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Nowadays high-precision computations are still more desired. Either for simulation on a level of atoms where every digit is important and inaccurary in computation can cause invalid result or numerical approximations in partial differential equations solving where a small deviation causes a result to be useless. The computations are carried over data types with precision of order hundred to thousand digits, or even more. This creates pressure on time complexity of problem solving and so it is essential to find very efficient methods for computation. Every complex physical problem is usually described by a system of equations frequently containing elementary functions like sinus, cosines or exponentials. The aim of the work is to design and implement methods that for a given precision, arbitrary elementary function and a point compute its value in the most efficent way. The core of the work is an application of methods based on AGM (arithmetic-geometric mean) with a time complexity of order $O(M(n)\log_2{n})$ 9(expresed for multiplication $M(n)$). The complexity can not be improved. There are many libraries supporting multi-precision atithmetic, one of which is GMP and is about to be used for efficent method implementation. In the end all implemented methods are compared with existing ones.

Simulationsmodelle sind ein etabliertes Instrument zur Analyse von Auswirkungen exogener Schocks in komplexen Systemen. Die in jüngster Zeit gestiegene verfügbare Rechenleistung und -geschwindigkeit hat die Entwicklung detaillierterer und komplexerer Simulationsmodelle befördert. Dieser Trend hat jedoch Bedenken hinsichtlich der Unsicherheit solcher Modellergebnisse aufgeworfen und daher viele Nutzer von Simulationsmodellen dazu motiviert, Unsicherheiten in ihren Simulationen zu integrieren. Eine Möglichkeit dies systematisch zu tun besteht darin, stochastische Elemente in die Modellgleichungen zu integrieren, wodurch das jeweilige Modell zu einem Problem (mehrfacher) numerischer Integrationen wird. Da es für solche Probleme meist keine analytischen Lösungen gibt, werden numerische Approximationsmethoden genutzt. Die derzeit zur Quantifizierung von Unsicherheiten in Simulationsmodellen genutzt en Techniken, sind entweder rechenaufwändig (Monte Carlo [MC] -basierte Methoden) oder liefern Ergebnisse von heterogener Qualität (Gauß-Quadraturen [GQs]). In Anbetracht der Bedeutung von effizienten Methoden zur Quantifizierung von Unsicherheit im Zeitalter von „big data“ ist es das Ziel dieser Doktorthesis, Methoden zu entwickeln, die die Näherungsfehler von GQs verringern und diese Methoden einer breiteren Forschungsgemeinschaft zugänglich machen. Zu diesem Zweck werden zwei neuartige Methoden zur Quantifizierung von Unsicherheiten entwickelt und in vier verschiedene, große partielle und allgemeine Gleichgewichtsmodelle integriert, die sich mit Agrarumweltfragen befassen. Diese Arbeit liefert methodische Entwicklungen und ist von hoher Relevanz für angewandte Simulationsmodellierer. Obwohl die Methoden in großen Simulationsmodellen für Agrarumweltfragen entwickelt und getestet werden, sind sie nicht durch Modelltyp oder Anwendungsgebiet beschränkt, sondern können ebenso in anderen Zusammenhängen angewandt werden. / Simulation models are an established tool for assessing the impacts of exogenous shocks in complex systems. Recent increases in available computational power and speed have led to simulation models with increased levels of detail and complexity. However, this trend has raised concerns regarding the uncertainty of such model results and therefore motivated many users of simulation models to consider uncertainty in their simulations. One way is to integrate stochastic elements into the model equations, thus turning the model into a problem of (multiple) numerical integration. As, in most cases, such problems do not have analytical solutions, numerical approximation methods are applied. The uncertainty quantification techniques currently used in simulation models are either computational expensive (Monte Carlo [MC]-based methods) or produce results of varying quality (Gaussian quadratures [GQs]). Considering the importance of efficient uncertainty quantification methods in the era of big data, this thesis aims to develop methods that decrease the approximation errors of GQs and make these methods accessible to the wider research community. For this purpose, two novel uncertainty quantification methods are developed and integrated into four different large-scale partial and general equilibrium models addressing agro-environmental issues. This thesis provides method developments and is of high relevance for applied simulation modelers who struggle to apply computationally burdensome stochastic modeling methods. Although the methods are developed and tested in large-scale simulation models addressing agricultural issues, they are not restricted to a model type or field of application.

