Baudet, G.M.: Asynchronous iterative methods for multiprocessors. Journal of the ACM 25 (1978) 226--244  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[4] and the preliminary tests in [4] indicate that this method leads quickly to cheap solutions of limited accuracy. Due to the rapid development and increasing usage of parallel computers and distributed computing, it is now important to adapt the methods to the new architectures. Baudet s [1] experimental results on systems of linear equations show a considerable advantage for iterative methods on parallel computers with no synchronization. This leads to experimentation with totally asynchronous [2] block iterative methods for the solution of linear least squares problems. The block ....

Baudet, G. M.: Asynchronous Iterative Methods for Multiprocessors. Journal of the ACM. 25 (1978) 226--244

....so that the parallel computation can proceed in the correct order. Third the tasks should be able to deal with the failure of one or more of the tasks they communicate with in a manner consistent with the computation being performed. The approach we took combines two ideas: chaotic relaxation [10,11] and meshless methods[12] The first is to avoid having to synchronize across all the nodes between iterations. The second is to be able to adapt to faults without requiring a replacement node to fill in the hole in the grid. Together these two ideas form the basis for a naturally fault tolerant ....

G. M. Baudet, "Asynchronous Iterative Methods for Multiprocessors", Journal of the ACM, Volume 25, Issue 2, pp. 226-244, April 1978.

....Systems of equations. 75 IOAN LAZ AR that all processors have to wait at some synchronization point before proceeding to the next iteration. The asynchronous nonlinear multisplitting methods were considered in [1] and [12] i.e. methods where no synchronization barrier is present (see [5, 3, 7] for some general discussions on asynchronous methods) Bahi et al..l [1] studied asynchronous nonlinear multisplitting methods in a general context for nonlinear fixed point problems, while Szyld and Xu [12] studied these methods for problems of the form (1) and extended the study to the case of ....

....(10) s(k) min s i (k) We obtain immediately from (7) and (8) 11) s(k) k and lim s(k) #. Suppose that (7) 9) are satisfied, then we can define an increasing sequence l#N having the properties (12) 0#s(k)#k k0 (13) #s(k)#k k . L . The proofs given by Baudet [3] and El Tarazi [13] for general asynchronous iterations use the sequence defined above. This sequence says that the asynchronous iteration (6) updates all block components at least once at the steps k 0 , k 1 , If k l 1 k l = L for all l, we get a synchronous block Gauss Seidel ....

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Gerard M. Baudet. Asynchronous iterative methods for multiprocessors. J. Association for Computing Machinery, 25:226--244, 1978.

.... of the chaotic iteration above (see also Section 3) Recent contributions to fixpoint theory provide efficient strategies for vector iteration, e.g. by using demand driven evaluation strategies (cf. VWL94, Jr94] The vector approach has been further generalized towards asynchronous iterations [Bau78, Cou77, UD89, Wei93], where f J i may use components of a choice of earlier vectors x j , with j i, of the iteration. Despite its power the vector iteration approach turns out too restrictive in two aspects. First, the functions involved in the fixpoint iteration may be such that they cannot be regarded as ....

G'erard Baudet. Asynchronous iterative methods for multiprocessors. Journal of the ACM, 25(2):226--244, April 1978.

....are synchronous in the sense that all processors have to wait at some synchronization point before proceeding to the next iteration. In this paper we study asynchronous nonlinear multisplitting methods for the solution of (1) i.e. methods where no synchronization barrier is present; see, e.g. [4], 6] 15] for some general discussion on asynchronous methods. We present here a framework which is different than that in the recent paper [1] Our paper deals with local convergence and, in a sense, is more specialized. In our special situation, our hypotheses are more general than those in ....

....the asynchronous nonlinear multisplitting method (ANM) we use a mathematical model describing the Computational Model 3.1. To that end, consider a counter k, which is updated every time a new vector is computed by some processor and let x , l = 1; L, be the initial guess; see, e.g. [4], 6] 15] 23] and the references given therein. Then we write l = T l i P L j=1 E j x r(j;k) if l 2 J k k Gamma1 l otherwise. 6) for k = 1; 2; where the sets J k and the sequence r(j; k) j = 1; L, k = 1; 2; satisfy the following standard conditions ....

G'erard M. Baudet. Asynchronous iterative methods for multiprocessors. Journal of the Association for Computing Machinery, 25:226--244, 1978.

