Minimizing Communication in Linear Algebra

In 1981 Hong and Kung proved a lower bound on the amount of communication needed to perform dense, matrix-multiplication using the conventional $O(n^3)$ algorithm, where the input matrices were too large to fit in the small, fast memory. In 2004 Irony, Toledo and Tiskin gave a new proof of this result and extended it to the parallel case. In both cases the lower bound may be expressed as $\Omega$(#arithmetic operations / $\sqrt{M}$), where M is the size of the fast memory (or local memory in the parallel case). Here we generalize these results to a much wider variety of algorithms, including LU factorization, Cholesky factorization, $LDL^T$ factorization, QR factorization, algorithms for eigenvalues and singular values, i.e., essentially all direct methods of linear algebra. The proof works for dense or sparse matrices, and for sequential or parallel algorithms. In addition to lower bounds on the amount of data moved (bandwidth) we get lower bounds on the number of messages required to move it (latency). We illustrate how to extend our lower bound technique to compositions of linear algebra operations (like computing powers of a matrix), to decide whether it is enough to call a sequence of simpler optimal algorithms (like matrix multiplication) to minimize communication, or if we can do better. We give examples of both. We also show how to extend our lower bounds to certain graph theoretic problems. We point out recently designed algorithms for dense LU, Cholesky, QR, eigenvalue and the SVD problems that attain these lower bounds; implementations of LU and QR show large speedups over conventional linear algebra algorithms in standard libraries like LAPACK and ScaLAPACK. Many open problems remain.Comment: 27 pages, 2 table

Investigating Single Precision Floating General Matrix Multiply in Heterogeneous Hardware

The fundamental operation of matrix multiplication is ubiquitous across a myriad of disciplines. Yet, the identification of new optimizations for matrix multiplication remains relevant for emerging hardware architectures and heterogeneous systems. Frameworks such as OpenCL enable computation orchestration on existing systems, and its availability using the Intel High Level Synthesis compiler allows users to architect new designs for reconfigurable hardware using C/C++. Using the HARPv2 as a vehicle for exploration, we investigate the utility of several of the most notable matrix multiplication optimizations to better understand the performance portability of OpenCL and the implications for such optimizations on this and future heterogeneous architectures. Our results give targeted insights into the applicability of best practices that were for existing architectures when used on emerging heterogeneous systems

Architectural explorations for streaming accelerators with customized memory layouts

