [10] from O(m log m) to O(n log n + m), where m is the number of tetrahedra and n is the number of vertices in the domain of F.The second part of the paper introduces a new divide-and-conquer algorithm that computes the Augmented Contour Tree with improved efficiency. The central part of the scheme computes the output Contour Tree by merging two intermediate Contour Trees and is independent of the interpolant. In this way we confine any knowledge regarding a specific interpolant to an oracle that computes the tree for a single cell. We have implemented this oracle for the trilinear interpolant and plan to replace it with higher order interpolants when needed. The complexity of the scheme is O(n + t log n), where t is the number of critical points of F. For the first time we can compute the Contour Tree in linear time in many practical cases when t = O(n1 - e).Lastly, we report the running times for a parallel implementation of our algorithm, showing good scalability with the number of processors.