25 research outputs found
Functionals of exponential Brownian motion and divided differences
No full textWe provide a surprising new application of classical approximation theory to a fundamental asset-pricing model of mathematical finance. Specifically, we calculate an analytic value for the correlation coefficient between
exponential Brownian motion and its time average, and we find the use of divided differences greatly elucidates formulae, providing a path to several new results. As applications, we find that this correlation coefficient is always at least 1/p2 and, via the Hermite–Genocchi integral relation, demonstrate that all moments of the time average are certain divided differences of the exponential function. We also prove that these moments agree with the somewhat more complex formulae obtained by Oshanin and Yor
Rapid evaluation of radial basis functions
No full textOver the past decade, the radial basis function method has been shown to produce high quality solutions to the multivariate scattered data interpolation problem. However, this method has been associated with very high computational cost, as compared to alternative methods such as finite element or multivariate spline interpolation. For example. the direct evaluation at M locations of a radial basis function interpolant with N centres requires O(M N) floating-point operations. In this paper we introduce a fast evaluation method based on the Fast Gauss Transform and suitable quadrature rules. This method has been applied to the Hardy multiquadric, the inverse multiquadric and the thin-plate spline to reduce the computational complexity of the interpolant evaluation to O(M + N) floating point operations. By using certain localisation properties of conditionally negative definite functions this method has several performance advantages against traditional hierarchical rapid summation methods which we discuss in detail
Preconditioned conjugate gradients, radial basis functions, and Toeplitz matrices
No full textAbstractRadial basis functions provide highly useful and flexible interpolants to multivariate functions. Further, they are beginning to be used in the numerical solution of partial differential equations. Unfortunately, their construction requires the solution of a dense linear system. Therefore, much attention has been given to iterative methods. In this paper, we present a highly efficient preconditioner for the conjugate gradient solution of the interpolation equations generated by gridded data. Thus, our method applies to the corresponding Toeplitz matrices. The number of iterations required to achieve a given tolerance is independent of the number of variables
On Shifted Cardinal Interpolation by Gaussians and Multiquadrics
No full textAbstractA radial basis function approximation is a linear combination of translates of a fixed functionϕ: Rd→R. Such functions possess many useful and interesting properties when the translates are integers andϕis radially symmetric. We study the closely related problem for which the fixed function is the shifted Gaussianϕ=G(·−α), whereG(x)=exp(−λ‖x‖22) andα∈Rd. Specifically, we exploit the theory of elliptic functions to establish the invertibility of the Toeplitz operator[formula]whenαhas no half-integer components; it is singular otherwise. This implies the existence of ashifted Gaussian cardinal function, that is, a linear combinationχof integer translates of the shifted Gaussian satisfyingχ(j)=δ0j. We also study shifted cardinal functions when the parameterλtends to zero. In particular, we discover their uniform convergence to the sinc function when the shift vectorαpossesses no half-integer components. Our methods are based in part on similar results established by the first author when the basis function is the Hardy multiquadric. Several intriguing links with the theory of shifted B-spline cardinal interpolation are described in the finale
Norm Estimates for Inverses of Toeplitz Distance Matrices
No full textAbstractA radial basis function approximation has the form [formula] where φ: [0,∞)→ R is some given function, (yj)n1 are real coefficients, and the centres (xj)n1 are points in Rd. For a wide class of functions φ, it is known that the interpolation matrix A = (φ(||xj − xk||2))nj,k=1 is invertible. Further, several recent papers have provided upper bounds on ||A−1||2, where the points (xj)n1 satisfy the condition ||xj − xk||2 ≥ δ, j ≠ k, for some positive constant δ. In this paper, we provide the least upper bound on ||A−1||2 when the points (xj)n1 form any subset of the integer lattice Ld, and when φ is a conditionally negative definite function of order 1, a large set of functions which includes the multiquadric. Specifically, for any set of points (xj)n1 ⊂ Ld, we provide the inequality [formula] where e = [1, . . . , 1]T ∈ Rd and where φ̂ is the generalized Fourier transform of φ. We provide a constructive proof that no smaller bound is valid and comment on the relevance of the method of analysis to the problem of estimating all the eigenvalues of such an interpolation matrix
Scaling radial basis functions via euclidean distance matrices