....: ii) The set fi j 2 J i g is unbounded for all = 1; p. iii) lim i 1 r(k; i) 1 for all k = 1; p. The representation (3) together with (i) iii) can be found (sometimes with some minor variations) in most of the literature on asynchronous methods, including [1] [4], 6] 12] 14] 19] 21] 23] Chazan and Miranker [11] proved the following result, where, as before, we assume that T is nonnegative. Theorem 1. If ae(T ) 1 the asynchronous method (3) converges to the solution of (1) If ae(T ) 1, an initial vector x and a sequence fr(k; i)g, k = ....

G'erard M. Baudet. Asynchronous iterative methods for multiprocessors. Journal of the Association for Computing Machinery, 25:226--244, 1978.

....of a new iterate at the ith step. As is customary in the description and analysis of asynchronous algorithms, we assume that the subscripts r( i) and the sets J i satisfy the following conditions. They appear as classical conditions in convergence results for asynchronous iterations; see e.g. [5], 8] 14] 17] r( i) i for all = 1; 2; L; i = 1; 2; 21) lim r( i) 1 for all = 1; 2; L: 22) The set fi j 2 J i g is unbounded for all = 1; 2; L: 23) With this notation, the asynchronous counterpart of Algorithm 2 can be described by the ....

G. M. Baudet. Asynchronous iterative methods for multiprocessors. Journal of the Association for Computing Machinery, 25:226--244, 1978.

....Steihaug [4] and their preliminary tests indicate that this leads quickly to cheap solutions of limited accuracy. Due to the rapid development and increasing usage of parallel computers and distributed computing it has become important to adapt these methods to the new architectures. Baudet s [1] experimental results on systems of linear equations show a considerable advantage for iterative methods on parallel computers with no synchronization at all. This statement has led us to experiment with totally asynchronous [2] block iterative methods for the solution of linear least squares ....

.... attempt to solve the following Newton step equation [9] 0 A I A 0 0 Z 0 X A 0 Deltay Deltaz Gammar c Gammar b GammaX Ze oee A (5) where X =diag(x) Z =diag(z) r b = Ax Gamma b, r c = A y z Gamma c, is the duality gap, oe is an algorithm dependent parameter between [0,1], and e is the vector of all ones. Equation (5) can be reformulated eliminating Deltaz to give Z A c Gamma A e Ax Gamma b ; 6) Deltaz = X (oee Z Deltax) Gamma z; which is known as the augmented system. Let M = X Z) Then, from (6) we get AMA ....

G. M. Baudet, Asynchronous Iterative Methods for Multiprocessors, J. of the ACM 25, pp. 226--244 (1978).

....of the network problem solving process. In fact, additional mechanisms required to handle hardware, communication, and processing errors may be unnecessary with the FA C approach, since uncertainty resolving mechanisms are already a part of the distributed system s problemsolving structure [4, 17, 40]. In FA C distributed systems, it may be difficult to determine which alternative tasks are globally the most beneficial to perform without extensive inter node communication. This control uncertainty is due to differences between the natural distribution of control information among the nodes ....

....3] When this equilibrium is achieved, a coherent set of local views has been constructed. The MSYS problem solving technique is an example of a more general problem solving paradigm, called iterative refinement, that is contained in different forms in many types of problem solving systems [4, 50, 63, 64]. We feel that knowledge based AI approaches to problem solving provide a basis for the development of design methodologies for FA C distributed systems. The mechanisms used in these problem solving systems to resolve error from incorrect and incomplete data and knowledge can also be used to ....

[Article contains additional citation context not shown here]

Gerard M. Baudet. Asynchronous iterative methods for multiprocessors. Journal of the ACM, 25(2):226--244, April 1978.

....of the associative nets, that has been studied at the IEF [11, 10] It is an SIMD multiprocessor that implements asynchronous communication. It is based on the mesh topology. 3.5 Related works on asynchronism Several works concern the asynchronous computations for matrix like calculations. In [2], the author define the asynchronous iterations and shows that they give the same result as the synchronous one for the so called contractant operators of R n . These results have been extended in [21] for operators of S n , where S is either a finite or infinite set. Several experimentations ....

Baudet, G. M. Asynchronous iterative methods for multiprocessors. Journal of the ACM 25, 2 (April 1978), 226--244.

