Robust Interpolation Research Articles

Fast migration/inversion with multivalued ray fields: Part 2—Applications to the 3D SEG/EAGE salt model

Three‐dimensional prestack depth migration is the convenient approach for seismic imaging in the case of strong lateral variations of the velocity. Because of computing limitations, it has been limited to single‐arrival kinematic Kirchhoff migration until recently. This approach fails in the case of complex media characterized by multiarrival traveltimes. We present numerical strategies for extending in three dimensions first‐arrival kinematic Kirchhoff migration to multiarrival quantitative ray‐based migration (preserved amplitude migration). We rely on wavefront construction in a smooth velocity model to compute the multivalued traveltime and amplitude maps, and the CPU efficiency of migration itself is ensured by efficient and robust interpolation or extrapolation strategies. We present an application to the synthetic 3D SEG/EAGE salt model. Taking into account multiarrivals clearly improves subsalt imaging at the price of quite limited computing costs (a 20% increase in our case, with respect to a preserved‐amplitude single‐arrival migration).

Grid transfer operators for highly variable coefficient problems in two‐level non‐overlapping domain decomposition methods

AbstractWe propose a robust interpolation scheme for non‐overlapping two‐level domain decomposition methods applied to two‐dimensional elliptic problems with discontinuous coefficients. This interpolation is used to design a preconditioner closely related to the BPS scheme proposed in [Bramble et al. (Math. Comput. 1986; 47(175):103)]. Through numerical experiments, we show on structured and unstructured finite element problems that the new preconditioning scheme reduces to the BPS method on smooth problems but outperforms it on problems with discontinuous coefficients. In particular it maintains good scalable convergence behaviour even when the jumps in the coefficients are not aligned with subdomain interfaces. Copyright © 2003 John Wiley & Sons, Ltd.

Robust array interpolation using second-order cone programming

We study Friedlander's (1993) array interpolation technique, whose main shortcoming in multisource scenarios is that it does not provide sufficient robustness against sources arriving outside specified interpolation sectors. In this letter, we develop a new robust interpolation approach by minimizing the interpolation error inside the sectors of interest while setting multiple stopband constraints outside these sectors to prevent performance degradation effects caused by out-of-sector sources. Computationally efficient convex formulations of the robust interpolation matrix design problem using second-order cone programming are derived.

Improving time-domain measurements with a network analyzer using a robust rational interpolation technique

A method to efficiently and accurately compute a time-domain waveform from a network-analyzer frequency-domain measurement is presented in this paper. The method is based on a robust interpolation technique to construct a pole-residue representation of the response of the device-under-test. First, the rational function is expressed in terms of Chebyshev polynomials, instead of the usual power series, to accurately determine the poles of the network over a wide frequency range. The properties of a passive system are then utilized to efficiently calculate the residues. The resulting pole-residue model is analytically transformed to obtain the time-domain response in any time window, beyond the limitations of the discrete Fourier transform (DFT) technique. Unlike the DFT technique, the method does not require a large number of equally spaced harmonically related frequency points. The parametric model can also be used to economically store large measurement data. The proposed procedure is computationally inexpensive and less sensitive to numerical instability. To illustrate the validity of the method, examples of frequency- and time-domain measurements of a Beatty structure and simulation data of a low-pass Butterworth filter are given.

Multiresolution Approximate Inverse Preconditioners

We introduce a new preconditioner for elliptic PDEs on unstructured meshes. Using a wavelet-inspired basis we compress the inverse of the matrix, allowing an effective sparse approximate inverse by solving the sparsity vs. accuracy conflict. The key issue in this compression is to use second generation wavelets which can be adapted to the unstructured mesh, the true boundary conditions, and even the PDE coefficients. We also show how this gives a new perspective on multiresolution algorithms such as multigrid, interpreting the new preconditioner as a variation on node-nested multigrid. In particular, we hope the new preconditioner will combine the best of both worlds: fast convergence when multilevel methods can succeed but with robust performance for more difficult problems. The rest of the paper discusses the core issues for the preconditioner: ordering and construction of a factored approximate inverse in the multiresolution basis, robust interpolation on unstructured meshes, automatic mesh coarsening, and purely algebraic alternatives. Some exploratory numerical experiments suggest the superiority of the new basis over the standard basis for several tough problems, including discontinuous anisotropic coefficients, strong convection, and indefinite reaction problems on unstructured meshes, with scalability like hierarchical basis methods achieved.