El concepto básico de la arquitectura mono-nucleo en los procesadores de propósito general se ajusta bien a un modelo de programación secuencial. La integración de multiples núcleos en un solo chip ha permitido a los procesadores correr partes del programa en paralelo. Sin embargo, la explotación del enorme paralelismo disponible en muchas aplicaciones de alto rendimiento y de los datos correspondientes es difícil de conseguir usando unicamente multicores de propósito general. La aparición de aceleradores tipo streaming y de los correspondientes modelos de programación han mejorado esta situación proporcionando arquitecturas orientadas al proceso de flujos de datos. La idea básica detrás del diseño de estas arquitecturas responde a la necesidad de procesar conjuntos enormes de datos. Estos dispositivos de alto rendimiento orientados a flujos permiten el procesamiento rapido de datos mediante el uso eficiente de computación paralela y comunicación entre procesos. Los aceleradores streaming orientados a flujos, igual que en otros procesadores, consisten en diversos componentes micro-arquitectonicos como por ejemplo las estructuras de memoria, las unidades de computo, las unidades de control, los canales de Entrada/Salida y controles de Entrada/Salida, etc. Sin embargo, los requisitos del flujo de datos agregan algunas características especiales e imponen otras restricciones que afectan al rendimiento. Estos dispositivos, por lo general, ofrecen un gran número de recursos computacionales, pero obligan a reorganizar los conjuntos de datos en paralelo, maximizando la independiencia para alimentar los recursos de computación en forma de flujos. La disposición de datos en conjuntos independientes de flujos paralelos no es una tarea sencilla. Es posible que se tenga que cambiar la estructura de un algoritmo en su conjunto o, incluso, puede requerir la reescritura del algoritmo desde cero. Sin embargo, todos estos esfuerzos para la reordenación de los patrones de las aplicaciones de acceso a datos puede que no sean muy útiles para lograr un rendimiento óptimo. Esto es debido a las posibles limitaciones microarquitectonicas de la plataforma de destino para los mecanismos hardware de prefetch, el tamaño y la granularidad del almacenamiento local, y la flexibilidad para disponer de forma serial los datos en el interior del almacenamiento local. Las limitaciones de una plataforma de streaming de proposito general para el prefetching de datos, almacenamiento y demas procedimientos para organizar y mantener los datos en forma de flujos paralelos e independientes podría ser eliminado empleando técnicas a nivel micro-arquitectonico. Esto incluye el uso de memorias personalizadas especificamente para las aplicaciones en el front-end de una arquitectura streaming. El objetivo de esta tesis es presentar exploraciones arquitectónicas de los aceleradores streaming con diseños de memoria personalizados. En general, la tesis cubre tres aspectos principales de tales aceleradores. Estos aspectos se pueden clasificar como: i) Diseño de aceleradores de aplicaciones específicas con diseños de memoria personalizados, ii) diseño de aceleradores con memorias personalizadas basados en plantillas, y iii) exploraciones del espacio de diseño para dispositivos orientados a flujos con las memorias estándar y personalizadas. Esta tesis concluye con la propuesta conceptual de una Blacksmith Streaming Architecture (BSArc). El modelo de computación Blacksmith permite la adopción a nivel de hardware de un front-end de aplicación específico utilizando una GPU como back-end. Esto permite maximizar la explotación de la localidad de datos y el paralelismo a nivel de datos de una aplicación mientras que proporciona un flujo mayor de datos al back-end. Consideramos que el diseño de estos procesadores con memorias especializadas debe ser proporcionado por expertos del dominio de aplicación en la forma de plantillas.The basic concept behind the architecture of a general purpose CPU core conforms well to a serial programming model. The integration of more cores on a single chip helped CPUs in running parts of a program in parallel. However, the utilization of huge parallelism available from many high performance applications and the corresponding data is hard to achieve from these general purpose multi-cores. Streaming accelerators and the corresponding programing models improve upon this situation by providing throughput oriented architectures. The basic idea behind the design of these architectures matches the everyday increasing requirements of processing huge data sets. These high-performance throughput oriented devices help in high performance processing of data by using efficient parallel computations and streaming based communications. The throughput oriented streaming accelerators ¿ similar to the other processors ¿ consist of numerous types of micro-architectural components including the memory structures, compute units, control units, I/O channels and I/O controls etc. However, the throughput requirements add some special features and impose other restrictions for the performance purposes. These devices, normally, offer a large number of compute resources but restrict the applications to arrange parallel and maximally independent data sets to feed the compute resources in the form of streams. The arrangement of data into independent sets of parallel streams is not an easy and simple task. It may need to change the structure of an algorithm as a whole or even it can require to write a new algorithm from scratch for the target application. However, all these efforts for the re-arrangement of application data access patterns may still not be very helpful to achieve the optimal performance. This is because of the possible micro-architectural constraints of the target platform for the hardware pre-fetching mechanisms, the size and the granularity of the local storage and the flexibility in data marshaling inside the local storage. The constraints of a general purpose streaming platform on the data pre-fetching, storing and maneuvering to arrange and maintain it in the form of parallel and independent streams could be removed by employing micro-architectural level design approaches. This includes the usage of application specific customized memories in the front-end of a streaming architecture. The focus of this thesis is to present architectural explorations for the streaming accelerators using customized memory layouts. In general the thesis covers three main aspects of such streaming accelerators in this research. These aspects can be categorized as : i) Design of Application Specific Accelerators with Customized Memory Layout ii) Template Based Design Support for Customized Memory Accelerators and iii) Design Space Explorations for Throughput Oriented Devices with Standard and Customized Memories. This thesis concludes with a conceptual proposal on a Blacksmith Streaming Architecture (BSArc). The Blacksmith Computing allow the hardware-level adoption of an application specific front-end with a GPU like streaming back-end. This gives an opportunity to exploit maximum possible data locality and the data level parallelism from an application while providing a throughput natured powerful back-end. We consider that the design of these specialized memory layouts for the front-end of the device are provided by the application domain experts in the form of templates. These templates are adjustable according to a device and the problem size at the device's configuration time. The physical availability of such an architecture may still take time. However, simulation framework helps in architectural explorations to give insight into the proposal and predicts potential performance benefits for such an architecture

