Decomposition Of Kernels Research Articles

Spectral Decomposition of Discrepancy Kernels on the Euclidean Ball, the Special Orthogonal Group, and the Grassmannian Manifold

To numerically approximate Borel probability measures by finite atomic measures, we study the spectral decomposition of discrepancy kernels when restricted to compact subsets of mathbb {R}^d. For restrictions to the Euclidean ball in odd dimensions, to the rotation group textrm{SO}(3), and to the Grassmannian manifold mathcal {G}_{2,4}, we compute the kernels’ Fourier coefficients and determine their asymptotics. The L_2-discrepancy is then expressed in the Fourier domain that enables efficient numerical minimization based on the nonequispaced fast Fourier transform. For textrm{SO}(3), the nonequispaced fast Fourier transform is publicly available, and, for mathcal {G}_{2,4}, the transform is derived here. We also provide numerical experiments for textrm{SO}(3) and mathcal {G}_{2,4}.

A note on the fractional Hardy inequality

We give a direct proof of fractional Hardy inequality by means of Littlewood–Paley decomposition and properties of singular homogeneous kernels of degree -d. A refinement when q>2 is proved.

Spatial Distributions of the Dose Created Phantom Pencil Beam of Mono-Energy and Bremsstrahlung Photons in a Water with Energies from 0.25 to 20 MeV

Purpose: Critical analysis of existing and obtaining more accurate data on the spatial dose distributions created in the water phantom by pencil beams (PB) of monoenergetic and bremsstrahlung photons with energies from 0.25 to 20.0 MeV, and approximation of these distributions for the purpose of calculating doses in radiation therapy. Material and Methods: Using the Monte Carlo method, the EGSnrc program and the MATLAB mathematical package, these distributions were calculated for monoenergetic photons in the energy range from 0.25 to 19.75 MeV in increments of 0.5 MeV, for bremsstrahlung photons with a maximum energy of 4.0, 6.0, 10.0, 15.0, 18.0 MeV and for the gamma-radiation spectrum of the therapeutic apparatus ROCUS. The calculation results are converted into the so-called dose kernel of photon pencil beam. The obtained dose kernel values are compared with previously published data and the observed discrepancies are discussed. Depths in water were studied from 1.0 to 40 cm in increments of 0,5 cm and along the radius from 0.02 to 46.0 cm with an uneven grid. For bremsstrahlung and photons with the spectrum of the Rocus apparatus, the possibility of approximating dose kernel values using approximation formulas convenient for calculating doses in radiation therapy has been investigated. Results: On the basis of the results obtained, a new version of the library of dose kernels of a pencil photon beam for water was created, which differs from previous versions by the use for calculating a better description and modeling of the physical processes of the interaction of photons and charged particles with matter, more adequate data on the interaction cross sections and significantly lower values of statistical uncertainties of the results. For bremsstrahlung and photons with the spectrum of the Rocus apparatus, a mathematical model of dose kernels of a pencil beam is proposed, which includes decomposition of the dose kernels into components of the primary and scattered doses, approximation formulas and empirical coefficients convenient for integration. The values of empirical coefficients are determined by fitting to the results of the calculation of dose kernels using a combination of the random search method and the nonlinear regression method. Conclusion: The results obtained in this work will improve the algorithms and increase the accuracy of dose calculation when planning remote therapy with photon beams.

Omnibus goodness of fit test based on quadratic distance

This study considers a goodness of fit test based on the quadratic distance (QD) in composite hypotheses. Lindsay et al. [Quadratic distances on probabilities: a unified approach. Ann Statist. 2008;36:983–1006] established a general theory of QD measures for the goodnees of fit test. Using the spectral decomposition of centred kernels, they verified that the QD test asymptotically follows a sum of weighed chi-square distributions. In this study special attention is paid to a smoothing kernel-based QD test and its bootstrap version. Their performances are compared via Monte Carlo simulations with those of the Bickel-Rosenblatt test and those of the Fisher's dispersion test for the normality and the testing for the Poisson distribution in IID samples and AR(1) models. The comparison results demonstrate the validity of our method.