Estimating the Inverse Autocorrelation Function from Outlier Contaminated Data

We show how a number of commonly used estimates of the inverse autocorrelation function can be modified to deal with outlier contaminated data. The robust analogues of the orthogonal and interpolation based techniques appear to be new, and provide an alternative to the robust autoregressive approach. We examine the performance of these techniques in a large scale numerical experiment. This shows significant improvements in performance in outlier contaminated data when robust techniques are used. While there was no uniformly best robust technique, our experiments support the use of the autoregressive approach to avoid catastrophic reductions in performance, and robust interpolation for short series corrupted by few outliers.

Coarse-to-fine planar object identification using invariant curve features and B-spline modeling

This paper presents a hybrid algorithm for coarse-to-fine matching of affine-invariant object features and B-spline object curves, and simultaneous estimation of transformation parameters. For coarse-matching, two dissimilar measures are exploited by using the significant corners of object boundaries to remove candidate objects with large dissimilarity to a target object. For fine-matching, a robust point interpolation approach and a simple gradient-based algorithm are applied to B-spline object curves under MMSE criterion. The combination of coarse and fine-matching steps reduces the computational cost without degrading the matching accuracy. The proposed algorithm is evaluated using affine transformed objects.

Variational extensions of the mean spherical approximation

In a previous work we have proposed a method to study complex systems with objects of arbitrary size. For certain specific forms of the atomic and molecular interactions, surprisingly simple and accurate theories (The Variational Mean Spherical Scaling Approximation, VMSSA) [(Velazquez, Blum J. Chem. Phys. 110 (1990) 10 931; Blum, Velazquez, J. Quantum Chem. (Theochem), in press)] can be obtained. The basic idea is that if the interactions can be expressed in a rapidly converging sum of (complex) exponentials, then the Ornstein–Zernike equation (OZ) has an analytical solution. This analytical solution is used to construct a robust interpolation scheme, the variation mean spherical scaling approximation (VMSSA). The Helmholtz excess free energy Δ A=Δ E− TΔ S is then written as a function of a scaling matrix Γ. Both the excess energy Δ E( Γ) and the excess entropy Δ S( Γ) will be functionals of Γ. In previous work of this series the form of this functional was found for the two- (Blum, Herrera, Mol. Phys. 96 (1999) 821) and three-exponential closures of the OZ equation (Blum, J. Stat. Phys., submitted for publication). In this paper we extend this to M Yukawas, a complete basis set: We obtain a solution for the one-component case and give a closed-form expression for the MSA excess entropy, which is also the VMSSA entropy.

Subsampling, symmetrization, and robust interpolation

The recently developed subsampling methodology has been shown to be valid for the construction of large-sample confidence regions for a general unknown parameter 9 under very minimal conditions. Nevertheless, in some specific cases—e.g. in the case of the sample mean of i.i.d. data—it has been noted that the subsampling distribution estimator underperforms as compared to alternative estimators such as the bootstrap or the asymptotic normal distribution (with estimated variance). In the present report we introduce a (partially) symmetrized.

An Energy-minimizing Interpolation for Robust Multigrid Methods

We propose a robust interpolation for multigrid based on the concepts of energy minimization and approximation. It can handle PDE coefficients of various types on structured or unstructured grids under one framework. The formulation is general; it can be applied to any dimension. We demonstrate numerically the effectiveness of the multigrid method in two dimensions by applying it to a discontinuous coefficient problem, an oscillatory coefficient problem, and an anisotropic problem. Empirically, the convergence rate is independent of the coefficients of the underlying PDE, in addition to being independent of the mesh size. The proposed method is primarily designed for second-order elliptic PDEs, with possible extensions to other classes of problems such as integral equations.

Spatial interpolation of climatic Normals: test of a new method in the Canadian boreal forest

Spatial interpolation of climatic data is frequently required to provide input for plant growth models. As no single method is optimal for all regions, it is important to compare the results obtained using alternative methods applied to each data set. For estimating 30-year averages (Normals) of monthly temperature and precipitation at specific sites in western Canada, we examined four forms of kriging and three simple alternatives. One of the alternatives was a novel technique, termed `gradient-plus-inverse distance squared' (GIDS), which combines multiple linear regression and distance-weighting. Based on the mean absolute errors from cross-validation tests, the methods were ranked GIDS > detrended kriging > nearest neighbour >co-kriging > inverse distance squared > universal kriging > ordinary kriging for interpolating monthly temperature, and GIDS > co-kriging>inverse distance squared > nearest neighbour > ordinary kriging > detrended kriging > universal kriging for interpolating monthly precipitation. GIDS gave the lowest errors, which averaged 0.5°C for monthly temperature and 3.6 mm, or 11%, for monthly precipitation. These errors were comparable with those from optimal methods in other studies. GIDS errors were also more consistent for a wide range of data variability than the other methods. The performance of kriging may have been constrained by the limited number of stations (32) in the study region, but if so, this is an unavoidable limitation in regions with sparse coverage of climate stations. Compared with kriging, GIDS was simple to apply and avoided the subjectivity involved in defining variogram models and neighbourhoods. We conclude that GIDS is a simple, robust and accurate interpolation method for use in our region, and that it should be applicable elsewhere, subject to careful comparison with other methods.