<p>Examensarbetet handlar om en jämförelsestudie av beräkningseffektivitet med avseende på energi- och tidsförbrukning mellan grafikkort och processorer i persondatorer och PlayStation 3.</p><p>Problemet studeras för att göra allmänheten uppmärksam på att det går att lösa en del av energiproblematiken med beräkningar genom att öka energieffektiviteten av beräkningsenheterna.</p><p>Undersökningen har genomförts på ett explorativt sätt och studerar förhållandet mellan processorer, grafikkort och vilken som presterar bäst i vilket sammanhang. Prestandatest genomförs med molekylberäkningsprogrammet F@H och med filkomprimeringsprogrammet WinRAR. Testerna utförs på MultiCore- och SingleCorePCs och PS3s av olika karaktär. I vissa test mäts effektförbrukning för att kunna räkna ut hur energieffektiva vissa system är.</p><p>Resultatet visar tydligt hur den genomsnittliga effektförbrukningen och energieffektiviteten för olika testsystem skiljer sig vid belastning, viloläge och olika typer beräkningar.</p> / <p>The thesis is a comparative study of computational efficiency in terms of energy and time consumption of graphics cards and processors in personal computers and Playstation3’s.</p><p>The problem is studied in order to make the public aware that it is possible to solve some of the energy problems with computations by increasing energy efficiency of the computational units.</p><p>The audit was conducted in an exploratory way, studying the relationship between the processors, graphics cards and which one performs best in which context. Performance tests are carried out by the molecule calculating F@H-program and the file compression program WinRAR. Tests performed on MultiCore and SingleCore PC’s and PS3’s with different characteristics. In some tests power consumption is measured in order to figure out how energy-efficient certain systems are.</p><p>The results clearly show how the average power consumption and energy efficiency for various test systems at differ at load, sleep and various calculations.</p><p> </p>

Examensarbetet handlar om en jämförelsestudie av beräkningseffektivitet med avseende på energi- och tidsförbrukning mellan grafikkort och processorer i persondatorer och PlayStation 3. Problemet studeras för att göra allmänheten uppmärksam på att det går att lösa en del av energiproblematiken med beräkningar genom att öka energieffektiviteten av beräkningsenheterna. Undersökningen har genomförts på ett explorativt sätt och studerar förhållandet mellan processorer, grafikkort och vilken som presterar bäst i vilket sammanhang. Prestandatest genomförs med molekylberäkningsprogrammet F@H och med filkomprimeringsprogrammet WinRAR. Testerna utförs på MultiCore- och SingleCorePCs och PS3s av olika karaktär. I vissa test mäts effektförbrukning för att kunna räkna ut hur energieffektiva vissa system är. Resultatet visar tydligt hur den genomsnittliga effektförbrukningen och energieffektiviteten för olika testsystem skiljer sig vid belastning, viloläge och olika typer beräkningar. / The thesis is a comparative study of computational efficiency in terms of energy and time consumption of graphics cards and processors in personal computers and Playstation3’s. The problem is studied in order to make the public aware that it is possible to solve some of the energy problems with computations by increasing energy efficiency of the computational units. The audit was conducted in an exploratory way, studying the relationship between the processors, graphics cards and which one performs best in which context. Performance tests are carried out by the molecule calculating F@H-program and the file compression program WinRAR. Tests performed on MultiCore and SingleCore PC’s and PS3’s with different characteristics. In some tests power consumption is measured in order to figure out how energy-efficient certain systems are. The results clearly show how the average power consumption and energy efficiency for various test systems at differ at load, sleep and various calculations.

Telemetry system, Optimisation, Sensoric networks, Smart Grid, Internet of Things, Sensors, Information security, Cryptography, Cryptography algorithms, Cryptosystem, Confidentiality, Integrity, Authentication, Data freshness, Non-Repudiation.