Separable Basis Functions Research Articles

Efficient Multidimensional Functional Data Analysis Using Marginal Product Basis Systems

In areas ranging from neuroimaging to climate science, advances in data storage and sensor technology have led to a proliferation in multidimensional functional datasets. A common approach to analyzing functional data is to first map the discretely observed functional samples into continuous representations, and then perform downstream statistical analysis on these smooth representations. It is well known that many of the traditional approaches used for 1D functional data representation are plagued by the curse of dimensionality and quickly become intractable as the dimension of the domain increases. In this article, we propose a computational framework for learning continuous representations from a sample of multidimensional functional data that is immune to several manifestations of the curse. The representations are constructed using a set of separable basis functions that are defined to be optimally adapted to the data. We show that the resulting estimation problem can be solved efficiently by the tensor decomposition of a carefully defined reduction transformation of the observed data. Roughness-based regularization is incorporated using a class of differential operator-based penalties. Relevant theoretical properties are also discussed. The advantages of our method over competing methods are thoroughly demonstrated in simulations. We conclude with a real data application of our method to a clinical diffusion MRI dataset. Supplementary materials for this article are available online.

Fast Overlapping Block Processing Algorithm for Feature Extraction

In many video and image processing applications, the frames are partitioned into blocks, which are extracted and processed sequentially. In this paper, we propose a fast algorithm for calculation of features of overlapping image blocks. We assume the features are projections of the block on separable 2D basis functions (usually orthogonal polynomials) where we benefit from the symmetry with respect to spatial variables. The main idea is based on a construction of auxiliary matrices that virtually extends the original image and makes it possible to avoid a time-consuming computation in loops. These matrices can be pre-calculated, stored and used repeatedly since they are independent of the image itself. We validated experimentally that the speed up of the proposed method compared with traditional approaches approximately reaches up to 20 times depending on the block parameters.

Probing inflation with precision bispectra

Calculating the primordial bispectrum predicted by a model of inflation and comparing it to what we see in the sky is very computationally intensive, necessitating layers of approximations and limiting the models which can be constrained. Exploiting the inherent separability of the tree level in-in formalism using expansions in separable basis functions provides a means by which to obviate some of these difficulties. Here, we develop this approach further into a practical and efficient numerical methodology which can be applied to a much wider and more complicated range of bispectrum phenomenology, making an important step forward towards observational pipelines which can directly confront specific models of inflation. We describe a simple augmented Legendre polynomial basis and its advantages, then test the method on single-field inflation models with non-trivial phenomenology, showing that our calculation of these coefficients is fast and accurate to high orders.

Assessing the effect of herbicide diuron on river biofilm: A statistical model

River biofilm communities are the first ones to be exposed to all toxic discharges received via run off from agricultural fields. Hence, changes in river biofilm community structure and growth pattern are considered as indicator of overall health of lotic ecosystem. Toxicants have effect on biofilm biomass, photosynthetic efficiency and chlorophyll a concentrations. Mathematical models may be applied to estimate the overall vigor of riverine ecosystems considering biofilms as indicators. Herein, previous empirical data of Ricart et al. (2009) on long term effects of environmentally relevant concentrations of diuron on biofilm communities of the River Llobregat, Spain was considered as our model inputs. Our objective is to understand the influence of diuron, chlorophyll a concentrations and photosynthetic efficiency on biovolume using a statistical model. The non-linear relationships between biovolume (dependent variable) and diuron, chlorophyll a concentrations and photosynthetic efficiency (independent variables) were represented by constructing three separate basis functions based on day 8 empirical data. Biovolume, due to nonlinear influence as yielded by the basis functions were used in a multiple linear regression model to estimate the net biovolume. Model validation was done based on day 29 empirical data. The experimentally determined biovolume and our model estimated biovolume showed similar trends. Also, diuron and photosynthetic efficiency had significant (p < 0.05) influence on biovolume. Since, the predominance of diatoms as biofilms within periphytic layers is very common in lotic systems, estimation of changes in diatom biovolume will be significant to assess the effect of herbicides. Diatom biovolume of any day (for example day 22) mentioned in the experimental study may be determined by this model, without the requirement of tedious manual biovolume calculation. Our model will be useful in numerous other studies undertaken on the toxic effect of pollutants on biofilms to quickly and accurately estimate the biofilm biovolume.