Reflection Full Waveform Inversion with Decoupled Elastic-wave Equations in Inhomogeneous Medium

Reflection full-waveform inversion (RFWI) can reduce the nonlinearity of inversion providing an accurate initial velocity model for full-waveform inversion (FWI) through the tomographic components (low-wavenumber). However, elastic-wave reflection full-waveform inversion (ERFWI) is more vulnerable to the problem of local minimum due to the complicated multi-component wavefield. Our algorithm first divides kernels of ERFWI into four subkernels based on the theory of decoupled elastic-wave equations. Then we try to construct the tomographic components of ERFWI with only single-component wavefields, similarly to acoustic inversions. However, the S-wave velocity is still vulnerable to the coupling effects because of P-wave components contained in the S-wavefield in an inhomogeneous medium. Therefore we develop a method for decoupling elastic- wave equations in an inhomogeneous medium, which is applied to the decomposition of kernels in ERFWI. The new decoupled system can improve the accuracy of S-wavefield and hence further reduces the high-wavenumber crosstalk in the subkernel of S-wave velocity after kernels are decomposed. The numerical examples of Sigsbee2A model demonstrate that our ERFWI method with decoupled elastic-wave equations can efficiently recover the low-wavenumber components of the model and improve the precision of the S-wave velocity.

Elastic reflection waveform inversion based on the decomposition of sensitivity kernels

Full-waveform inversion (FWI) has the potential to obtain an accurate velocity model. Nevertheless, it depends strongly on the low-frequency data and the initial model. When the starting model is far from the real model, FWI tends to converge to a local minimum. Based on a scale separation of the model (into the background model and reflectivity model), reflection waveform inversion (RWI) can separate out the tomography term in the conventional FWI kernel and invert for the long-wavelength components of the velocity model by smearing the reflected wave residuals along the transmission (or “rabbit-ear”) paths. We have developed a new elastic RWI method to build the P- and S-wave velocity macromodels. Our method exploits a traveltime-based misfit function to highlight the contribution of tomography terms in the sensitivity kernels and a sensitivity kernel decomposition scheme based on the P- and S-wave separation to suppress the high-wavenumber artifacts caused by the crosstalk of different wave modes. Numerical examples reveal that the gradients of the background models become sufficiently smooth owing to the decomposition of sensitivity kernels and the traveltime-based misfit function. We implement our elastic RWI in an alternating way. At each loop, the reflectivity model is generated by elastic least-squares reverse time migration, and then the background model is updated using the separated traveltime kernels. Our RWI method has been successfully applied in synthetic and real reflection seismic data. Inversion results demonstrate that the proposed method can retrieve preferable low-wavenumber components of the P- and S-wave velocity models, which are reliable to serve as a starting model for conventional elastic FWI. Also, our method with a two-stage inversion workflow, first updating the P-wave velocity using the PP kernels and then updating the S-wave velocity using the PS kernels, is feasible and robust even when P- and S-wave velocities have different structures.

Observation-Based Decomposition of Radiative Perturbations and Radiative Kernels.

The Clouds and the Earth's Radiant Energy System (CERES)-partial radiative perturbation [PRP (CERES-PRP)] methodology applies partial-radiative-perturbation-like calculations to observational datasets to directly isolate the individual cloud, atmospheric, and surface property contributions to the variability of the radiation budget. The results of these calculations can further be used to construct radiative kernels. A suite of monthly mean observation-based inputs are used for the radiative transfer, including cloud properties from either the diurnally resolved passive-sensor-based CERES synoptic (SYN) data or the combination of the CloudSat cloud radar and Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) lidar. The CloudSat/CALIPSO cloud profiles are incorporated via a clustering method that obtains monthly mean cloud properties suitable for accurate radiative transfer calculations. The computed fluxes are validated using the TOA fluxes observed by CERES. Applications of the CERES-PRP methodology are demonstrated by computing the individual contributions to the variability of the radiation budget over multiple years and by deriving water vapor radiative kernels. The calculations for the former are used to show that an approximately linear decomposition of the total flux anomalies is achieved. The observation-based water vapor kernels were used to investigate the accuracy of the GCM-based NCAR CAM3.0 water vapor kernel. Differences between our observation-based kernel and the NCAR one are marginally larger than those inferred by previous comparisons among different GCM kernels.

Multi-scale kernels for Nyström based extension schemes

Nonlinear dimensionality reduction methods often include the construction of kernels for embedding the high-dimensional data points. Standard methods for extending the embedding coordinates (such as the Nyström method) also rely on spectral decomposition of kernels. It is desirable that these kernels capture most of the data sets’ information using only a few leading modes of the spectrum.In this work we propose multi-scale kernels, which are constructed as combinations of Gaussian kernels, to be used for kernel-based extension schemes. We review the kernels’ spectral properties and show that their first few modes capture more information compared to the standard Gaussian kernel. Their application is demonstrated on a synthetic data-set and also applied to a real-life example that models daily electricity profiles and predicts the average day-ahead behavior.