....the pioneering work by Chazan and Miranker [6] the condition ae(jH j) 1 is known as a necessary and sufficient condition for the convergence of asynchronous iterations. However, the precise mathematical model for the asynchronous iterations underlying the Chazan Miranker result (see [6] and [3] for a slight modification) is very general. Virtually any practical situation can in fact be modelled mathematically by assuming slightly more restricitive hypotheses than in the Chazan Miranker model, and this has indeed been done repeatedly in the literature on asynchronous iterations. Examples ....

....as a whole class of iterative methods derived from the total step method (3) One now allows that only certain components of the iterate are updated at a given time step and that more than just the previous iterate may be used in the updated process. The precise definition is as follows, see [3, 5]. 4 Definition 1 For k = 1; 2; let J k f1; ng and (s 1 (k) s n (k) 2 N n 0 be such that s i (k) k Gamma 1 for i = 1; n; k = 1; 2; 7) lim k 1 s i (k) 1 for i = 1; n; 8) for every i 2 f1; ng the set fk j i = J k g is ....

[Article contains additional citation context not shown here]

G'erard M. Baudet. Asynchronous iterative methods for multiprocessors. J. ACM, 25:226--244, 1978.

....of a new iterate at the ith step. As is customary in the description and analysis of asynchronous algorithms, we assume that the subscripts r(#, i) and the sets J i satisfy the following conditions. They appear as classical conditions in convergence results for asynchronous iterations; see e.g. [5], 8] 15] 20] r(#, i) i for all # =1, 2, L, i=1, 2, 21) lim i## r(#, i) # for all # =1, 2, L. 22) The set i # # J i is unbounded for all # =1, 2, L. 23) With this notation, the asynchronous counterpart of Algorithm 2 can be described by the following algorithm. ....

Baudet, G.M. (1978): Asynchronous iterative methods for multiprocessors. Journal of the Association for Computing Machinery 25, 226--244

....of inputs) Asynchronous iterations satisfying these conditions are called totally asynchronous [5] Such asynchronous iterations only exclude starving computations. There has been a considerable amount of work in the literature on the convergence conditions for totally asynchronous iterations [2, 4, 5, 6, 7, 9, 13, 14, 15, 18, 20, 21, 22, 23, 25, 26, 27]. The most general result can be stated as follows [5, 7, 27] x i t ( F i u i t ( 4 Proposition 1. Let X(k) be a sequence of sets such that . for all k. for all k, and furthermore all sequences z(k) such that z(k) X(k) for all k converge to x, where x is the unique ....

Baudet, G.M., "Asynchronous iterative methods for multiprocessors," Journal of the Association for Computing Machinery (JACM), 25(2), pp. 226-244, April 1978.

....iteration step by the next variables during the image scanning, is known to improve convergence speed, but is inherently sequential and directionally asymmetric. In between, and retaining advantages from both, stand the asynchronous updating modes that have been studied for relaxation algorithms [Bau78]. The idea is to allow at each iteration step the updating of a random subset of state variables from delayed values of other variables dating back to past iterations rather than the immediately preceding one. Thus it corresponds to a less constrained updating order. In fact, a local calculation ....

G. Baudet, "Asynchronous iterative methods for multiprocessors", Journal of the Association for Computing Machinery, Vol. 25, No. 2, April 1978, pp. 226-244.

....points: x # F (x) This is su#ciently general to include Newton s method. He provided theoretical convergence results for the case of contracting operators, namely operators F (x) with contracting Lipschitz matrices: F (x) F (y) # A x y for some nonnegative A such that #(A) 1. Baudet (1978) relaxed various theoretical hypotheses of the earlier papers and performed experiments on a 6 processor C.mmp at CMU; and so forth. Bersekas (1990) is a book length monograph on aynchronous methods. We do not need to contemplate full asynchrony to derive a latencytolerant benefit from such ....

Baudet (1978) Asynchronous Iterative Methods for Multiprocessors, J. of the ACM 25, pp. 226--244.