Functionally-fitted block &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \theta $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-methods for ordinary differential equations

We propose a new family of functionally-fitted block \begin{document}$ \theta $\end{document} -methods for numerically solving ordinary differential equations which integrates a chosen set of linearly independent functions exactly. The advantage of such variable coefficient methods is that the basis functions can be chosen to exploit specific properties of the problem that may be known in advance. The basic theory for the proposed methods is established. First, we derive a sufficient condition to ensure the existence of the functionally-fitted block \begin{document}$ \theta $\end{document} -methods, and discuss the independence on integration time for a set of separable basis functions. We then obtain some basic characteristics of the methods by Taylor series expansions, and show that the order of accuracy of \begin{document}$ r $\end{document} -dimensional functionally-fitted block \begin{document}$ \theta $\end{document} -method is at least \begin{document}$ r $\end{document} for ordinary differential equations. Numerical experiments are conducted to illustrate the efficiency of the functionally-fitted block \begin{document}$ \theta $\end{document} -methods.

Benchmark calculation for tunnelling through a multidimensional asymmetric double well potential

A benchmark calculation is presented for the quantum dynamics of tunnelling through a multidimensional asymmetric double well potential. A model Hamiltonian is used with a 1-dimensional tunnelling mode coupled to an (M−1)-dimensional harmonic bath, a system-bath problem. The benchmark calculation uses a basis set expansion of the wavefunction, with separate basis functions for the system and bath. Indistinguishability of configurations is exploited to greatly reduce the expense of the calculation, and a fully converged result is achieved. Comparison is offered to existing quantum dynamical methods that have tested this model problem, and further benchmark results not previously studied are presented.

Communication: practical and rigorous reduction of the many-electron quantum mechanical Coulomb problem to O(N(2/3)) storage.

It is tacitly accepted that, for practical basis sets consisting of N functions, solution of the two-electron Coulomb problem in quantum mechanics requires storage of O(N(4)) integrals in the small N limit. For localized functions, in the large N limit, or for planewaves, due to closure, the storage can be reduced to O(N(2)) integrals. Here, it is shown that the storage can be further reduced to O(N(2/3)) for separable basis functions. A practical algorithm, that uses standard one-dimensional Gaussian-quadrature sums, is demonstrated. The resulting algorithm allows for the simultaneous storage, or fast reconstruction, of any two-electron Coulomb integral required for a many-electron calculation on processors with limited memory and disk space. For example, for calculations involving a basis of 9171 planewaves, the memory required to effectively store all Coulomb integrals decreases from 2.8 Gbytes to less than 2.4 Mbytes.

Functionally-fitted block methods for ordinary differential equations

Functionally-fitted methods generalize collocation techniques to integrate exactly a chosen set of linearly independent functions. In this paper, we propose a new type of functionally-fitted block methods for ordinary differential equations. The basic theory for the proposed methods is established. First, we derive two sufficient conditions to ensure the existence of the functionally-fitted block methods, discuss their equivalence to collocation methods in a special case and independence on integration time for a set of separable basis functions. We then obtain some basic characteristics of the methods by Taylor series expansions, and show that the order of accuracy of the r-point functionally-fitted block method is at least r for general ordinary differential equations. Experimental results are conducted to demonstrate the validity of the functionally-fitted block methods.

Reduced order borehole induction modelling

This article presents a new reduced order model based on proper orthogonal decomposition (POD) for solving the electromagnetic equation for borehole modelling applications. The method aims to accurately and efficiently predict the electromagnetic fields generated by an array induction tool – an instrument that transmits and receives electrical signals along different positions within a borehole. The motivation for this approach is in the generation of an efficient ‘forward model’ (which provides solutions to the electromagnetic equation) for the purpose of improving the efficiency of inversion calculations (which typically require a large number of forward solutions) that are used to determine surrounding material properties. This article develops a reduced order model for this purpose as it can be significantly more efficient to compute than standard models, for example, those based on finite elements. It is shown here how the POD basis functions are generated from the snapshot solutions of a high resolution model, and how the discretised equations can be generated efficiently. The novelty is that this is the first time such a POD model reduction approach has been developed for this application, it is also unique in its use of separate POD basis functions for the real and complex solution fields. A numerical example for predicting the electromagnetic field is used to demonstrate the accuracy of the POD method for use as a forward model. It is shown that the method retains accuracy whilst reducing the costs of the computation by several orders of magnitude in comparison to an established method.