Nonlinear channel modeling and identification using baseband Volterra–Parafac models

Baseband Volterra models are very useful for representing nonlinear communication channels. These models present the specificity to include only odd-order nonlinear terms, with kernels characterized by a double symmetry. The main drawback is their parametric complexity. In this paper, we develop a new class of Volterra models, called baseband Volterra–Parafac models, with a reduced parametric complexity, by using a doubly symmetric Parafac decomposition of high order Volterra kernels viewed as tensors. Three adaptive algorithms are then proposed for estimating the parameters of these models. Some Monte Carlo simulation results are presented to compare the performance of the proposed estimation algorithms, in the case of third-order baseband Volterra systems excited by PSK and QAM inputs.

Lebesgue type decompositions for nonnegative forms

A nonnegative form t on a complex linear space is decomposed with respect to another nonnegative form w : it has a Lebesgue decomposition into an almost dominated form and a singular form. The part which is almost dominated is the largest form majorized by t which is almost dominated by w . The construction of the Lebesgue decomposition only involves notions from the complex linear space. An important ingredient in the construction is the new concept of the parallel sum of forms. By means of Hilbert space techniques the almost dominated and the singular parts are identified with the regular and a singular parts of the form. This decomposition addresses a problem posed by B. Simon. The Lebesgue decomposition of a pair of finite measures corresponds to the present decomposition of the forms which are induced by the measures. T. Ando's decomposition of a nonnegative bounded linear operator in a Hilbert space with respect to another nonnegative bounded linear operator is a consequence. It is shown that the decomposition of positive definite kernels involving families of forms also belongs to the present context. The Lebesgue decomposition is an example of a Lebesgue type decomposition, i.e., any decomposition into an almost dominated and a singular part. There is a necessary and sufficient condition for a Lebesgue type decomposition to be unique. This condition is inspired by the work of Ando concerning uniqueness questions.

Stable Convergence of Multiple Wiener-Itô Integrals

We prove sufficient conditions ensuring that a sequence of multiple Wiener-Itô integrals (with respect to a general Gaussian process) converges stably to a mixture of normal distributions. Note that stable convergence is stronger than convergence in distribution. Our key tool is an asymptotic decomposition of contraction kernels, realized by means of increasing families of projection operators. We also use an infinite-dimensional Clark-Ocone formula, as well as a version of the correspondence between “abstract” and “concrete” filtered Wiener spaces, in a spirit similar to that of Üstünel and Zakai (J. Funct. Anal. 143, 10–32, [1997]).

A grid-based approach for enterprise-scale data mining

We describe a grid-based approach for enterprise-scale data mining, which is based on leveraging parallel database technology for data storage, and on-demand compute servers for parallelism in the statistical computations. This approach is targeted towards the use of data mining in highly-automated vertical business applications, where the data is stored on one or more relational database systems, and an independent set of high-performance compute servers or a network of low-cost, commodity processors is used to improve the application performance and overall workload management. The goal of this paper is to describe an algorithmic decomposition of data mining kernels between the data storage and compute grids, which makes it possible to exploit the parallelism on the respective grids in a simple way, while minimizing the data transfer between these grids. This approach is compatible with existing standards for data mining task specification and results reporting, so that larger applications using these data mining algorithms do not have to be modified to benefit from this grid-based approach.

Parallel multigrid summation for the [formula omitted]-body problem

An Θ ( n ) parallel multigrid summation method (MG) for the N-body problem is presented. The method was originally devised for vacuum boundary conditions. Here, it is extended to periodic boundary conditions and implemented in parallel using force decomposition and MPI. MG is based on a hierarchical decomposition of computational kernels on multiple grids. For low accuracy calculations, appropriate for molecular dynamics, a sequential implementation is as fast or faster than particle mesh Ewald (PME). Our parallel implementation is more scalable than PME. The method can be combined with multiple time stepping integrators to produce a powerful simulation protocol for simulation of biological molecules and other materials. The parallel implementation is tested on both a Linux cluster with Myrinet interconnect and a shared memory computer. It is available as open-source at http://protomol.sourceforge.net. An auxiliary tool allows the automatic selection of optimal parameters for MG, and is available at http://mdsimaid.cse.nd.edu.