A novel interpolation method for electric potential fields in the heart during excitation.

In mapping the electrical activity of the heart, interpolation of electric potentials plays two important roles. First, it permits the estimation of potentials in regions that could not be sampled or where signal quality was poor, and second, it supports the construction of isopotential lines and surfaces for visualization. The difficulty in developing robust interpolation techniques for cardiac applications lies in the abrupt change in potential in the vicinity of the activation wave front. Despite the resulting nonlinearities in spatial potential distributions, simple linear interpolation methods are the current standard and the resulting errors due to aliasing can be large if electrode spacing does not lie on the order of 0.5-2 mm--the thickness of the activation wave front. We have developed a novel interpolation method that is based on two observations specific to the spread of excitation in the heart: (1) that propagation velocity changes smoothly within a region large enough to contain several measurement electrodes and (2) that electrogram morphology varies very little in the neighborhood of each sample point except for a time shift in the potential wave forms. The resulting interpolation scheme breaks the interpolation of one highly nonlinear variable--extracellular potential--into two separate interpolations of variables with much less drastic spatial variation--activation time and electrogram morphology. We have applied this method to potentials originally recorded at 1.5 mm spacing and then subsampled at a range of densities for testing of the interpolation. The results based both on reconstruction of isopotential contour maps and statistical comparison showed significant improvement of this novel approach over standard linear techniques. The applications of the new method include improved determination of electrophysiological parameters such as spatial gradients of potential and the path of cardiac activation and recovery, estimation of electrograms at desired locations, and visualization of electric potential distributions.

Robust Solutions to Least-Squares Problems with Uncertain Data

We consider least-squares problems where the coefficient matrices A,b are unknown but bounded. We minimize the worst-case residual error using (convex) second-order cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpreted as a Tikhonov regularization procedure, with the advantage that it provides an exact bound on the robustness of solution and a rigorous way to compute the regularization parameter. When the perturbation has a known (e.g., Toeplitz) structure, the same problem can be solved in polynomial-time using semidefinite programming (SDP). We also consider the case when A,b are rational functions of an unknown-but-bounded perturbation vector. We show how to minimize (via SDP) upper bounds on the optimal worst-case residual. We provide numerical examples, including one from robust identification and one from robust interpolation.

Accurate frequency-domain modeling and efficient circuit simulation of high-speed packaging interconnects

The paper describes an efficient frequency-domain modeling and simulation method of a coupled interconnect system using scattering parameters. First, low-order rational approximations of the multiport scattering parameters are derived over a wide frequency range using a robust interpolation technique. The method applies frequency normalization, shift, and Householder QR orthogonalization to improve the stability and the accuracy when solving the resulting systems of equations. For interconnects characterized with frequency-dependent parasitic parameters, the order of the rational of approximation is reduced by using appropriate reference system. Then, the generated multiport pole-residue models are incorporated into a circuit simulator using recursive convolution. Thus, the method avoids explicit convolution, numerical transform, and artificial filtering of a large number of points that are often necessary in conventional approaches. Examples with experimental and simulated results are given to illustrate the method.

On interpolating between probability distributions

Uncertainty in the simulation of physical systems is common and may be due to parameter variations or noise. When the uncertainty is random in nature, probability distributions for the quantities of interest are obtained. Sometimes, knowing only the mean and variance is sufficient; at other times, safety and reliability considerations require knowledge of the complete distribution. Computing these distributions is often a time-consuming task and needs to be repeated when some system parameters are changed. In this paper, formulas for interpolating between probability density and mass functions in spaces of arbitrary dimensionality are presented. It is found that these formulas give accurate results even when the functions one is interpolating between are not that “close”. As the mesh used in interpolation is refined, the accuracy of the interpolated quantities increases; accordingly, in addition to the more complicated and robust interpolation formulas meant for the case of a coarse mesh, simplified versions that result in good accuracy when the mesh is fine are also given. Savings in computational effort up to a factor of one hundred are common even when the more complicated interpolation formulas are used. This means that interpolation is a lucrative alternative to Monte Carlo simulation and even to the Generalized Cell Mapping (GCM) method when complete probability distributions, as opposed to only the low-order statistics, are needed. It is expected that this technique will relieve much of the burden of repeated, time-consuming simulations as certain relevant parameters are varied. In addition, the given formulas may be interpreted as general-purpose algorithms for the blending of shapes, thereby leading to applications beyond what is considered in this paper.