No full textAbstractA radial basis function approximation is typically a linear combination of shifts of a radially symmetric function, possibly augmented by a polynomial of suitable degree, that is, it takes the forms(x)=∑k=1nckφ(∥x−xk∥)+p(x),x∈ℝdIn the mid 1980s, Micchelli, building on pioneering work of Schoenberg in the 1930s and 1940s, provided simple sufficient conditions on ƒ that imply radial basis functions can interpolate scattered data. However, when the data density varies locally, several authors, such as Hon and Kansa [1], have suggested scaling the translates. In other words, it can be advantageous to replace the Euclidean norm by some more general distance functional Δ(·,·), ), that iss(x)=∑k=1nckφ(Δ(x,xk))+p(x),x∈ℝdThis distance functional A need not be a metric, but we shall require that Δ be symmetric and satisfy Δ (χ, χ) = 0, for all χ ∈ ℝd. Unfortunately, the Micchelli-Schoenberg theory does not obviously apply in this more general setting, but some papers have observed that interpolation is well defined if the distance functional is a sufficiently small perturbation of the Euclidean norm. However, in this study we follow a different approach which returns to the roots of Schoenberg's work. Specifically, we use Schoenberg's classification of Euclidean distance matrices to provide a simple technique which, given a suggested distance functional Δ, calculates a perturbed distance functional Δ for which the underlying interpolation matrix is invertible, when the function θ is strictly positive definite (i.e., a Mercer kernel) or strictly conditionally positive (or negative) definite of order one. As a simple by-product of this method, we can also apply the Narcowich-Ward [2] norm estimate results easily, since the minimum distance between points is now under our control via Δ
The interpolation theory of radial basis functions
No full textSIGLEAvailable from British Library Document Supply Centre- DSC:D062627 / BLDSC - British Library Document Supply CentreGBUnited Kingdo
The asymptotic cardinal function of the multiquadratic ϕ(r) = (r2 + c2)12as c→∞
No full textAbstractA radial basis function approximation has the form s(x) = ΣjϵRd yj φ(‖x−xj‖2), x ϵ Rd, where φ: [0, ∞) → R is some given function, (yj)jϵZd are real coefficients, and the centres(xj)jϵZd are points in Rd. It is known that radial basis function approximations using the multiquadric φ(r) = (r2 + c2)12 possess many useful and interesting properties when the centres form an infinite regular lattice. We analyse the limiting case as c → ∞ and identify a class of functions that arise as uniform limits of the multiquadric interpolants. In the univariate case, we observe that the cardinal function for the multiquadratic becomes the sinc function as c → ∞. The limit of the multivariate cardinal function is also identified
Low-Speed Model Support Interference - Elements of an Expert System
No full textWind tunnel support interference is one of the constraints affecting the quality of wind tunnel measurements. Several methods to determine the interference are experimental- empirical- and numerical methods. Experimental methods are often time consuming and costly. This also holds for empirical methods as they are founded on a vast number of experimental data sets. CFD is also found to be time consuming and sometimes computationally expensive. Future guidelines for the treatment of support interference aim at providing engineers more alternatives. Such alternatives require however an extensive knowledge on experiments and CFD: it requires the engineer to be an expert in the field, something that is often impossible. Engineers should therefore be guided by an expert system (a computer program that represents and reasons with knowledge of some specialist subject with a view to solving problems or giving advice) in their dealings with support interference. In this thesis such an application-based expert system is considered. The system focuses on low-speed model support interference on single sting mounted models carrying an internal balance. The research objective of this thesis is stated as: ''To identify the necessary elements for the design of an expert system for support interference on sting mounted models carrying internal balances applicable to low-speed wind tunnels.'' In this thesis the necessary components of such an expert system are identified through a study on the elements of its knowledge base and a study on a feasible structure of the system in terms of its applications. Considering the study on the elements of its knowledge base, experimental- and numerical research is carried out to gain intelligibility on support interference. It is shown that a support break down facilitating the treatment of disturbances of individual support parts spanning a certain setup is a systematic method to analyze support interference. The order of magnitude and the nature of the disturbances are not compromised when this approach is adopted provided that the amount of separate parts is kept to a minimum. This approach enables the crucial study on the disturbances of the model sting that causes the complete spectrum of support disturbances. Advantages of studying the sting include the possibility to generalize the research results to a wider class of support structures, allow for a qualitative analysis on the nature of near-field and far-field effects but also a qualitative and quantitative validation of several methods applied to determine support interference. Comparing measurements (balance measurements and 5-hole probe measurements) to calculations (panel code- and Euler calculations) on model sting near-field and far-field effects shows that without knowing the specific details of a complex interference flow field, it is not