Number Of Basis Elements Research Articles

Application of Bernstein Polynomial Multiwavelets for Solving Non Linear Variational Problems with Moving and Fixed Boundaries

Background: In this article, an efficient direct method has been proposed in order to solve physically significant variational problems. The proposed technique finds its basis in Bernstein polynomials multiwavelets (BPMWs). The mechanism of the proposed method is to transform the variational problem into an algebraic equation system through the use of BPMWs. Objective: Since the necessary condition of extremization consists of a differential equation that cannot be easily integrated into complex cases, an approximated numerical solution becomes a necessity. Our primary objective was to establish a wavelet-based method for solving variational problems of physical interest. Besides being computationally more effective, the proposed approach yields relatively more accurate results than other comparable methods. The approach employs fewer basis elements, which in turn increase the simplicity, decrease the calculation time, and furnishes better results. Methods: An operational matrix of integration, which is based on the BPMWs, is presented. We substituted the approximated values of x , unknown function ξ (x) and their derivative functions ( ), ( ),..... ( ) ξ ′ x ξ ′′ x ξ n x with BPMWs operational matrix of integration and BPMWs. On substituting the respective values in the given variational problem, it gets converted into a system of algebraic equations. The obtained system is further solved using the Lagrange multiplier. Results: The results obtained yield a greater degree of convergence as compared to other existing numerical methods. Numerical illustrations based on physical variational problems and the comparisons of outcomes with exact solutions demonstrate that the proposed method yields better efficiency, applicability, and accuracy. Conclusion: The proposed method gives better results than other comparable methods, even with the use of a fewer number of basis elements. The large order of matrices, such as 32, 64, and 512, obtained by using other available methods is far too high to achieve accuracy in results in comparison to the ones we obtained by using matrices of relatively lower orders, such as 7, 8 and 13, in the proposed method. This method can also be used for extremization functional occurring in electrical circuits and mechanical physical problems.

Relations between elliptic modular graphs

We consider certain elliptic modular graph functions that arise in the asymptotic expansion around the non-separating node of genus two string invariants that appear in the integrand of the D8ℛ4 interaction in the low momentum expansion of the four graviton amplitude in type II superstring theory. These elliptic modular graphs have links given by the Green function, as well its holomorphic and anti-holomorphic derivatives. Using appropriate auxiliary graphs at various intermediate stages of the analysis, we show that each graph can be expressed solely in terms of graphs with links given only by the Green function and not its derivatives. This results in a reduction in the number of basis elements in the space of elliptic modular graphs.

The asymptotic error of chaos expansion approximations for stochastic differential equations

In this paper we present a numerical scheme for stochastic differential equations based upon the Wiener chaos expansion. The approximation of a square integrable stochastic differential equation is obtained by cutting off the infinite chaos expansion in chaos order and in number of basis elements. We derive an explicit upper bound for the $L^2$ approximation error associated with our method. The proofs are based upon an application of Malliavin calculus.

A realistic, accurate and fast source modeling approach for the EEG forward problem

The aim of this paper is to advance electroencephalography (EEG) source analysis using finite element method (FEM) head volume conductor models that go beyond the standard three compartment (skin, skull, brain) approach and take brain tissue inhomogeneity (gray and white matter and cerebrospinal fluid) into account. The new approach should enable accurate EEG forward modeling in the thin human cortical structures and, more specifically, in the especially thin cortices in children brain research or in pathological applications. The source model should thus be focal enough to be usable in the thin cortices, but should on the other side be more realistic than the current standard mathematical point dipole. Furthermore, it should be numerically accurate and computationally fast. We propose to achieve the best balance between these demands with a current preserving (divergence conforming) dipolar source model. We develop and investigate a varying number of current preserving source basis elements n (n=1,…,n=5). For validation, we conducted numerical experiments within a multi-layered spherical domain, where an analytical solution exists. We show that the accuracy increases along with the number of basis elements, while focality decreases. The results suggest that the best balance between accuracy and focality in thin cortices is achieved with n=4 (or in extreme cases even n=3) basis functions, while in thicker cortices n=5 is recommended to obtain the highest accuracy. We also compare the current preserving approach to two further FEM source modeling techniques, namely partial integration and St. Venant, and show that the best current preserving source model outperforms the competing methods with regard to overall balance. For all tested approaches, FEM transfer matrices enable high computational speed. We implemented the new EEG forward modeling approaches into the open source duneuro library for forward modeling in bioelectromagnetism to enable its broader use by the brain research community. This library is build upon the DUNE framework for parallel finite elements simulations and integrates with high-level toolboxes like FieldTrip. Additionally, an inversion test has been implemented using the realistic head model to demonstrate and compare the differences between the aforementioned source models.