Robust interpolation of random fields homogeneous in time and isotropic on a sphere, which are observed with noise

We study the problem of optimal linear estimation of the functional $$A_N \xi = \sum\limits_{k = 0}^{\rm N} {\int\limits_{S_n } {a(k,x)\xi (k,x)m_n (dx),} }$$ , which depends on unknown values of a random field ξ(k, x),k∃Z,x∃Sn homogeneous in time and isotropic on a sphereSn, by observations of the field ξ(k,x)+η(k,x) with k∃ Z{0, 1, ...,N},x∃Sn (here, η (k, x) is a random field uncorrelated with ξ(k, x), homogeneous in time, and isotropic on a sphere Sn). We obtain formulas for calculation of the mean square error and spectral characteristic of the optimal estimate of the functionalANξ. The least favorable spectral densities and minimax (robust) spectral characteristics are found for optimal estimates of the functionalANξ.

Robust waveform interpolation for the entire speech signal

In waveform interpolation (WI), a speech signal is reconstructed by the concatenation of infinitesimal segments of an evolving characteristic waveform, which is obtained by interpolation over time [W. B. Kleijn, IEEE Trans. Speech Audio Process. 1, 386–399 (1993)]. WI leads to efficient coding of voiced speech, but current implementations switch to CELP for nonperiodic signals. The WI paradigm is extended to provide an effective basis for the coding of voiced and unvoiced speech and background noise. Prototype waveforms are extracted every 2.5 ms. At this high rate the WI analysis–synthesis system results in transparent speech quality. Each prototype waveform is decomposed into a slowly evolving waveform (SEW), obtained by convolution in time with a 40-ms smoothing window, and a remainder, the rapidly evolving waveform (REW). Because of its low bandwidth, a low bit rate suffices for the SEW (additional processing lowers the bit rate further), while the REW requires only a rough statistical description (e.g., its phase spectrum can be randomized). The new paradigm facilitates efficient, robust speech coding in the range 2–8 kb/s.

A breakpoint analysis procedure based on temporal decomposition

Temporal decomposition (TD), which is an analysis procedure based on a linear model of the effects of coarticulation, yields a linear approximation of a time sequence of speech parameters in terms of a series of time-overlapping interpolation functions and an associated series of data vectors. The number and positions of these interpolation functions show a high correspondence with phonetic events present in the speech signal. A new and more robust interpolation scheme for TD that gives it a geometric interpretation as a breakpoint analysis procedure in a multidimensional parameter space is described, where breakpoints are connected by straight line segments. The interpolation scheme can be viewed as a generalization of segmentation, allowing for a gradual transition from one segment towards the next. >

Multidimensional vibrational quantum Monte Carlo technique using robust interpolation from static or growing sets of discrete potential energy points

A simple algorithm for the calculation of fully anharmonic vibrational ground states and their properties from clamped nucleus electronic energy calculations is presented. The method is based on the diffusion quantum Monte Carlo formalism but it does not require an analytical representation of the Born—Oppenheimer potential surface. Instead, it can be coupled to any electronic structure solver. In a static variant, the algorithm exploits a pre-existing data set of potential energy points, whereas in the dynamical formulation it iteratively samples arbitrarily sized batches of new optimal nuclear configurations, whose electronic energies can be evaluated in parallel on a network of processors. The method is well suited for molecular systems involving light nuclei and other applications where the classical nuclear motion of “ab initio molecular dynamics” methods is inadequate. Test calculations for (HF) 2 and (HF) 3 demonstrate accuracy and favourable dimensional scaling of the algorithm.

Robust prediction and interpolation for vector stationary processes

Robust multivariate prediction and interpolation problems for statistically contaminated vector valued second order stationary processes are considered. The statistical contamination is modeled by requiring that the spectral density matrices of the processes lie within certain nonparametric classes. Both prediction and interpolation are then formalized as games whose saddle point solutions are sought. Finally, such solutions are found and analyzed, for two specific multivariate spectral classes.