MaxEVA: Maximizing the Efficiency of Matrix Multiplication on Versal AI Engine

The increasing computational and memory requirements of Deep Learning (DL) workloads has led to outstanding innovations in hardware architectures. An archetype of such architectures is the novel Versal AI Engine (AIE) by AMD/Xilinx. The AIE comprises multiple programmable processors optimized for vector-based algorithms. An AIE array consisting of 400 processor cores, operating at 1.25 GHz is able to deliver a peak throughput of 8 TFLOPs for 32-bit floating-point (fp32), and 128 TOPs for 8-bit integer (int8) precision. In this work, we propose MaxEVA: a novel framework to efficiently map Matrix Multiplication (MatMul) workloads on Versal AIE devices. Our framework maximizes the performance and energy efficiency of MatMul applications by efficiently exploiting features of the AIE architecture and resolving performance bottlenecks from multiple angles. When demonstrating on the VC1902 device of the VCK190 board, MaxEVA accomplishes up to 5.44 TFLOPs and 77.01 TOPs throughput for fp32 and int8 precisions, respectively. In terms of energy efficiency, MaxEVA attains up to 124.16 GFLOPs/W for fp32, and 1.16 TOPs/W for int8. Our proposed method substantially outperforms the state-of-the-art approach by exhibiting up to 2.19x throughput gain and 20.4% higher energy efficiency. The MaxEVA framework provides notable insights to fill the knowledge gap in effectively designing MatMul-based DL workloads on the new Versal AIE devices.Comment: Accepted as full paper at FPT 202

Architectural explorations for streaming accelerators with customized memory layouts

El concepto básico de la arquitectura mono-nucleo en los procesadores de propósito general se ajusta bien a un modelo de programación secuencial. La integración de multiples núcleos en un solo chip ha permitido a los procesadores correr partes del programa en paralelo. Sin embargo, la explotación del enorme paralelismo disponible en muchas aplicaciones de alto rendimiento y de los datos correspondientes es difícil de conseguir usando unicamente multicores de propósito general. La aparición de aceleradores tipo streaming y de los correspondientes modelos de programación han mejorado esta situación proporcionando arquitecturas orientadas al proceso de flujos de datos. La idea básica detrás del diseño de estas arquitecturas responde a la necesidad de procesar conjuntos enormes de datos. Estos dispositivos de alto rendimiento orientados a flujos permiten el procesamiento rapido de datos mediante el uso eficiente de computación paralela y comunicación entre procesos. Los aceleradores streaming orientados a flujos, igual que en otros procesadores, consisten en diversos componentes micro-arquitectonicos como por ejemplo las estructuras de memoria, las unidades de computo, las unidades de control, los canales de Entrada/Salida y controles de Entrada/Salida, etc. Sin embargo, los requisitos del flujo de datos agregan algunas características especiales e imponen otras restricciones que afectan al rendimiento. Estos dispositivos, por lo general, ofrecen un gran número de recursos computacionales, pero obligan a reorganizar los conjuntos de datos en paralelo, maximizando la independiencia para alimentar los recursos de computación en forma de flujos. La disposición de datos en conjuntos independientes de flujos paralelos no es una tarea sencilla. Es posible que se tenga que cambiar la estructura de un algoritmo en su conjunto o, incluso, puede requerir la reescritura del algoritmo desde cero. Sin embargo, todos estos esfuerzos para la reordenación de los patrones de las aplicaciones de acceso a datos puede que no sean muy útiles para lograr un rendimiento óptimo. Esto es debido a las posibles limitaciones microarquitectonicas de la plataforma de destino para los mecanismos hardware de prefetch, el tamaño y la granularidad del almacenamiento local, y la flexibilidad para disponer de forma serial los datos en el interior del almacenamiento local. Las limitaciones de una plataforma de streaming de proposito general para el prefetching de datos, almacenamiento y demas procedimientos para organizar y mantener los datos en forma de flujos paralelos e independientes podría ser eliminado empleando técnicas a nivel micro-arquitectonico. Esto incluye el uso de memorias personalizadas especificamente para las aplicaciones en el front-end de una arquitectura streaming. El objetivo de esta tesis es presentar exploraciones arquitectónicas de los aceleradores streaming con diseños de memoria personalizados. En general, la tesis cubre tres aspectos principales de tales aceleradores. Estos aspectos se pueden clasificar como: i) Diseño de aceleradores de aplicaciones específicas con diseños de memoria personalizados, ii) diseño de aceleradores con memorias personalizadas basados en plantillas, y iii) exploraciones del espacio de diseño para dispositivos orientados a flujos con las memorias estándar y personalizadas. Esta tesis concluye con la propuesta conceptual de una Blacksmith Streaming Architecture (BSArc). El modelo de computación Blacksmith permite la adopción a nivel de hardware de un front-end de aplicación específico utilizando una GPU como back-end. Esto permite maximizar la explotación de la localidad de datos y el paralelismo a nivel de datos de una aplicación mientras que proporciona un flujo mayor de datos al back-end. Consideramos que el diseño de estos procesadores con memorias especializadas debe ser proporcionado por expertos del dominio de aplicación en la forma de plantillas.The basic concept behind the architecture of a general purpose CPU core conforms well to a serial programming model. The integration of more cores on a single chip helped CPUs in running parts of a program in parallel. However, the utilization of huge parallelism available from many high performance applications and the corresponding data is hard to achieve from these general purpose multi-cores. Streaming accelerators and the corresponding programing models improve upon this situation by providing throughput oriented architectures. The basic idea behind the design of these architectures matches the everyday increasing requirements of processing huge data sets. These high-performance throughput oriented devices help in high performance processing of data by using efficient parallel computations and streaming based communications. The throughput oriented streaming accelerators ¿ similar to the other processors ¿ consist of numerous types of micro-architectural components including the memory structures, compute units, control units, I/O channels and I/O controls etc. However, the throughput requirements add some special features and impose other restrictions for the performance purposes. These devices, normally, offer a large number of compute resources but restrict the applications to arrange parallel and maximally independent data sets to feed the compute resources in the form of streams. The arrangement of data into independent sets of parallel streams is not an easy and simple task. It may need to change the structure of an algorithm as a whole or even it can require to write a new algorithm from scratch for the target application. However, all these efforts for the re-arrangement of application data access patterns may still not be very helpful to achieve the optimal performance. This is because of the possible micro-architectural constraints of the target platform for the hardware pre-fetching mechanisms, the size and the granularity of the local storage and the flexibility in data marshaling inside the local storage. The constraints of a general purpose streaming platform on the data pre-fetching, storing and maneuvering to arrange and maintain it in the form of parallel and independent streams could be removed by employing micro-architectural level design approaches. This includes the usage of application specific customized memories in the front-end of a streaming architecture. The focus of this thesis is to present architectural explorations for the streaming accelerators using customized memory layouts. In general the thesis covers three main aspects of such streaming accelerators in this research. These aspects can be categorized as : i) Design of Application Specific Accelerators with Customized Memory Layout ii) Template Based Design Support for Customized Memory Accelerators and iii) Design Space Explorations for Throughput Oriented Devices with Standard and Customized Memories. This thesis concludes with a conceptual proposal on a Blacksmith Streaming Architecture (BSArc). The Blacksmith Computing allow the hardware-level adoption of an application specific front-end with a GPU like streaming back-end. This gives an opportunity to exploit maximum possible data locality and the data level parallelism from an application while providing a throughput natured powerful back-end. We consider that the design of these specialized memory layouts for the front-end of the device are provided by the application domain experts in the form of templates. These templates are adjustable according to a device and the problem size at the device's configuration time. The physical availability of such an architecture may still take time. However, simulation framework helps in architectural explorations to give insight into the proposal and predicts potential performance benefits for such an architecture.Postprint (published version

