Basic Linear Algebra Subprograms

Basic Linear Algebra Subprograms (BLAS) est un ensemble de fonctions standardisées (interface de programmation) réalisant des opérations de base de l'algèbre linéaire telles que des additions de vecteurs, des produits scalaires ou des multiplications de matrices. Ces fonctions ont d'abord été publiées en 1979 et sont utilisées dans des bibliothèques plus développées comme LAPACK. Largement utilisées pour le calcul haute performance, ces fonctions ont été développées de manière très optimisée par des constructeurs de calculateurs comme Intel et AMD, ou encore par d'autres auteurs ( BLAS et - une version portable de BLAS - en sont des exemples). Les tests de performance LINPACK utilisent massivement la fonction multiplication de matrices générales (DGEMM) de BLAS. Nécessité de BLAS Jusqu'au début des années 1970, les codes de calcul rapide utilisaient des routines optimisées codées directement en langage machine pour l'évaluation de certaines fonctions. Entre 1972 et 1978, o

Efficient Online Processing for Advanced Analytics

Mohamed Elsayed Mohamed Ahmed El Seidy

With the advent of emerging technologies and the Internet of Things, the importance of online data analytics has become more pronounced. Businesses and companies are adopting approaches that provide responsive analytics to stay competitive in the global marketplace. Online analytics allow data analysts to promptly react to patterns or to gain preliminary insights from early results that aid in research, decision making, and effective strategy planning. The growth of data-velocity in a variety of domains including, high-frequency trading, social networks, infrastructure monitoring, and advertising require adopting online engines that can efficiently process continuous streams of data. This thesis presents foundations, techniques, and systems' design that extend the state-of-the-art in online query processing to efficiently support relational joins with arbitrary join-predicates (beyond traditional equi-joins); and to support other data models (beyond relational) that target machine learning and graph computations. The thesis is divided into two parts: We first present a brief overview of Squall, our open-source online query processing engine that supports SQL-like queries on top of streams. Then, we focus on extending Squall to support efficient theta-join processing. Scalable distributed join processing requires a partitioning policy that evenly distributes the processing load while minimizing the size of maintained state and duplicated messages. Efficient load-balance demands apriori-statistics which are not available in the online setting. We propose a novel operator that continuously adjusts itself to the data dynamics, through adaptive dataflow routing and state repartitioning. It is also resilient to data-skew, maintains high throughput rates, avoids blocking during state repartitioning, and behaves as a black-box dataflow operator with provable performance guarantees. Our evaluation demonstrates that the proposed operator outperforms the state-of-the-art static partitioning schemes in resource utilization, throughput, and execution time up to 7x. In the second part, we present a novel framework that supports the Incremental View Maintenance (IVM) of workloads expressed as linear algebra programs. Linear algebra represents a concrete substrate for advanced analytical tasks including, machine learning, scientific computation, and graph algorithms. Previous works on relational calculus IVM are not applicable to matrix algebra workloads. This is because a single entry change to an input-matrix results in changes all over the intermediate views, rendering IVM useless in comparison to re-evaluation. We present Lago, a unified modular compiler framework that supports the IVM of a broad class of linear algebra programs. Lago automatically derives and optimizes incremental trigger programs of analytical computations, while freeing the user from erroneous manual derivations, low-level implementation details, and performance tuning. We present a novel technique that captures

\Delta

changes as low-rank matrices. Low-rank matrices are representable in a compressed factored form that enables cheaper computations. Lago automatically propagates the factored representation across program statements to derive an efficient trigger program. Moreover, Lago extends its support to other domains that use different semi-ring configurations, e.g., graph applications. Our evaluation results demonstrate orders of magnitude (10x-10

EPFL2017

Numerical methods for option pricing: polynomial approximation and high dimensionality