General CMB and primordial trispectrum estimation

We present trispectrum estimation methods which can be applied to general non-separable primordial and CMB trispectra. We present a general optimal estimator for the connected part of the trispectrum, for which we derive a quadratic term to incorporate the effects of inhomogeneous noise and masking. We describe a general algorithm for creating simulated maps with given arbitrary (and independent) power spectra, bispectra and trispectra. We propose a universal definition of the trispectrum parameter $T_{NL}$, so that the integrated bispectrum on the observational domain can be consistently compared between theoretical models. We define a shape function for the primordial trispectrum, together with a shape correlator and a useful parametrisation for visualizing the trispectrum. We derive separable analytic CMB solutions in the large-angle limit for constant and local models. We present separable mode decompositions which can be used to describe any primordial or CMB bispectra on their respective wavenumber or multipole domains. By extracting coefficients of these separable basis functions from an observational map, we are able to present an efficient estimator for any given theoretical model with a nonseparable trispectrum. The estimator has two manifestations, comparing the theoretical and observed coefficients at either primordial or late times. These mode decomposition methods are numerically tractable with order $l^5$ operations for the CMB estimator and approximately order $l^6$ for the general primordial estimator (reducing to order $l^3$ in both cases for a special class of models). We also demonstrate how the trispectrum can be reconstructed from observational maps using these methods.

General CMB and primordial bispectrum estimation: Mode expansion, map making, and measures ofFNL

We present a detailed implementation of two bispectrum estimation methods which can be applied to general non-separable primordial and CMB bispectra. The method exploits bispectrum mode decompositions on the domain of allowed wavenumber or multipole values. Concrete mode examples constructed from symmetrised tetrahedral polynomials are given, demonstrating rapid convergence for known bispectra. We use these modes to generate simulated CMB maps of high resolution (l > 2000) given an arbitrary primordial power spectrum and bispectrum or an arbitrary late-time CMB angular power spectrum and bispectrum. By extracting coefficients for the same separable basis functions from an observational map, we are able to present an efficient and general f_NL estimator for a given theoretical model. The estimator has two versions comparing theoretical and observed coefficients at either primordial or late times, thus encompassing a wider range of models, including secondary anisotropies, lensing and cosmic strings. We provide examples and validation of both f_NL estimation methods by direct comparison with simulations in a WMAP-realistic context. In addition, we show how the full bispectrum can be extracted from observational maps using these mode expansions, irrespective of the theoretical model under study. We also propose a universal definition of the bispectrum parameter F_NL for more consistent comparison between theoretical models. We obtain WMAP5 estimates of f_NL for the equilateral model from both our primordial and late-time estimators which are consistent with each other, as well as with results already published in the literature. These general bispectrum estimation methods should prove useful for the analysis of nonGaussianity in the Planck satellite data, as well as in other contexts.

Fast Analysis of Large Antenna Arrays Using the Characteristic Basis Function Method and the Adaptive Cross Approximation Algorithm

The characteristic basis function method (CBFM) has been hybridized with the adaptive cross approximation (ACA) algorithm to construct a reduced matrix equation in a time-efficient manner and to solve electrically large antenna array problems in-core, with a solve time orders of magnitude less than those in the conventional methods. Various numerical examples are presented that demonstrate that the proposed method has a very good accuracy, computational efficiency and reduced memory storage requirement. Specifically, we analyze large 1-D and 2-D arrays of electrically interconnected tapered slot antennas (TSAs). The entire computational domain is subdivided into many smaller subdomains, each of which supports a set of characteristic basis functions (CBFs). We also present a novel scheme for generating the CBFs that do not conform to the edge condition at the truncated edge of each subdomain, and provide a minor overlap between the CBFs in adjacent subdomains. As a result, the CBFs preserve the continuity of the surface current across the subdomain interfaces, thereby circumventing the need to use separate ldquoconnectionrdquo basis functions.

