2,492 research outputs found
B-spline techniques for volatility modeling
This paper is devoted to the application of B-splines to volatility modeling,
specifically the calibration of the leverage function in stochastic local
volatility models and the parameterization of an arbitrage-free implied
volatility surface calibrated to sparse option data. We use an extension of
classical B-splines obtained by including basis functions with infinite
support. We first come back to the application of shape-constrained B-splines
to the estimation of conditional expectations, not merely from a scatter plot
but also from the given marginal distributions. An application is the Monte
Carlo calibration of stochastic local volatility models by Markov projection.
Then we present a new technique for the calibration of an implied volatility
surface to sparse option data. We use a B-spline parameterization of the
Radon-Nikodym derivative of the underlying's risk-neutral probability density
with respect to a roughly calibrated base model. We show that this method
provides smooth arbitrage-free implied volatility surfaces. Finally, we sketch
a Galerkin method with B-spline finite elements to the solution of the partial
differential equation satisfied by the Radon-Nikodym derivative.Comment: 25 page
Grid sensitivity for aerodynamic optimization and flow analysis
After reviewing relevant literature, it is apparent that one aspect of aerodynamic sensitivity analysis, namely grid sensitivity, has not been investigated extensively. The grid sensitivity algorithms in most of these studies are based on structural design models. Such models, although sufficient for preliminary or conceptional design, are not acceptable for detailed design analysis. Careless grid sensitivity evaluations, would introduce gradient errors within the sensitivity module, therefore, infecting the overall optimization process. Development of an efficient and reliable grid sensitivity module with special emphasis on aerodynamic applications appear essential. The organization of this study is as follows. The physical and geometric representations of a typical model are derived in chapter 2. The grid generation algorithm and boundary grid distribution are developed in chapter 3. Chapter 4 discusses the theoretical formulation and aerodynamic sensitivity equation. The method of solution is provided in chapter 5. The results are presented and discussed in chapter 6. Finally, some concluding remarks are provided in chapter 7
Nuclear halo of a 177 MeV proton beam in water: theory, measurement and parameterization
The dose distribution of a monoenergetic pencil beam in water consists of an
electromagnetic "core", a "halo" from charged nuclear secondaries, and a much
larger "aura" from neutral secondaries. These regions overlap, but each has
distinct spatial characteristics.
We have measured the core/halo using a 177MeV test beam offset in a water
tank. The beam monitor was a fluence calibrated plane parallel ionization
chamber (IC) and the field chamber, a dose calibrated Exradin T1, so the dose
measurements are absolute (MeV/g/p). We performed depth-dose scans at ten
displacements from the beam axis ranging from 0 to 10cm. The dose spans five
orders of magnitude, and the transition from halo to aura is clearly visible.
We have performed model-dependent (MD) and model-independent (MI) fits to the
data. The MD fit separates the dose into core, elastic/inelastic nuclear,
nonelastic nuclear and aura terms, and achieves a global rms measurement/fit
ratio of 15%. The MI fit uses cubic splines and the same ratio is 9%.
We review the literature, in particular the use of Pedroni's parametrization
of the core/halo. Several papers improve on his Gaussian transverse
distribution of the halo, but all retain his T(w), the radial integral of the
depth-dose multiplying both the core and halo terms and motivating measurements
with large "Bragg peak chambers" (BPCs).
We argue that this use of T(w), which by its definition includes energy
deposition by nuclear secondaries, is incorrect. T(w) should be replaced in the
core term, and in at least part of the halo, by a purely electromagnetic mass
stopping power. BPC measurements are unnecessary, and irrelevant to
parameterizing the pencil beam.Comment: 55 pages, 4 tables, 29 figure
Manifold interpolation and model reduction
One approach to parametric and adaptive model reduction is via the
interpolation of orthogonal bases, subspaces or positive definite system
matrices. In all these cases, the sampled inputs stem from matrix sets that
feature a geometric structure and thus form so-called matrix manifolds. This
work will be featured as a chapter in the upcoming Handbook on Model Order
Reduction (P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A.
Schilders, L.M. Silveira, eds, to appear on DE GRUYTER) and reviews the
numerical treatment of the most important matrix manifolds that arise in the
context of model reduction. Moreover, the principal approaches to data
interpolation and Taylor-like extrapolation on matrix manifolds are outlined
and complemented by algorithms in pseudo-code.Comment: 37 pages, 4 figures, featured chapter of upcoming "Handbook on Model
Order Reduction
Designing Smooth Motions of Rigid Objects: Computing Curves in Lie Groups
Consider the problem of designing the path of a camera in 3D. As we may identify each camera position with a member of the Euclidean motions, SE(3), the problem may be recast mathematically as constructing interpolating curves on the (non-Euclidean) space SE(3). There exist many ways to formulate this problem, and indeed many solutions. In this thesis we shall examine solutions based on simple geometric constructions, with the goal of discovering well behaved and computable solutions. In affine spaces there exist elegant solutions to the problem of curve design, which are collectively known as the techniques of Computer Aided Geometric Design (CAGD). The approach of this thesis will be the generalization of these methods and an examination of computation on matrix Lie groups. In particular, the Lie groups SO(3) and SE(3) will be examined in some detail
Exponential Parameterized Cubic B-Spline Curves And Surfaces
The use of B-spline interpolation function for curves and surfaces has been developed
for many reasons. One reason is the higher degree of continuity and smoothness.
A general B-Spline is a polynomial curve and its shape is determined by the
control points. To interpolate data points, various works have been done by previous
researchers who studies B-Spline parameterization. In this thesis, we develop a new
way for interpolating cubic B-Spline curve by taking the first and the second derivative
at endpoints and only the first derivative at inner points. The proposed method is the
extension in the B-spline interpolation technique of using arbitrary derivatives at end
points. In developing B-spline curve interpolation method, an algorithm is presented
for interpolating data points. The algorithm computes knot values for parameterization
methods. These knot values are used in constructing a matrix of B-Spline basis
function and derivative of the basis function. Then, we solve it for control points by
using the LU decomposition method, such that the curve will pass through the given
data points. Selection of proper parametrization technique is critical for curve and
surface reconstruction process. The parametrization method used in this study is an
exponential parameterization method with a = 0:8. The main advantage of developing
B-spline curve interpolation method is that we can generate different shapes of
curves by setting different direction at all data points. As an application, we applied
the proposed method in curve reconstruction on a road map from given data points
and driving directions, and also for path planning in autonomous vehicle with given
starting and goal position
High-order adaptive methods for computing invariant manifolds of maps
The author presents efficient and accurate numerical methods for computing invariant manifolds of maps which arise in the study of dynamical systems. In order to decrease the number of points needed to compute a given curve/surface, he proposes using higher-order interpolation/approximation techniques from geometric modeling. He uses B´ezier curves/triangles, fundamental objects in curve/surface design, to create adaptive methods. The methods are based on tolerance conditions derived from properties of B´ezier curves/triangles. The author develops and tests the methods for an ordinary parametric curve; then he adapts these methods to invariant manifolds of planar maps. Next, he develops and tests the method for parametric surfaces and then he adapts this method to invariant manifolds of three-dimensional maps
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