Search results for "Diffusion"

Itô-Stratonovitch Formula for the Wave Equation on a Torus

2010

We give an Ito-Stratonovitch formula for the wave equation on a torus, where we have no stochastic process associated to this partial differential equation. This gives a generalization of the classical Ito-Stratonovitch equation for diffusion in semi-group theory established by ourself in [18], [20].

Random Walk and Diffusion

2014

The concept of random walk as introduced by Einstein is introduced. It is shown that a random walk on a lattice can be descrbed by a difference equation, which becomes a partial differential equation (diffusion equation) in the continuum limit. The equation is solved with the help of Fourier and Laplace transformations.

Gear classification and fault detection using a diffusion map framework

2013

Adaptive framework for network traffic classification using dimensionality reduction and clustering

2012

Information security has become a very important topic especially during the last years. Web services are becoming more complex and dynamic. This offers new possibilities for attackers to exploit vulnerabilities by inputting malicious queries or code. However, these attack attempts are often recorded in server logs. Analyzing these logs could be a way to detect intrusions either periodically or in real time. We propose a framework that preprocesses and analyzes these log files. HTTP queries are transformed to numerical matrices using n-gram analysis. The dimensionality of these matrices is reduced using principal component analysis and diffusion map methodology. Abnormal log lines can then …

Gear classification and fault detection using a diffusion map framework

2015

This article proposes a system health monitoring approach that detects abnormal behavior of machines. Diffusion map is used to reduce the dimensionality of training data, which facilitates the classification of newly arriving measurements. The new measurements are handled with Nyström extension. The method is trained and tested with real gear monitoring data from several windmill parks. A machine health index is proposed, showing that data recordings can be classified as working or failing using dimensionality reduction and warning levels in the low dimensional space. The proposed approach can be used with any system that produces high-dimensional measurement data. peerReviewed

A posteriori error estimates for time-dependent reaction-diffusion problems based on the Payne-Weinberger inequality

2015

We consider evolutionary reaction-diffusion problem with mixed Dirichlet--Robin boundary conditions. For this class of problems, we derive two-sided estimates of the distance between any function in the admissible energy space and exact solution of the problem. The estimates (majorants and minorants) are explicitly computable and do not contain unknown functions or constants. Moreover, it is proved that the estimates are equivalent to the energy norm of the deviation from the exact solution.

Reduced Order Models for Pricing European and American Options under Stochastic Volatility and Jump-Diffusion Models

2017

Abstract European options can be priced by solving parabolic partial(-integro) differential equations under stochastic volatility and jump-diffusion models like the Heston, Merton, and Bates models. American option prices can be obtained by solving linear complementary problems (LCPs) with the same operators. A finite difference discretization leads to a so-called full order model (FOM). Reduced order models (ROMs) are derived employing proper orthogonal decomposition (POD). The early exercise constraint of American options is enforced by a penalty on subset of grid points. The presented numerical experiments demonstrate that pricing with ROMs can be orders of magnitude faster within a give…

Reduced Order Models for Pricing American Options under Stochastic Volatility and Jump-diffusion Models

2016

American options can be priced by solving linear complementary problems (LCPs) with parabolic partial(-integro) differential operators under stochastic volatility and jump-diffusion models like Heston, Merton, and Bates models. These operators are discretized using finite difference methods leading to a so-called full order model (FOM). Here reduced order models (ROMs) are derived employing proper orthogonal decomposition (POD) and non negative matrix factorization (NNMF) in order to make pricing much faster within a given model parameter variation range. The numerical experiments demonstrate orders of magnitude faster pricing with ROMs. peerReviewed

IMEX schemes for pricing options under jump–diffusion models

2014

We propose families of IMEX time discretization schemes for the partial integro-differential equation derived for the pricing of options under a jump-diffusion process. The schemes include the families of IMEX-midpoint, IMEX-CNAB and IMEX-BDF2 schemes. Each family is defined by a convex combination parameter [email protected]?[0,1], which divides the zeroth-order term due to the jumps between the implicit and explicit parts in the time discretization. These IMEX schemes lead to tridiagonal systems, which can be solved extremely efficiently. The schemes are studied through Fourier stability analysis and numerical experiments. It is found that, under suitable assumptions and time step restric…

Online anomaly detection using dimensionality reduction techniques for HTTP log analysis

2015

Modern web services face an increasing number of new threats. Logs are collected from almost all web servers, and for this reason analyzing them is beneficial when trying to prevent intrusions. Intrusive behavior often differs from the normal web traffic. This paper proposes a framework to find abnormal behavior from these logs. We compare random projection, principal component analysis and diffusion map for anomaly detection. In addition, the framework has online capabilities. The first two methods have intuitive extensions while diffusion map uses the Nyström extension. This fast out-of-sample extension enables real-time analysis of web server traffic. The framework is demonstrated using …