Free functions with symmetry

In 1936, Margarete C. Wolf showed that the ring of symmetric free polynomials in two or more variables is isomorphic to the ring of free polynomials in infinitely many variables. We show that Wolf’s theorem is a special case of a general theory of the ring of invariant free polynomials: every ring of invariant free polynomials is isomorphic to a free polynomial ring. Furthermore, we show that this isomorphism extends to the free functional calculus as a norm-preserving isomorphism of function spaces on a domain known as the row ball. We give explicit constructions of the ring of invariant free polynomials in terms of representation theory and develop a rudimentary theory of their structures. Specifically, we obtain a generating function for the number of basis elements of a given degree and explicit formulas for good bases in the abelian case.

Localized adaptive bounds for approximation-based backstepping

Recent research has established the utility of adaptive bounds on model uncertainty in adaptive approximation-based control. Such bounds have utility both for robust control law design and for self-organizing approximators that could adjust the number of basis elements N by adding additional approximation resources in the regions where the approximation error bound is large. Existing adaptive bounding methods utilize algorithms with global forgetting. In this article, we investigate methods to develop bounds on approximation accuracy that involve local forgetting. The importance of local versus global forgetting is motivated in the text and illustrated with an example.

A Haar-like Construction for the Ornstein Uhlenbeck Process

The classical Haar construction of Brownian motion uses a binary tree of triangular wedge-shaped functions. This basis has compactness properties which make it especially suited for certain classes of numerical algorithms. We present a similar basis for the Ornstein-Uhlenbeck process, in which the basis elements approach asymptotically the Haar functions as the index increases, and preserve the following properties of the Haar basis: all basis elements have compact support on an open interval with dyadic rational endpoints; these intervals are nested and become smaller for larger indices of the basis element, and for any dyadic rational, only a finite number of basis elements is nonzero at that number. Thus the expansion in our basis, when evaluated at a dyadic rational, terminates in a finite number of steps. We prove the covariance formulae for our expansion and discuss its statistical interpretation.

Reinforcement Learning with Orthonormal Basis Adaptation Based on Activity-Oriented Index Allocation

An orthonormal basis adaptation method for function approximation was developed and applied to reinforcement learning with multi-dimensional continuous state space. First, a basis used for linear function approximation of a control function is set to an orthonormal basis. Next, basis elements with small activities are replaced with other candidate elements as learning progresses. As this replacement is repeated, the number of basis elements with large activities increases. Example chaos control problems for multiple logistic maps were solved, demonstrating that the method for adapting an orthonormal basis can modify a basis while holding the orthonormality in accordance with changes in the environment to improve the performance of reinforcement learning and to eliminate the adverse effects of redundant noisy states.

Valuation portfolio in non-life insurance

The purpose of this paper is the construction of the Valuation Portfolio (VaPo) for a non-life insurance company. The VaPo represents the obligations of the insurer for the whole period of his insurance contracts. These obligations are not simply measured by a one-dimensional figure (as standard for reserves in practice), but are expressed as a portfolio of financial instruments. Hence, the actuarial reserves become multidimensional. The financial instruments in the VaPo are a properly chosen basis to represent the future cashflows resulting from the insurance contracts. In this setup, financial and technical risks are clearly separated. The financial fluctuations derive from the basis elements, the technical fluctuations are covered by an increased number of basis elements (VaPo protected against technical risks). We show how this protection can be calculated in the case of a non-life insurance company.

