By Eric F. Van de Velde
Mathematics is enjoying an ever extra vital function within the actual and organic sciences, scary a blurring of barriers among medical dis ciplines and a resurgence of curiosity within the glossy in addition to the classical recommendations of utilized arithmetic. This renewal of curiosity, either in examine and educating, has resulted in the institution of the sequence: Texts in utilized Mathe matics (TAM). the improvement of latest classes is a usual end result of a excessive point of pleasure at the examine frontier as more moderen recommendations, akin to numerical and symbolic computers, dynamical platforms, and chaos, combine with and toughen the normal tools of utilized arithmetic. therefore, the aim of this textbook sequence is to fulfill the present and destiny wishes of those advances and inspire the educating of recent classes. TAM will put up textbooks compatible to be used in complex undergraduate and starting graduate classes, and may supplement the utilized Mathematical Sciences (AMS) sequence, so as to specialise in complicated textbooks and study point monographs. Preface A winning concurrent numerical simulation calls for physics and math ematics to improve and research the version, numerical research to strengthen answer tools, and machine technological know-how to increase a concurrent implemen tation. No unmarried path can or may still disguise some of these disciplines. in its place, this direction on concurrent medical computing makes a speciality of a subject that's not lined or is insufficiently lined by way of different disciplines: the algorith mic constitution of numerical methods.
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Extra resources for Concurrent Scientific Computing
For a coarse-grained computation with a load-balanced distribution, we have that! , and Sp ~ P/(l + ~ TA + Tc(l)). TA 2I Hence, a near-maximum speed-up is achieved if D « 2! or, equivalently, if P log2 P « 2M. 8, which was derived for program Inner-Product-3. Program Inner-Product-4 also outperforms program Illller-Product-3 for fine-grained computations. With one vector component per process, the execution time is TM = TA + (TA + Tc(l)) log2 M. For large M, this leads to a speed-up SM ~ 2M log2(2M) 1 + 2M 1 Te(1) TA The upper bound is obtained by setting Tc(l) = O.
D We can now remove the assumption that P = 2D. The general case, 2D < P < 2D + 1 , is reduced to the former by an incomplete recursivedoubling step. In this step, process p with p 2: 2D communicates its w-value to process p - 2D , where a pairwise sum is made. Subsequently, the global sum is computed by plain recursive doubling between all 2D processes. Finally, the computed sum, known only to processes JI with p < 2D , is communicated to processes 2D through P - 1. This procedure is precisely formulated in program Recursive-Doubling-3.
3. In 36 1. The Basics this case, superlinear speed-up is a rare occurrence, but it is theoretically and practically possible. There is no controversy here. In more theoretical papers, the term superlinear speed-up is used for computations below the line of perfect speed-up. We shall refer to such computations as having superperJect speed-up. The controversy really centers around superperfect speed-up. Theorem 1 excludes any possibility of superperfect speed-up and, in spite of its elementary proof, some energy is spent to construct and refute counterexamples of the theorem.