1,163 research outputs found
Smoothing the payoff for efficient computation of Basket option prices
We consider the problem of pricing basket options in a multivariate Black
Scholes or Variance Gamma model. From a numerical point of view, pricing such
options corresponds to moderate and high dimensional numerical integration
problems with non-smooth integrands. Due to this lack of regularity, higher
order numerical integration techniques may not be directly available, requiring
the use of methods like Monte Carlo specifically designed to work for
non-regular problems. We propose to use the inherent smoothing property of the
density of the underlying in the above models to mollify the payoff function by
means of an exact conditional expectation. The resulting conditional
expectation is unbiased and yields a smooth integrand, which is amenable to the
efficient use of adaptive sparse grid cubature. Numerical examples indicate
that the high-order method may perform orders of magnitude faster compared to
Monte Carlo or Quasi Monte Carlo in dimensions up to 35
Brain tumor visualization for magnetic resonance images using modified shape-based interpolation method
3D visualization plays an essential role in medical diagnosis and setting treatment plans especially for brain cancer. There have been many attempts for brain tumor reconstruction and visualization using various techniques. However, this problem is still considered unsolved as more accurate results are needed in this critical field. In this paper, a sequence of 2D slices of brain magnetic resonance Images was used to reconstruct a 3D model for the brain tumor. The images were automatically segmented using a wavelet multi-resolution expectation maximization algorithm. Then, the inter-slice gaps were interpolated using the proposed modified shape-based interpolation method. The method involves three main steps; transferring the binary tumor images to distance images using a suitable distance function, interpolating the distance images using cubic spline interpolation and thresholding the interpolated values to get the reconstructed slices. The final tumor is then visualized as a 3D isosurface. We evaluated the proposed method by removing an original slice from the input images and interpolating it, the results outperform the original shape-based interpolation method by an average of 3% reaching 99% of accuracy for some slice images
Scattered Data Interpolation on Embedded Submanifolds with Restricted Positive Definite Kernels: Sobolev Error Estimates
In this paper we investigate the approximation properties of kernel
interpolants on manifolds. The kernels we consider will be obtained by the
restriction of positive definite kernels on , such as radial basis
functions (RBFs), to a smooth, compact embedded submanifold \M\subset \R^d.
For restricted kernels having finite smoothness, we provide a complete
characterization of the native space on \M. After this and some preliminary
setup, we present Sobolev-type error estimates for the interpolation problem.
Numerical results verifying the theory are also presented for a one-dimensional
curve embedded in and a two-dimensional torus
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
Airborne LiDAR for DEM generation: some critical issues
Airborne LiDAR is one of the most effective and reliable means of terrain data collection. Using LiDAR data for DEM generation is becoming a standard practice in spatial related areas. However, the effective processing of the raw LiDAR data and the generation of an efficient and high-quality DEM remain big challenges. This paper reviews the recent advances of airborne LiDAR systems and the use of
LiDAR data for DEM generation, with special focus on LiDAR data filters, interpolation methods, DEM resolution, and LiDAR data reduction. Separating LiDAR points into ground and non-ground is the most critical and difficult step for
DEM generation from LiDAR data. Commonly used and most recently developed LiDAR filtering methods are presented. Interpolation methods and choices of suitable interpolator and DEM resolution for LiDAR DEM generation are discussed in detail. In order to reduce the data redundancy and increase the efficiency in terms of storage
and manipulation, LiDAR data reduction is required in the process of DEM generation. Feature specific elements such as breaklines contribute significantly to DEM quality. Therefore, data reduction should be conducted in such a way that critical elements are kept while less important elements are removed. Given the highdensity
characteristic of LiDAR data, breaklines can be directly extracted from LiDAR data. Extraction of breaklines and integration of the breaklines into DEM generation are presented
Reduced-order modelling for high-speed aerial weapon aerodynamics
In this work a high-fidelity low-cost surrogate of a computational fluid dynamics
analysis tool was developed. This computational tool is composed of general and physics-
based approximation methods by which three dimensional high-speed aerodynamic flow-
field predictions are made with high efficiency and an accuracy which is comparable with
that of CFD. The tool makes use of reduced-basis methods that are suitable for both
linear and non-linear problems, whereby the basis vectors are computed via the proper
orthogonal decomposition (POD) of a training dataset or a set of observations.
The surrogate model was applied to two flow problems related to high-speed weapon
aerodynamics. Comparisons of surrogate model predictions with high-fidelity CFD
simulations suggest that POD-based reduced-order modelling together with response
surface methods provide a reliable and robust approach for efficient and accurate
predictions. In contrast to the many modelling efforts reported in the literature, this
surrogate model provides access to information about the whole flow-field.
In an attempt to reduce the up-front cost necessary to generate the training dataset
from which the surrogate model is subsequently developed, a variable-fidelity POD-
based reduced-order modelling method is proposed in this work for the first time. In this
model, the scalar coefficients which are obtained by projecting the solution vectors onto
the basis vectors, are mapped between spaces of low and high fidelities, to achieve high-
fidelity predictions with complete flow-field information. In general, this technique offers an automatic way of fusing variable-fidelity data through interpolation and extrapolation
schemes together with reduced-order modelling (ROM).
Furthermore, a study was undertaken to investigate the possibility of modelling the
transonic flow over an aerofoil using a kernel PODâbased reduced-order modelling
method. By using this type of ROM it was noticed that the weak non-linear features of
the transonic flow are accurately modelled using a small number of basis vectors. The
strong non-linear features are only modelled accurately by using a large number of basis
vectors
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