....n [i] if i = 2 J n F (X n ) A Omega X n Phi B if i 2 J n (3.3) Informally, Equation 3.3 reads as only selected processors (those of J n ) compute a new value using the last produced result of their direct ancestors. Such equations are known in the literature as asynchronous iterations (see [9, 31]) 16 3.4.4 The Fully Distributed Demon Under control of the fully distributed demon, activated processors do not necessaryly write their output registers within the same round. Thus Equation 3.3 does not hold in this context. Hence, a processor may compute its output value using its last read ....

....(X Dn [1] 1] X Dn [N ] N ] t Delta = A Omega (X Dn [1] 1] XDn [N ] N ] t Phi B if i 2 J n (3.4) 3.4. 5 Conditions for Convergence of Asynchronous Iterations Asynchronous iterations have been extensively studied for optimization purpose on parallel computers (see [9, 19, 23, 30, 31, 32]) Under particular conditions, asynchronous iterations (Equations 3.3 and 3.4) converge to the same result as synchronous iterations (Equation 3.2) while reducing data dependency. In [31] Uresin and Dubois give several sufficient conditions ensuring the convergence of asynchronous iterations. ....

G. M. Baudet. Asynchronous Iterative Methods for Multiprocessors. Journal of the ACM, Vol. 25, No. 2, pp. 226--244, April 1978. 24

....[4] and the preliminary tests in [4] indicate that this method leads quickly to cheap solutions of limited accuracy. Due to the rapid development and increasing usage of parallel computers and distributed computing, it is now important to adapt the methods to the new architectures. Baudet s [1] experimental results on systems of linear equations show a considerable advantage for iterative methods on parallel computers with no synchronization. This leads to experimentation with totally asynchronous [2] block iterative methods for the solution of linear least squares problems. The block ....

Baudet, G. M.: Asynchronous Iterative Methods for Multiprocessors. Journal of the ACM. 25 (1978) 226--244

....INTRODUCTION In this paper we use a block iterative method for solving sparse linear least squares problems. A general framework for this method is introduced by Dennis and Steihaug [4] and their preliminary tests indicate that this leads quickly to cheap solutions of limited accuracy. Baudet s [1] experimental results on systems of linear equations show a considerable advantage for iterative methods on parallel computers with no synchronization at all. His statement has led us to experiment with totally asynchronous [2] block iterative methods for the solution of linear least squares ....

G. M. Baudet, Asynchronous Iterative Methods for Multiprocessors, J. of the ACM 25, p. 226 (1978).

.... R n , the solution of F(x) x is obtained by a similar computational model, Daniel B. Szyld 5 where at step 3, the processor solves for x (or its approximation) in the equation F (x 1 ; x ; xL ) x . This model then relates to the work by many authors; see, e.g. [6], 10] 14] 26] 37] 42] 45] 47] 48] 56] 62] and the references given therein. See also further extensions in Section 6. 3 First Mathematical Models The first mathematical model analyzing the convergence of the asynchronous iteration dates from 1959. At that time there were no ....

....condition (ii 0 ) or equivalently (ii) some authors call this sequence admissible; see, e.g. 18] 20] In these papers, a sequence satisfying (iii 0 ) is called regulated. Condition (iii) is more general than (iii 0 ) since no uniform bound d is required, as pointed out in, e.g. [6], 11] 21] see also [60] for an analysis of a condition other than (iii 0 ) Most convergence results in this paper correspond to the more general assumption (iii) We should mention though that there are models where the additional assumption on uniformity is required; see, e.g. 11, Ch. ....

G'erard M. Baudet. Asynchronous iterative methods for multiprocessors. Journal of the Association for Computing Machinery, 25:226--244, 1978.

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Baudet, G.M.: Asynchronous iterative methods for multiprocessors. Journal of the ACM 25 (1978) 226--244

No context found.

Gerard M. Baudet. Asynchronous iterative methods for multiprocessors. Journal of the ACM, 25(2):226--244, April 1978.

No context found.

Gerard M. Baudet. Asynchronous iterative methods for multiprocessors. J. ACM, 25(2):226--244, April 1978.

No context found.

G.M. Baudet, "Asynchronous iterative methods for multiprocessors," J. ACM, vol. 25, no. 2, pp. 226-244, 1978.

No context found.

G. Baudet, Asynchronous iterative methods for multiprocessors, JACM, vol. 25, no. 2, pp. 226-244, 1978.

No context found.

G.M. Baudet. Asynchronous Iterative Methods for Multiprocessors. Journal of the ACM, 2:226--244, 1978.

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