The Münster Skeleton Library ‚Muesli‘:A Comprehensive Overview

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Achieving High Speed CFD simulations: Optimization, Parallelization, and FPGA Acceleration for the unstructured DLR TAU Code

Today, large scale parallel simulations are fundamental tools to handle complex problems. The number of processors in current computation platforms has been recently increased and therefore it is necessary to optimize the application performance and to enhance the scalability of massively-parallel systems. In addition, new heterogeneous architectures, combining conventional processors with specific hardware, like FPGAs, to accelerate the most time consuming functions are considered as a strong alternative to boost the performance. In this paper, the performance of the DLR TAU code is analyzed and optimized. The improvement of the code efficiency is addressed through three key activities: Optimization, parallelization and hardware acceleration. At first, a profiling analysis of the most time-consuming processes of the Reynolds Averaged Navier Stokes flow solver on a three-dimensional unstructured mesh is performed. Then, a study of the code scalability with new partitioning algorithms are tested to show the most suitable partitioning algorithms for the selected applications. Finally, a feasibility study on the application of FPGAs and GPUs for the hardware acceleration of CFD simulations is presented

Accelerated volumetric reconstruction from uncalibrated camera views

While both work with images, computer graphics and computer vision are inverse problems. Computer graphics starts traditionally with input geometric models and produces image sequences. Computer vision starts with input image sequences and produces geometric models. In the last few years, there has been a convergence of research to bridge the gap between the two fields. This convergence has produced a new field called Image-based Rendering and Modeling (IBMR). IBMR represents the effort of using the geometric information recovered from real images to generate new images with the hope that the synthesized ones appear photorealistic, as well as reducing the time spent on model creation. In this dissertation, the capturing, geometric and photometric aspects of an IBMR system are studied. A versatile framework was developed that enables the reconstruction of scenes from images acquired with a handheld digital camera. The proposed system targets applications in areas such as Computer Gaming and Virtual Reality, from a lowcost perspective. In the spirit of IBMR, the human operator is allowed to provide the high-level information, while underlying algorithms are used to perform low-level computational work. Conforming to the latest architecture trends, we propose a streaming voxel carving method, allowing a fast GPU-based processing on commodity hardware