Algebraic Kernels Research Articles

Error analysis of reiterated projection methods for Hammerstein integral equations

In this paper, we consider the Newton-iteration scheme based on Galerkin and multi-Galerkin operators to solve non-linear integral equations of the Fredholm-Hammerstein type for both smooth and weakly singular algebraic and logarithmic type kernels. By choosing as space of piecewise polynomials subspace of degree , we show that for a smooth kernel, Galerkin and multi-Galerkin approximate solution in the kth Newton-iteration scheme converges with the orders and , respectively, where h is the norm of the partition. For weakly singular kernels, we show that the Galerkin approximate solution in the kth Newton-iteration scheme converges with the orders , for algebraic kernels and for logarithmic kernels. Also the multi-Galerkin approximate solution in the kth Newton-iteration scheme converges with the orders for algebraic kernels and for logarithmic kernels. Numerical results are given for the justification of our theoretical results.

Integral representation for three-loop banana graph

It has recently been shown that two-loop kite-type diagrams can be computed analytically in terms of iterated integrals with algebraic kernels. This result was obtained using a new integral representation for two-loop sunset subgraphs. In this paper, we have developed a similar representation for a three-loop banana integral in $d = 2-2\varepsilon$ dimensions. This answer can be generalized up to any given order in the $\varepsilon$-expansion and can be calculated numerically both below and above the threshold. We also demonstrate how this result can be used to compute more complex three-loop integrals containing the three-loop banana as a subgraph.

Massive kite diagrams with elliptics

We present the results for two-loop massive kite master integrals with elliptics in terms of iterated integrals with algebraic kernels. The key ingredients are new integral representations for sunset subgraphs in d=4−2ε and d=2−2ε dimensions together with differential equations for considered kite master integrals in A+Bε form. The obtained results can be easily generalized to all orders in ε-expansion and show that the class of functions defined as iterated integrals with algebraic kernels may be large enough for writing down results for a large class of massive Feynman diagrams.

Parallel finite element technique using Gaussian belief propagation

The computational efficiency of Finite Element Methods (FEMs) on parallel architectures is severely limited by conventional sparse iterative solvers. Conventional solvers are based on a sequence of global algebraic operations that limits their parallel efficiency. Traditionally, sophisticated programming techniques tailored to specific CPU architectures are used to improve the poor performance of sparse algebraic kernels. The introduced FEM Multigrid Gaussian Belief Propagation (FMGaBP) algorithm is a novel technique that eliminates all global algebraic operations and sparse data-structures. The algorithm is based on reformulating the FEM into a distributed variational inference problem on graphical models. We present new formulations for FMGaBP, which enhance its computation and communication complexities. A Helmholtz problem is used to validate the FMGaBP formulation for 2D, 3D and higher FEM degrees. Implementation techniques for multicore architectures that exploit the parallel features of FMGaBP are presented showing speedups compared to open-source libraries, specifically deal.II and Trilinos. FMGaBP is also implemented on manycore architectures in this work; Speedups of 4.8X, 2.3X and 1.5X are achieved on an NVIDIA Tesla C2075 compared to the parallel CPU implementation of FMGaBP on dual-core, quad-core and 12-core CPUs respectively.

Systematic derivation of time and power models for linear algebra kernels on multicore architectures

The power wall asks for a holistic effort from the high performance and scientific communities to develop power-aware tools and applications which ultimately drive the design of energy-efficient hardware. Toward this goal, we introduce a systematic methodology to derive reliable time and power models for algebraic kernels employing a bottom-up approach. This strategy helps to understand the contribution of the different kernels to the total energy consumption of applications, as well as to distinguish between the cost of fine-grain components such as arithmetic, memory access, and overheads introduced by, e.g., multithreading or reductions.To study and validate our methodology, we initially focus on two key memory-bound BLAS-1 vector kernels: the dot product and the axpy operation. Subsequently, we show how these kernels can be composed to accurately predict the energy consumption of more heterogeneous algorithms, such as the Conjugate Gradient method, while tackling the elaborate memory hierarchy and the high degree of concurrency of today's processors; in particular, the evaluation of the models on the IBM® Blue Gene/Q supercomputer, as well as on the IBM® Power 755 server, reveals that average power consumption is captured at high accuracy, yet the models and the methodology are universal to be portable to any general-purpose multicore architecture.

Memory Trace Compression and Replay for SPMD Systems Using Extended PRSDs

Concurrency levels in large-scale supercomputers are rising exponentially, and shared-memory nodes with hundreds of cores and non-uniform memory access latencies are expected within the next decade. However, even current petascale systems with tens of cores per node suffer from memory bottlenecks. As core counts increase, memory issues will become critical for the performance of large-scale supercomputers. Trace analysis tools are thus vital for diagnosing the root causes of memory problems. However, existing memory tracing tools are expensive due to prohibitively large trace sizes, or they collect only statistical summaries and omit potentially valuable information. In this paper, we present ScalaMemTrace, a novel technique for collecting memory traces in a scalable manner. ScalaMemTrace builds on prior trace methods with aggressive compression techniques to allow lossless representation of memory traces for dense algebraic kernels, with near-constant trace size irrespective of the problem size or the number of threads. We further introduce a replay mechanism for ScalaMemTrace traces, and discuss the results of our prototype implementation on the x86_64 architecture.