Asymptotic law for the distribution of prime numbers in the context of free Abelian semigroups

A refinement of B.M. Bredikhin’s theorems concerning the asymptotic behavior of the number of basis elements of a free Abelian semigroup with countable basis is presented under the assumption that the semigroup is a locally finite part of the semigroup of nonnegative real numbers. Standard analytic tools are used, including the Fourier-Laplace transform; however, we also explicitly use the fact that some specific functions, including the function t → log ζ (σ + it), are positive definite, and apply the algebraic approach to the Tauberian theorems due to G. E. Shilov; this enables one to substantially simplify the exposition as compared with the elementary approach used by Bredikhin and to establish much more (including an analog of the Dirichlet theorem). In the course of our consideration, we discuss some related problems, including the Mobius inversion formulas in the context of Abelian semigroups and some crucial inequalities for positive-definite functions; these inequalities are close in content to the topic under consideration.

Modeling Inverse Covariance Matricesby Basis Expansion

This paper proposes a new covariance modeling technique for Gaussian mixture models. Specifically the inverse covariance (precision) matrix of each Gaussian is expanded in a rank-1 basis i.e., /spl Sigma//sub j//sup -1/=P/sub j/=/spl Sigma//sub k=1//sup D//spl lambda//sub k//sup j/a/sub k/a/sub k//sup T/, /spl lambda//sub k//sup j//spl isin//spl Ropf/,a/sub k//spl isin//spl Ropf//sup d/. A generalized EM algorithm is proposed to obtain maximum likelihood parameter estimates for the basis set {a/sub k/a/sub k//sup T/}/sub k=1//sup D/ and the expansion coefficients {/spl lambda//sub k//sup j/}. This model, called the extended maximum likelihood linear transform (EMLLT) model, is extremely flexible: by varying the number of basis elements from D=d to D=d(d+1)/2 one gradually moves from a maximum likelihood linear transform (MLLT) model to a full-covariance model. Experimental results on two speech recognition tasks show that the EMLLT model can give relative gains of up to 35% in the word error rate over a standard diagonal covariance model, 30% over a standard MLLT model.

An orthonormal set of Stokes profiles

A family of well-known orthonormal functions, the set of Hermite functions, is proposed as a suitable basis for expanding the Stokes profiles of any spectral line. An expansion series thus provides different degrees of approximation to the Stokes spectrum, depending on the number of basis elements used (or on the number of coefficients). Hence, an usually large number of wavelength samples, may be substituted by a few such coefficients, thus reducing considerably the size of data files and the analysis of observable information. Moreover, since the set of Hermite functions is an universal basis, it promises to help in modern inversion techniques of the radiative transfer equation that infer the solar physical quantities from previously compiled look-up tables or artificial neural networks. These features appear to be particularly important in modern solar applications producing huge amounts of spectropolarimetric data and on near-future, on-line applications aboard spacecrafts.

General Orthonormal Bases for Robust Identification in $H_\infty$

In this paper, the problem of system identification in $H_{\infty}$ with general orthonormal basis functions is investigated. A two-stage algorithm is shown to be robust, provided that the number of basis elements as a function of the amount of data does not increase faster than $O(N^{\frac{2}{5}})$. Worst-case identification error bounds in the $H_{\infty}$ norm are derived. The algorithm also works on nonuniformly spaced frequency response measurements. An example is provided to illustrate the application of the algorithm.

On the L2 convergence of collocation for the generalized airfoil equation

In this paper we establish the L 2 convergence of a polynomial collocation method for the solution of a class of Cauchy singular linear integral equations, which we term the generalized airfoil equation. Previous numerical results have shown that if the right hand side is smooth then convergence is rapid, with 6 decimal accuracy achievable using 8–10 basis elements. Practical problems in aerodynamics dictate that this equation be solved for discontinuous data. The convergence rate is numerically demonstrated to be O( 1 N ) , where N is the number of basis elements used. Simple extrapolation is shown to be effective in accelerating the convergence, 4–5 decimal accuracy being achieved using 16 basis elements.

Singularities of Shannon functions in the case of automation bases

It is shown that in the realization of functions of logical algebra by means of circuits in automaton bases, the asymptotic behavior of the Shannon function can depend on many parameters of an arbitrarily large number of basis elements.