justified (from the viewpoint of accuracy) to determine model sting near-field and far-field effects using methods at low levels of complexity and intrinsic accuracy. Significant calculation offsets (out of the bounds of experimental accuracy) are caused by the action of the balance cavity and slit, vorticity and viscosity. In depth understanding of the limitations of these numerical methods (panel code, Euler) can only be developed when the interference flow field itself is understood both qualitatively and quantitatively. Navier-Stokes calculations are used for this purpose. Calculations provide a qualitative image of the interference flow field that complies with measurements. Quantitatively, the Navier-Stokes calculations are not able to determine the values of the interference with the right trends and within typical measurement (balance) accuracy. Gained near-field flow knowledge is used for an assessment of the potential of various numerical and experimental methods in determining the near-field and far-field model sting effects on wind tunnel models at low speed for various sting placements. This knowledge is generalized such as to cover the treatment of the remaining support for typical sting mounted setups. Considering the numerical and experimental treatment of support disturbances of any support part it is concluded that classification parameters ''accuracy'' and ''effort'' (classifying the various methods for the determination of the interference) oppose each other: accurate methods demand a lot of implementation effort and vice versa. This opposition can not be solved by designing a custom-made model (that is both accurate and requires a low amount of implementation effort) for calculating model support interference. Such a model should calculate the disturbance effects fast (by incorporating only the disturbance factors of primary quantitative interest) with the right trends and magnitude. Solving for the confinements of such models implies an inevitable reduction in the applicability range of the model. Typical custom-made models are unsuitable for implementation in the expert system. They reveal the following rule of thumb: ''High accuracy (at a minimum equal to typical balance Delta-measurement accuracy) and low implementation effort (total measurement effort or modeling effort and computational effort) of a correction method for determining model support interference are currently incompatible when a wide range of applicability (freestream conditions, setups) is desired''. This rule of thumb necessitates a more elaborate definition of the expert system's requirements on speed and accuracy, resulting in an expert system with an application-based structure. The applications with given accuracy and speed assist in four stages defining a typical commercial wind tunnel measurement: negotiations at the client, test preparation phase, performing the measurements and finally the post test corrections. Next a closer look is taken at a feasible structure of the expert system. The proposed application-based structure fulfills the expert system's main requirements: advise on the test setup/correction methods, calculate the interference fast enough and accurate enough pre-test and on-line, correct for the interference on-line and off-line and allow easy plug-in of modules dealing with the problem of wall interference. Additional requirements relate to the use of the system (meet computer platform standards and be user friendly with professional interfaces). Typical necessary system elements are identified: the expanded knowledge base on model support interference has resulted in two basic expert applications (ESI and ASID, directly applicable for measurements in the LLF of DNW for which they are customized) and new methods (VOLAER and MVL) to approach the problem of wind tunnel wall- and support interference. These products are seen as basic elements of an expert system (generalizable to other wind tunnels). MVL (a method combining both uncorrected wind tunnel measurements and vortex-lattice calculations) proves to be particularly valuable as it predicts the interference of wind tunnel walls, support and includes secondary interference (when e.g. the support is traversed close to the wind tunnel walls). MVL is suitable for all support setups in all types of wind tunnels provided a vortex-lattice method is used enabling an accurate representation of the model aerodynamic derivatives (preferably including the effects of viscosity). MVL's prediction capabilities necessitates the use of multiple boundary conditions (interference values) in order to guarantee a stable solution thereby categorizing it as an interpolation tool with the potential of decreasing the amount of necessary experimental balance Delta-measurements. Currently, a very basic variant of an expert system is presented in this thesis and its necessary elements are identified. This is seen as a good initiative towards meeting the future needs. It is believed that the future needs can be met when further development of this expert system is stimulated. Increasing data availability and updating the applications is of utmost importance in this matter. To the authors opinion, the data availability can be expanded to exceed the companies thresholds and to span multiple companies and countries. In this light, cooperation might very well be seen as the most important future need of all.AerodynamicsAerospace Engineerin
