886,941 research outputs found
Spectral collocation methods
This review covers the theory and application of spectral collocation methods. Section 1 describes the fundamentals, and summarizes results pertaining to spectral approximations of functions. Some stability and convergence results are presented for simple elliptic, parabolic, and hyperbolic equations. Applications of these methods to fluid dynamics problems are discussed in Section 2
Spectral methods for volatility derivatives
In the first quarter of 2006 Chicago Board Options Exchange (CBOE)
introduced, as one of the listed products, options on its implied volatility
index (VIX). This created the challenge of developing a pricing framework that
can simultaneously handle European options, forward-starts, options on the
realized variance and options on the VIX. In this paper we propose a new
approach to this problem using spectral methods. We use a regime switching
model with jumps and local volatility defined in \cite{FXrev} and calibrate it
to the European options on the S&P 500 for a broad range of strikes and
maturities. The main idea of this paper is to "lift" (i.e. extend) the
generator of the underlying process to keep track of the relevant path
information, namely the realized variance. The lifted generator is too large a
matrix to be diagonalized numerically. We overcome this difficulty by applying
a new semi-analytic algorithm for block-diagonalization. This method enables us
to evaluate numerically the joint distribution between the underlying stock
price and the realized variance, which in turn gives us a way of pricing
consistently European options, general accrued variance payoffs and
forward-starting and VIX options.Comment: to appear in Quantitative Financ
Spectral Methods from Tensor Networks
A tensor network is a diagram that specifies a way to "multiply" a collection
of tensors together to produce another tensor (or matrix). Many existing
algorithms for tensor problems (such as tensor decomposition and tensor PCA),
although they are not presented this way, can be viewed as spectral methods on
matrices built from simple tensor networks. In this work we leverage the full
power of this abstraction to design new algorithms for certain continuous
tensor decomposition problems.
An important and challenging family of tensor problems comes from orbit
recovery, a class of inference problems involving group actions (inspired by
applications such as cryo-electron microscopy). Orbit recovery problems over
finite groups can often be solved via standard tensor methods. However, for
infinite groups, no general algorithms are known. We give a new spectral
algorithm based on tensor networks for one such problem: continuous
multi-reference alignment over the infinite group SO(2). Our algorithm extends
to the more general heterogeneous case.Comment: 30 pages, 8 figure
Spectral methods for CFD
One of the objectives of these notes is to provide a basic introduction to spectral methods with a particular emphasis on applications to computational fluid dynamics. Another objective is to summarize some of the most important developments in spectral methods in the last two years. The fundamentals of spectral methods for simple problems will be covered in depth, and the essential elements of several fluid dynamical applications will be sketched
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