Temporal blocking of finite-difference stencil operators with sparse "off-the-grid" sources

Stencil kernels dominate a range of scientific applications, including seismic and medical imaging, image processing, and neural networks. Temporal blocking is a performance optimization that aims to reduce the required memory bandwidth of stencil computations by re-using data from the cache for multiple time steps. It has already been shown to be beneficial for this class of algorithms. However, applying temporal blocking to practical applications' stencils remains challenging. These computations often consist of sparsely located operators not aligned with the computational grid (“off-the-grid”). Our work is motivated by modelling problems in which source injections result in wavefields that must then be measured at receivers by interpolation from the grided wavefield. The resulting data dependencies make the adoption of temporal blocking much more challenging. We propose a methodology to inspect these data dependencies and reorder the computation, leading to performance gains in stencil codes where temporal blocking has not been applicable. We implement this novel scheme in the Devito domain-specific compiler toolchain. Devito implements a domain-specific language embedded in Python to generate optimized partial differential equation solvers using the finite-difference method from high-level symbolic problem definitions. We evaluate our scheme using isotropic acoustic, anisotropic acoustic, and isotropic elastic wave propagators of industrial significance. After auto-tuning, performance evaluation shows that this enables substantial performance improvement through temporal blocking over highly-optimized vectorized spatially-blocked code of up to 1.6x

Automatic Data and Computation Mapping for Distributed-Memory Machines.

Distributed memory parallel computers offer enormous computation power, scalability and flexibility. However, these machines are difficult to program and this limits their widespread use. An important characteristic of these machines is the difference in the access time for data in local versus non-local memory; non-local memory accesses are much slower than local memory accesses. This is also a characteristic of shared memory machines but to a less degree. Therefore it is essential that as far as possible, the data that needs to be accessed by a processor during the execution of the computation assigned to it reside in its local memory rather than in some other processor\u27s memory. Several research projects have concluded that proper mapping of data is key to realizing the performance potential of distributed memory machines. Current language design efforts such as Fortran D and High Performance Fortran (HPF) are based on this. It is our thesis that for many practical codes, it is possible to derive good mappings through a combination of algorithms and systematic procedures. We view mapping as consisting of wo phases, alignment followed by distribution. For the alignment phase we present three constraint-based methods--one based on a linear programming formulation of the problem; the second formulates the alignment problem as a constrained optimization problem using Lagrange multipliers; the third method uses a heuristic to decide which constraints to leave unsatisfied (based on the penalty of increased communication incurred in doing so) in order to find a mapping. In addressing the distribution phase, we have developed two methods that integrate the placement of computation--loop nests in our case--with the mapping of data. For one distributed dimension, our approach finds the best combination of data and computation mapping that results in low communication overhead; this is done by choosing a loop order that allows message vectorization. In the second method, we introduce the distribution preference graph and the operations on this graph allow us to integrate loop restructuring transformations and data mapping. These techniques produce mappings that have been used in efficient hand-coded implementations of several benchmark codes

Compile-time optimization of near-neighbor communication for scalable shared-memory multiprocessors

Scalable shared-memory multiprocessor systems are typically NUMA (nonuniform memory access) machines, where the exploitation of the memory hierarchy is critical to achieving high performance. Iterative data parallel loops with near-neighbor communication account for many important numerical applications. In such loops, the communication of partial results stresses the memory system performance. In this paper, we develop data placement schemes that minimize communication time where the near-neighbor interaction is determined by a stencil. Under a given loop partition, our compile-time algorithm partitions global data into four classes for each processor, with each class requiring specific consistency maintenance requirements. The ADAPT (Automatic Data Allocation and Partitioning Tool) system was implemented to automatically partition parallel code segments for the BBN TC2000, a scalable shared-memory multiprocessor. ADAPT caches global arrays and maintains data consistency in software through instructions that flush data from private caches. Restructuring of a fluid flow code segment by ADAPT improved performance by a factor of more than 3 on the BBN TC2000. Features in current generation pipelined processors with multiple functional units permit the overlap of memory accesses with computation. Our experiments on the BBN TC2000 show that the degree of overlap is limited by architectural parameters, such as the number of CPU registers.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30342/1/0000744.pd

