Bulk Synchronous Parallelism (BSP) is a parallel programming model that abstracts from low-level program structures in favour of supersteps. A superstep consists of a set of independent local computations, followed by a global communication phase and a barrier synchronisation. Structuring programs in this way enables their costs to be accurately determined from a few simple architectural parameters, namely the permeability of the communication network to uniformly-random traffic and the time to synchronise. Although permutation routing and barrier synchronisations are widely regarded as inherently expensive, this is not the case. As a result, the structure imposed by BSP does not reduce performance, while bringing considerable benefits for application building. This paper answers the most common questions we are asked about BSP and justifies its claim to be a major step forward in parallel programming. Why Is Another Model Needed?In the 1980s, a large number of different types of parallel architectures were developed. This variety may have been necessary to thoroughly explore the design space but, in retrospect, it had a negative effect on the commercial development of parallel applications software. To achieve acceptable performance, software had to be carefully tailored to the specific architectural properties of each computer, making portability almost impossible. Each new generation of processors appeared in strikingly-different parallel architectural frameworks, forcing performancedriven software developers to redesign their applications from the ground up. Understandably, few were keen to join this process.Today, the number of parallel computation models and languages probably exceeds the number of different architectures with which parallel programmers had to contend ten years ago. Most make it hard to achieve portability, hard to achieve performance, or both. The two largest classes of models are based on message passing, and on shared memory. Those based on message passing are inadequate for three reasons. First, messages require paired actions at the sender and receiver, which it is difficult to ensure are correctly matched. Second, messages blend communication and synchronisation so that sender and receiver must be in appropriately-consistent states when the communication takes place. This is appallingly difficult to ensure in most models, and programs are prone to deadlock as a result. Third, the performance of such programs is impossible to predict because the interaction of large numbers of individual messages in the interconnection mechanism makes the variance in their delivery times large.The argument for shared-memory models is that they are easier to program because they provide the abstraction of a single, shared address space. A whole class of placement decisions are avoided. This is true, but is only half of the issue. When memory is shared, simultaneous access to the same location must be prevented. This requires either PRAM-style discipline by the programmer, or expensive lock mana...

BSPlib is a small communications library for bulk synchronous parallel (BSP) programming which consists of only 20 basic operations. This paper presents the full de®nition of BSPlib in C, motivates the design of its basic operations, and gives examples of their use. The library enables programming in two distinct styles: direct remote memory access (DRMA) using put or get operations, and bulk synchronous message passing (BSMP). Currently, implementations of BSPlib exist for a variety of modern architectures, including massively parallel computers with distributed memory, shared memory multiprocessors, and networks of workstations. BSPlib has been used in several scienti®c and industrial applications; this paper brie¯y describes applications in benchmarking, Fast Fourier Transforms (FFTs), sorting, and molecular dynamics.

In this paper, we establish existence, uniqueness, and scale-invariant estimates for fundamental solutions of non-homogeneous second order elliptic systems with bounded measurable coefficients in R n and for the corresponding Green functions in arbitrary open sets. We impose certain non-homogeneous versions of de Giorgi-Nash-Moser bounds on the weak solutions and investigate in detail the assumptions on the lower order terms sufficient to guarantee such conditions. CONTENTSV 44 Appendix D. Smoothing and Approximations 46 References 48 for some 0 < λ , Λ < ∞ and for all x in the domain. Note that, in particular, equations with complex bounded measurable coefficients fit into this scheme. We establish existence, uniqueness, as well as global scale-invariant estimates for the fundamental solution in R n and for the Dirichlet Green function in any connected, open set Ω ⊂ R n , where n ≥ 3.The key difficulty in our work is the lack of homogeneity of the system since this typically results in a lack of scale-invariant bounds. Here, the existence of solutions relies on a coercivity assumption, which controls the lower-order terms, and the validity of the Caccioppoli inequality. Furthermore, following many predecessors (see, e.g., [HK07], [KK10]), we require certain quantitative versions of the local boundedness of solutions. This turns out to be a delicate game, however, to impose local conditions which are sufficient for the construction of fundamental solutions and necessary for most prominent examples. Indeed, they have not been completely wellunderstood even in the case of real equations, due to the same type of difficulties: Solutions to nonhomogeneous equations can grow exponentially with the growth of the domain in the absence of a suitable control on the potential V, even if b = d = 0. This affects the construction of the fundamental solution. Let us discuss the details. The fundamental solutions and Green functions for homogeneous second order elliptic systems are fairly well-understood by now. We do not aim to review the vast literature addressing various situations with additional smoothness assumptions on the coefficients of the operator and/or the domain, and will rather comment on those works that are most closely related to ours. The analysis of Green functions for operators with bounded measurable coefficients goes back to the early 80's, [GW82] (see also [LSW63] for symmetric operators), in the case of homogeneous equations with real coefficients (N = 1). The case of homogeneous systems, and, respectively, equations with complex coefficients, has been treated much more recently in [HK07] and [KK10] under the assumptions of local boundedness and Hölder continuity of solutions, the so-called de Giorgi-Nash-Moser estimates. Later on, in [Ros13], the fundamental solution in R n was constructed using only the assumption of local boundedness, that is, without the requirement of Hölder continuity. In [Bar14], Barton constructed fundamental solutions, also in R n only, in the full generality of homogeneous elli...

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