Memory Trace Compression and Replay for SPMD Systems using Extended PRSDs?

Concurrency levels in large-scale supercomputers are rising exponentially, and shared-memory nodes with hundreds of cores and non-uniform memory access latencies are expected within the next decade. However, even current petascale systems with tens of cores per node suffer from memory bottlenecks. As core counts increase, memory issues become critical for the performance of large-scale supercomputers. Trace analysis tools are vital for diagnosing the root causes of memory problems. However, existing tools are expensive due to prohibitively large trace sizes, or they collect only statistical summaries that omit valuable information. In this paper, we present ScalaMemTrace, a novel technique for collecting memory traces in a scalable manner. ScalaMemTrace builds on prior trace methods with aggressive compression techniques to allow lossless representation of memory traces for dense algebraic kernels, with nearconstant trace size irrespective of the problem size or the number of threads. We further introduce a replay mechanism for ScalaMemTrace traces, and discuss the results of our prototype implementation on the x86 64 architecture.

INVOLUTIVE GRADINGS AND KERNELS IN JBW*-TRIPLES

Involutive automorphism, or bijective triple homorphisms of order two, on a JBW *-triple are in a one-to-one correspondence with involutive gradings and bicontractive projections. Such mappings are always isometric and weak*-continuous. This paper analyses the algebraic kernels of the 1-eigenspace B+ and -1-eigenspace B- of involutive automorphisms. It is shown that B+ and B- give rise to several decompositions of A and, remarkably, Ker (B+) and Ker (B-) are Peirce inner ideals.

Convergence Characteristics and Computational Cost of Two Algebraic Kernels in Vortex Methods with a Tree-Code Algorithm

We study the convergence characteristics of two algebraic kernels used in vortex calculations: the Rosenhead-Moore kernel, which is a low-order kernel, and the Winckelmans-Leonard kernel, which is a high-order kernel. To facilitate the study, a method of evaluating particle-cluster interactions is introduced for the Winckelmans-Leonard kernel. The method is based on Taylor series expansion in Cartesian coordinates, as initially proposed by Lindsay and Krasny [J. Comput. Phys., 172 (2001), pp. 879-907] for the Rosenhead-Moore kernel. A recurrence relation for the Taylor coefficients of the Winckelmans-Leonard kernel is derived by separating the kernel into two parts, and an error estimate is obtained to ensure adaptive error control. The recurrence relation is incorporated into a tree-code to evaluate vorticity-induced velocity. Next, comparison of convergence is made while utilizing the tree-code. Both algebraic kernels lead to convergence, but the Winckelmans-Leonard kernel exhibits a superior convergence rate. The combined desingularization and discretization error from the Winckelmans-Leonard kernel is an order of magnitude smaller than that from the Rosenhead-Moore kernel at a typical resolution. Simulations of vortex rings are performed using the two algebraic kernels in order to compare their performance in a practical setting. In particular, numerical simulations of the side-by-side collision of two identical vortex rings suggest that the three-dimensional evolution of vorticity at finite resolution can be greatly affected by the choice of the kernel. We find that the Winckelmans-Leonard kernel is able to perform the same task with a much smaller number of vortex elements than the Rosenhead-Moore kernel, greatly reducing the overall computational cost.

Various Constructions of Continuous Information Systems

In this paper, some classical constructions for continuous information systems in a more general sense are established. Some non-classical constructions for continuous information systems such as weak systems, algebraic kernels and algebraic retracts are also introduced.

Scalable solver software

AbstractTowards Optimal Petascale Simulations (TOPS) is a scalable solver software project based on domain decomposed parallelization to research, implement, and support in collaborations with users an open‐source package for large‐scale discretized PDE problems. Optimal complexity methods, such as multigrid/multilevel preconditioners, keep the time spent in dominant algebraic kernels close to linear in discrete problem size as the applications scale on massively parallel computers. Krylov accelerators and Jacobian‐free variants of Newton's method, as appropriate, are wrapped around the multilevel methods to deliver robustness in multirate, multiscale coupled systems, which are solved either implicitly or in more traditional forms of operator splitting. The TOPS software framework is being extended beyond direct computational simulation to computational optimization, including design, control, and inverse problems. We outline and illustrate the philosophy of TOPS. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Design and exploitation of a high-performance SIMD floating-point unit for Blue Gene/L

We describe the design of a dual-issue single-instruction, multiple-data-like (SIMD-like) extension of the IBM PowerPC® 440 floating-point unit (FPU) core and the compiler and algorithmic techniques to exploit it. This extended FPU is targeted at both the IBM massively parallel Blue Gene®/L machine and the more pervasive embedded platforms. We discuss the hardware and software codesign that was essential in order to fully realize the performance benefits of the FPU when constrained by the memory bandwidth limitations and high penalties for misaligned data access imposed by the memory hierarchy on a Blue Gene/L node. Using both hand-optimized and compiled code for key linear algebraic kernels, we validate the architectural design choices, evaluate the success of the compiler, and quantify the effectiveness of the novel algorithm design techniques. Our measurements show that the combination of algorithm, compiler, and hardware delivers a significant fraction of peak floating-point performance for compute-bound-kernels, such as matrix multiplication, and delivers a significant fraction of peak memory bandwidth for memorybound kernels, such as DAXPY, while remaining largely insensitive to data alignment.

Algebraic kernels of planar sets

Algebraic kernels of planar sets