Automatic Parallelization of Tiled Stencil Loop Nests on GPUs

This thesis attempts to design and implement a compiler framework based on the polyhedral model. The compiler automatically parallelizes loop nests; especially stencil kernels, into efficient GPU code by loop tiling transformations which the polyhedral model describes. To enhance parallel performance, we introduce three practically efficient techniques to process different types of loop nests. The experimental results of our compiler framework have demonstrated that these advanced techniques can outperform previous approaches. Firstly, we aim to find efficient tiling transformations without violating data dependences. How to select a tile's shape and size is an open issue that is performance-critical and influenced by GPU's hardware constraints. We propose an approach to determine the tile shapes out of consideration for improving two-level parallelism of GPUs. The new approach finds appropriate tiling hyperplanes by embedding parallelism-enhancing constraints into the polyhedral model to maximize intra-tile, i.e., intra-SM parallelism. This improves the load balance among the streaming processors (SPs), which execute a wavefront of loop iterations within a tile. We eliminate parallelism-hindering false dependences to optimize inter-tile, i.e., inter-SM parallelism. This improves the load balance among the streaming multiprocessors (SMs), which execute a wavefront of tiles. Furthermore, to avoid combinatorial explosion of tile size's configurations, we present a model-driven approach to automating tile size selection that is performance-critical for loop tiling transformations, especially for DOACROSS loop nests. Our tile size selection model accurately estimates the execution times of tiled loop nests running on GPUs. The selected tile sizes lead to the performance results that are close to the best observed for a range of problem sizes tested. Finally, to address the difficulty and low-performance of parallelizing widely used SOR stencil loop nests, we present a new tiled parallel SOR method, called MLSOR, which admits more efficient data-parallel SIMD execution on GPUs. Unlike the previous two approaches that are dependence-preserving, the basic idea is to algorithmically restructure a stencil kernel based on a non-dependence-preserving parallelization scheme to avoid pipelining for higher parallelism. The new approach can be implemented in compilers through a pattern matching pass to optimize SOR-like DOACROSS loop nests on GPUs

Automated cache optimisations of stencil computations for partial differential equations

This thesis focuses on numerical methods that solve partial differential equations. Our focal point is the finite difference method, which solves partial differential equations by approximating derivatives with explicit finite differences. These partial differential equation solvers consist of stencil computations on structured grids. Stencils for computing real-world practical applications are patterns often characterised by many memory accesses and non-trivial arithmetic expressions that lead to high computational costs compared to simple stencils used in much prior proof-of-concept work. In addition, the loop nests to express stencils on structured grids may often be complicated. This work is highly motivated by a specific domain of stencil computations where one of the challenges is non-aligned to the structured grid ("off-the-grid") operations. These operations update neighbouring grid points through scatter and gather operations via non-affine memory accesses, such as {A[B[i]]}. In addition to this challenge, these practical stencils often include many computation fields (need to store multiple grid copies), complex data dependencies and imperfect loop nests. In this work, we aim to increase the performance of stencil kernel execution. We study automated cache-memory-dependent optimisations for stencil computations. This work consists of two core parts with their respective contributions.The first part of our work tries to reduce the data movement in stencil computations of practical interest. Data movement is a dominant factor affecting the performance of high-performance computing applications. It has long been a target of optimisations due to its impact on execution time and energy consumption. This thesis tries to relieve this cost by applying temporal blocking optimisations, also known as time-tiling, to stencil computations. Temporal blocking is a well-known technique to enhance data reuse in stencil computations. However, it is rarely used in practical applications but rather in theoretical examples to prove its efficacy. Applying temporal blocking to scientific simulations is more complex. More specifically, in this work, we focus on the application context of seismic and medical imaging. In this area, we often encounter scatter and gather operations due to signal sources and receivers at arbitrary locations in the computational domain. These operations make the application of temporal blocking challenging. We present an approach to overcome this challenge and successfully apply temporal blocking.In the second part of our work, we extend the first part as an automated approach targeting a wide range of simulations modelled with partial differential equations. Since temporal blocking is error-prone, tedious to apply by hand and highly complex to assimilate theoretically and practically, we are motivated to automate its application and automatically generate code that benefits from it. We discuss algorithmic approaches and present a generalised compiler pipeline to automate the application of temporal blocking. These passes are written in the Devito compiler. They are used to accelerate the computation of stencil kernels in areas such as seismic and medical imaging, computational fluid dynamics and machine learning. \href{www.devitoproject.org}{Devito} is a Python package to implement optimised stencil computation (e.g., finite differences, image processing, machine learning) from high-level symbolic problem definitions. Devito builds on \href{www.sympy.org}{SymPy} and employs automated code generation and just-in-time compilation to execute optimised computational kernels on several computer platforms, including CPUs, GPUs, and clusters thereof. We show how we automate temporal blocking code generation without user intervention and often achieve better time-to-solution. We enable domain-specific optimisation through compiler passes and offer temporal blocking gains from a high-level symbolic abstraction. These automated optimisations benefit various computational kernels for solving real-world application problems.Open Acces

Compilation techniques for multicomputers

This thesis considers problems in process and data partitioning when compiling programs for distributed-memory parallel computers (or multicomputers). These partitions may be specified by the user through the use of language constructs, or automatically determined by the compiler. Data and process partitioning techniques are developed for two models of compilation. The first compilation model focusses on the loop nests present in a serial program. Executing the iterations of these loop nests in parallel accounts for a significant amount of the parallelism which can be exploited in these programs. The parallelism is exploited by applying a set of transformations to the loop nests. The iterations of the transformed loop nests are in a form which can be readily distributed amongst the processors of a multicomputer. The manner in which the arrays, referenced within these loop nests, are partitioned between the processors is determined by the distribution of the loop iterations. The second compilation model is based on the data parallel paradigm, in which operations are applied to many different data items collectively. High Performance Fortran is used as an example of this paradigm. Novel collective communication routines are developed, and are applied to provide the communication associated with the data partitions for both compilation models. Furthermore, it is shown that by using these routines the communication associated with partitioning data on a multicomputer is greatly simplified. These routines are developed as part of this thesis. The experimental context for this thesis is the development of a compiler for the Fujitsu AP1000 multicomputer. A prototype compiler is presented. Experimental results for a variety of applications are included