Analysis and simulation of spectral regrowth in radio frequency power amplifiers

This work presents a novel method for efficiently analyzing the relationship between spectral regrowth and physical distortion mechanisms in radio frequency power amplifiers. It utilizes a Volterra series model whose coefficients are computed from basic SPICE parameters. The analysis uses a decomposition of the Volterra kernels into simpler subsystems in order to greatly reduce the computation times. The method is applied to the design of several bipolar-transistor power amplifiers after a series-based model is developed for representing the increase in active device forward transit time at high collector current densities. A number of single-stage SiGe power amplifiers have been designed, fabricated, and tested using the IEEE 802.11b and IS-95 modulation schemes at different carrier frequencies, and these results are compared with the theoretical analysis.

Decomposition of Pascal's Kernels Mod p^s

Decomposition of Pascal's Kernels Mod p^s

The discrete fractional cosine and sine transforms

This paper is concerned with the definitions of the discrete fractional cosine transform (DFRCT) and the discrete fractional sine transform (DFRST). The definitions of DFRCT and DFRST are based on the eigen decomposition of DCT and DST kernels. This is the same idea as that of the discrete fractional Fourier transform (DFRFT); the eigenvalue and eigenvector relationships between the DFRCT, DFRST, and DFRFT can be established. The computations of DFRFT for even or odd signals can be planted into the half-size DFRCT and DFRST calculations. This will reduce the computational load of the DFRFT by about one half.

Spectral decomposition of time-frequency distribution kernels

This paper addresses the general problem of approximating a given time-frequency distribution (TFD) in terms of other distributions with desired properties. It relates the approximation of two time-frequency distributions to their corresponding kernel approximation. It is shown that the singular-value decomposition (SVD) of the time-frequency (t-f) kernels allows the expression of the time-frequency distributions in terms of weighted sum of smoothed pseudo Wigner-Ville distributions or modified periodograms, which are the two basic nonparametric power distributions for stationary and nonstationary signals, respectively. The windows appearing in the decomposition take zero and/or negative values and, therefore, are different than the time and lag windows commonly employed by these two distributions. The centrosymmetry and the time-support properties of the kernels along with the fast decay of the singular values lead to computational savings and allow for an efficient reduced rank kernel approximations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Regularized inversion of a two‐dimensional integral equation with applications in borehole induction measurements

Well bore measurements of conductivity, gravity, and surface measurements of magnetotelluric fields can be modeled as a two‐dimensional integral equation with additive measurement noise. The governing integral equation has the form of convolution in the first dimension and projection in the second dimension. However, these two operations are not in separable form. In these applications, given a set of measurements, efficient and robust estimation of the underlying physical property is required. For this purpose, a regularized inversion algorithm for the governing integral equation is presented in this paper. Singular value decomposition of the measurement kernels is used to exploit convolution‐projection structure of the integral equation, leading to a form where measurements are related to the physical property by a two‐stage operation: projection followed by convolution. On the other hand, estimation of the physical property can be carried out by a two‐stage inversion algorithm: deconvolution followed by back projection. A regularization method for the required multichannel deconvolution is given. Some important details of the algorithm are addressed in an application to wellbore induction measurements of conductivity.

Decomposition of pencil beam kernels for fast dose calculations in three-dimensional treatment planning.

A method for the calculation of three-dimensional dose distributions for high-energy photon beams is presented. The main features are (i) the calculation is fast enough to allow interactive three-dimensional treatment planning, and (ii) irregularly shaped or compensated fields, which are required to fit three-dimensional dose distributions to target volumes, are adequately taken into consideration. The method is based on the pencil beam convolution technique and shares its features concerning accuracy. A considerable gain in speed is achieved by decomposing the pencil beam kernel into three separated terms, thus reducing the required number of two-dimensional convolutions. The convolutions are performed in the frequency domain via the fast Hartley transform. Using these techniques, the calculation time for the convolutions is only about 8 s on a DEC VAX station 3100. This is one-fourth to one-third of the calculation time for the ray tracing through the three-dimensional CT data set, which has to be performed in any case. Results of the calculation are compared with measurements in a homogeneous phantom for 15 MV photons. Two irregular fields shaped with a multileaf collimator are considered. The deviations between measured and calculated absolute dose values are smaller than +/- 2%.

Decomposition and disintegration of positive definite kernels on convex *-semigroups

Decomposition and disintegration of positive definite kernels on convex *-semigroups