On the stability of functionally fitted Runge–Kutta methods

Classical collocation RK methods are polynomially fitted in the sense that they integrate an ODE problem exactly if its solution is an algebraic polynomial up to some degree. Functionally fitted RK (FRK) methods are collocation techniques that generalize this principle to solve an ODE problem exactly if its solution is a linear combination of a chosen set of arbitrary basis functions. Given for example a periodic or oscillatory ODE problem with a known frequency, it might be advantageous to tune a trigonometric FRK method targeted at such a problem. However, FRK methods lead to variable coefficients that depend on the parameters of the problem, the time, the stepsize, and the basis functions in a non-trivial manner that inhibits any in-depth analysis of the behavior of the methods in general. We present the class of so-called separable basis functions and show how to characterize the stability function of the methods in this particular class. We illustrate this explicitly with an example and we provide further insight for separable methods with symmetric collocation points.

On the stability function of functionally-fitted Runge--Kutta methods

Classical collocation Runge--Kutta methods are polynomially fitted in the sense that they integrate an ODE problem exactly if its solution is an algebraic polynomial up to some degree. Functionally fitted Runge--Kutta methods are collocation techniques that generalize this principle to solve an ODE problem exactly if its solution is a linear combination of a chosen set of arbitrary basis functions. Given for example a periodic or oscillatory ODE problem with a known frequency, it might be advantageous to tune a trigonometric functionally fitted Runge--Kutta method targeted at such a problem. However, functionally fitted Runge--Kutta methods lead to variable coefficients that depend on the parameters of the problem, the time, the step size, and the basis functions in a non-trivial manner that inhibits any in-depth analysis of the behavior of the methods in general. We present the class of so-called separable basis functions and show that it is possible to characterize the stability region of some special methods in this particular class. Explicit stability functions are given for some representative examples.

Inherent structures in the potential energy landscape of solid 4He

We study the potential energy landscape (many-atom potential energy as a function of atomic positions) of solid hcp 4He in the vicinity of the 0 K crystal structure using an accurate pair potential. At the melting point, the potential energy of the helium lattice is far above the minimum hcp interatomic potential energy. We confirm previous conclusions (based on less accurate potentials) that all of the classical phonon frequencies at the 0 K melting pressure are imaginary, indicating that the melting-point crystal corresponds to a local maximum in the potential landscape; a pressure of about 1300 bar, however, makes it a local minimum. We find that the atomic arrangements that lie at local minima in the potential landscape (‘‘inherent structures’’) are glassy and porous, and have much lower potential energy than the crystalline form at the same density. We have quantitatively characterized the glassy structures by their radial distribution functions and coordination number distributions; they qualitatively resemble inherent structures for classical monatomic liquids, but exhibit differences of detail. A model variational calculation has been carried out for the melting-density ground state. It utilizes separate basis functions for each of the inherent structures, predicts a large Lindemann ratio for the crystal, and indicates that the probability distribution is a maximum at the perfect lattice configuration.

Differential scattering and rotational excitation in low-energy positron-hydrogen-molecule collisions

Differential cross sections are calculated for positron-hydrogen-molecule scattering below the positronium formation threshold usingK-matrix elements obtained by the Kohn variational method. The internuclear separation of the hydrogen molecule is fixed at its equilibrium value, 1.4a0. The trial function used in the Kohn calculation is very flexible. It contains separable and Hylleraas-type basis functions and also basis functions to take into account longrange polarization. The cross sections for rotational excitation are also calculated, using the adiabatic-rotation approximation. Preliminary results are presented in this paper; full details will be published elsewhere.

A Tree-Structured Algorithm for Reducing Computation in Networks with Separable Basis Functions.

I describe a new algorithm for approximating continuous functions in high-dimensional input spaces. The algorithm builds a tree-structured network of variable size, which is determined both by the distribution of the input data and by the function to be approximated. Unlike other tree-structured algorithms, learning occurs through completely local mechanisms and the weights and structure are modified incrementally as data arrives. Efficient computation in the tree structure takes advantage of the potential for low-order dependencies between the output and the individual dimensions of the input. This algorithm is related to the ideas behind k-d trees (Bentley 1975), CART (Breiman et al. 1984), and MARS (Friedman 1988). I present an example that predicts future values of the Mackey-Glass differential delay equation.