Francesco Statti

Options are some of the most traded financial instruments and computing their price is a central task in financial mathematics and in practice. Consequently, the development of numerical algorithms for pricing options is an active field of research. In general, evaluating the price of a specific option relies on the properties of the stochastic model used for the underlying asset price. In this thesis we develop efficient and accurate numerical methods for option pricing in a specific class of models: polynomial models. They are a versatile tool for financial modeling and have useful properties that can be exploited for option pricing. Significant challenges arise when developing option pricing techniques. For instance, the underlying model might have a high-dimensional parameter space. Furthermore, treating multi-asset options yields high-dimensional pricing problems. Therefore, the pricing method should be able to handle high dimensionality. Another important aspect is the efficiency of the algorithm: in real-world applications, option prices need to be delivered within short periods of time, making the algorithmic complexity a potential bottleneck. In this thesis, we address these challenges by developing option pricing techniques that are able to handle low and high-dimensional problems, and we propose complexity reduction techniques. The thesis consists of four parts: First, we present a methodology for European and American option pricing. The method uses the moments of the underlying price process to produce monotone sequences of lower and upper bounds of the option price. The bounds are obtained by solving a sequence of polynomial optimization problems. As the order of the moments increases, the bounds become sharper and eventually converge to the exact price under appropriate assumptions. Second, we develop a fast algorithm for the incremental computation of nested block triangular matrix exponentials. This algorithm allows for an efficient incremental computation of the moment sequence of polynomial jump-diffusions. In other words, moments of order 0, 1, 2, 3... are computed sequentially until a dynamically evaluated criterion tells us to stop. The algorithm is based on the scaling and squaring technique and reduces the complexity of the pricing algorithms that require such an incremental moment computation. Third, we develop a complexity reduction technique for high-dimensional option pricing. To this end, we first consider the option price as a function of model and payoff parameters. Then, the tensorized Chebyshev interpolation is used on the parameter space to increase the efficiency in computing option prices, while maintaining the required accuracy. The high dimensionality of the problem is treated by expressing the tensorized interpolation in the tensor train format and by deriving an efficient way, which is based on tensor completion, to approximate the interpolation coefficients. Lastly, we propose a methodology for pricing single and multi-asset European options. The approach is a combination of Monte Carlo simulation and function approximation. We address the memory limitations that arise when treating very high-dimensional applications by combining the method with optimal sampling strategies and using a randomized algorithm to reduce the storage complexity of the approach. The obtained numerical results show the effectiveness of the algorithms developed in this thesis.

EPFL2019

Efficient Incremental Data Analysis

Milos Nikolic

Many data-intensive applications require real-time analytics over streaming data. In a growing number of domains -- sensor network monitoring, social web applications, clickstream analysis, high-frequency algorithmic trading, and fraud detections to name a few -- applications continuously monitor stream events to promptly react to certain data conditions. These applications demand responsive analytics even when faced with high volume and velocity of incoming changes, large numbers of users, and complex processing requirements. Developing suitable online analytics engine that meets these requirements is challenging. In this thesis, we study techniques for efficient online processing of complex analytical queries, ranging from standard database queries to complex machine learning and digital signal processing workflows. First, we focus on the problem of efficient incremental computation for database queries. We have developed a system, called DBToaster, that compiles declarative queries into high-performance stream processing engines that keep query results (views) fresh at very high update rates. At the heart of our system is a recursive query compilation algorithm that materializes a set of supporting higher-order delta views to achieve a substantially lower view maintenance cost. We study the trade-offs between single-tuple and batch incremental processing in local execution, and we present a novel approach for compiling view maintenance code into data-parallel programs optimized for distributed execution. DBToaster supports millions of complete view refreshes per second for a broad range of queries and outperforms commercial database and stream engines by orders of magnitude. We also study the incremental computation for queries written as iterative linear algebra, which can capture many machine learning and scientific calculations. We have developed a framework, called LINVIEW, for capturing deltas of linear algebra programs and understanding their computational cost. Linear algebra operations tend to cause an avalanche effect where even very local changes to the input matrices spread out and infect all of the intermediate results and the final view, causing incremental view maintenance to lose its performance benefit over re-evaluation. We develop techniques based on matrix factorizations to contain such epidemics of change and make incremental view maintenance of linear algebra practical and usually substantially cheaper than re-evaluation. We show, both analytically and experimentally, the usefulness of these techniques when applied to standard analytics tasks. Our last research question concerns the integration of general-purpose query processors and domain-specific operations to enable deep data exploration in both online and offline analysis. We advocate a deep integration of signal processing operations and general-purpose query processors. We demonstrate that in-situ processing of tempo-relational and signal data through a unified query language empowers users to express end-to-end workflows more succinctly inside one system while at the same time offering orders of magnitude better performance than existing popular data management systems.

EPFL2016