The instanton method and its numerical implementation in fluid mechanics

A precise characterization of structures occurring in turbulent fluid flows at high Reynolds numbers is one of the last open problems of classical physics. In this review we discuss recent developments related to the application of instanton methods to turbulence. Instantons are saddle point configurations of the underlying path integrals. They are equivalent to minimizers of the related Freidlin-Wentzell action and known to be able to characterize rare events in such systems. While there is an impressive body of work concerning their analytical description, this review focuses on the question on how to compute these minimizers numerically. In a short introduction we present the relevant mathematical and physical background before we discuss the stochastic Burgers equation in detail. We present algorithms to compute instantons numerically by an efficient solution of the corresponding Euler-Lagrange equations. A second focus is the discussion of a recently developed numerical filtering technique that allows to extract instantons from direct numerical simulations. In the following we present modifications of the algorithms to make them efficient when applied to two- or three-dimensional fluid dynamical problems. We illustrate these ideas using the two-dimensional Burgers equation and the three-dimensional Navier-Stokes equations

Instanton filtering for the stochastic Burgers equation

We address the question whether one can identify instantons in direct numerical simulations of the stochastically driven Burgers equation. For this purpose, we first solve the instanton equations using the Chernykh-Stepanov method [Phys. Rev. E 64, 026306 (2001)]. These results are then compared to direct numerical simulations by introducing a filtering technique to extract prescribed rare events from massive data sets of realizations. Using this approach we can extract the entire time history of the instanton evolution which allows us to identify the different phases predicted by the direct method of Chernykh and Stepanov with remarkable agreement

Quickest detection in coupled systems

This work considers the problem of quickest detection of signals in a coupled system of N sensors, which receive continuous sequential observations from the environment. It is assumed that the signals, which are modeled a general Ito processes, are coupled across sensors, but that their onset times may differ from sensor to sensor. The objective is the optimal detection of the first time at which any sensor in the system receives a signal. The problem is formulated as a stochastic optimization problem in which an extended average Kullback- Leibler divergence criterion is used as a measure of detection delay, with a constraint on the mean time between false alarms. The case in which the sensors employ cumulative sum (CUSUM) strategies is considered, and it is proved that the minimum of N CUSUMs is asymptotically optimal as the mean time between false alarms increases without bound.Comment: 6 pages, 48th IEEE Conference on Decision and Control, Shanghai 2009 December 16 - 1

Quickest detection in coupled systems

This work considers the problem of quickest detection of signals in a coupled system of $N$ sensors, which receive continuous sequential observations from the environment. It is assumed that the signals, which are modeled by general It\^{o} processes, are coupled across sensors, but that their onset times may differ from sensor to sensor. Two main cases are considered; in the first one signal strengths are the same across sensors while in the second one they differ by a constant. The objective is the optimal detection of the first time at which any sensor in the system receives a signal. The problem is formulated as a stochastic optimization problem in which an extended minimal Kullback-Leibler divergence criterion is used as a measure of detection delay, with a constraint on the mean time to the first false alarm. The case in which the sensors employ cumulative sum (CUSUM) strategies is considered, and it is proved that the minimum of $N$ CUSUMs is asymptotically optimal as the mean time to the first false alarm increases without bound. In particular, in the case of equal signal strengths across sensors, it is seen that the difference in detection delay of the $N$-CUSUM stopping rule and the unknown optimal stopping scheme tends to a constant related to the number of sensors as the mean time to the first false alarm increases without bound. Alternatively, in the case of unequal signal strengths, it is seen that this difference tends to zero.Comment: 29 pages. SIAM Journal on Control and Optimization, forthcomin

Static glow

Static Glow refers to the phenomenon by which data persists in the network long after the person it relates to is dead. Such as the so-called Facebook ‘ghost profiles’; accounts owned by the deceased that continue to appear in ‘Friends’, ‘Suggestions’, and somewhat more disturbingly, ‘Birthday reminders’. Example: Her static glow still lingers after 4 years. This article discusses the historical context of static glow, as well as its current and future significance

Erfolgsfaktoren des Softwarepricing - eine Benchmarkingstudie unter deutschen Softwareherstellern

Vor einigen Monaten wurde der deutsche So ftwarehersteller SAP mit einer unangenehmen Nachricht konfrontiert. Die Deutschsprachige SAP-Anwendergruppe (DSAG) trat an die Konzernleitung heran und bemängelte das aktuelle Preissystem für SAP BusinessObjects-Produkte (BO). Die Anschaffung der Software sei wegen hoher Anfangsinvestitionen und mangelnder Kostentransparenz für viele – vor allem mittelständische – Unternehmen weniger interessant. Die tatsächlichen Preise für einzelne Unternehmen sind aufgrund komplexer Preismodelle und verschiedener Rabatte, welche zudem kombiniert werden können, nur mit viel Aufwand zu errechnen. SAP nahm diese Kritik auf und hat beim Preismodell nachgebessert. Das Preismodell wurde durch die Einführung eines BO-Sta rterpakets zu einem Festpreis deutlich übersichtlicher. Aufwendige Rechnungen entfallen somit vollständig. Für Kunden, denen das Paket nicht ausreicht, wird optional ein Upgrade auf ein umfangreiches Business-Intelligence-Paket angeboten (vgl. Schulze 2009). Das Beispiel zeigt, wie wichtig und vor allem topaktuell das Thema Preismanagement in der Softwarebranche ist. Daher haben wir uns im Rahmen einer empirischen Studie intensiv mit dem Thema Softwarepricing beschäftigt. Ziel der Studie war es, durch ein Benchmarking unter deutschen Softwareherstellern Erfolgsfaktoren für Preismanagement in der Softwarebranche zu identifizieren