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No'1

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... be an arc of the dependency graph if and only if the function fj depends on xi. ... The degree of parallelism may depend on update ordering ... – PowerPoint PPT presentation

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Title: No'1


1
Applications Poissons equation
  • Find a function f0,12?R that satisfies
  • where g0,12?R is a known function and f has
    prescribed values on the boundary of the unit
    square.
  • Let

2
Applications Poissons equation (contd)
  • Assume that f is sufficiently smooth and the
    is a small scalar,
  • by Prop. A.33 in Appendix A.
  • By plugging (2) and (3) into (1),
  • A system of (N-1)2 linear equations in (N-1)2
    unknowns, i.e., can be represented in the form
    Axb.

3
Applications Poissons equation (contd)
  • JOR algorithm
  • where fi,j(t)fi,j are known, whenever i or j is
    equal to 0 or N.

4
Applications Power Control of CDMA Uplink
  • Assume K users in a cell, SINR per chip, denoted
    by SINRc, of user i is
  • where is the received energy per chip for
    user i and N0 is noise.
  • Since each bit is encoded onto a pseudonoise
    sequence of length Gi chips at the transmitter,
    the received energy per bit for user i is
    .

5
Applications Power Control of CDMA Uplink
(contd)
  • The SINR of user i, or equivalently the ratio of
    the received energy per bit to the interference
    and noise per chip (commonly called in
    the CDMA literature) is
  • where pi (joules/sec) is the transmit power of
    user i and gi is the attenuation of user is
    signal to base station.

6
Applications Power Control of CDMA Uplink
(contd)
  • To achieve equally reliable communication,
  • where is a certain threshold.
  • The data rate of user i, Ri (bits/sec), is
  • and Gi is called the processing gain of user i.

7
Applications Power Control of CDMA Uplink
(contd)
  • The power control problem of CMDA uplink is to
    find minimal nonnegative transmit power vector
    satisfying
  • That is, find nonnegative satisfying
  • A system of K linear equations in K unknowns,
    i.e., can be represented in the form Axb.

8
Applications Power Control of CDMA Uplink
(contd)
  • JOR algorithm
  • For each user i,
  • where ß, Gi, gi, N0 and W are given.

9
Parallelization of Iterative Methods Using
Dependency Graph
  • Consider a Jacobi-type iteration in the general
    form
  • The communication required for this iteration can
    be described by means of a directed graph
    G(N,A), called the dependency graph.
  • The set of nodes N is 1,,n, corresponding to
    the components of x. Let (i,j) be an arc of the
    dependency graph if and only if the function fj
    depends on xi.

10
Parallelization of Iterative Methods Using
Dependency Graph (contd)
  • The dependency over iterations can be described
    by means of a directed acyclic graph (DAG) where
    the nodes one of the form (i,t) and arcs are of
    the form ((i,t), (j,t1)).

t0
sweep
t1
t
t2
11
Parallelization of Iterative Methods Using
Dependency Graph (contd)
  • Consider a Gauss-Seidel type iteration in the
    general form
  • Often preferable since it incorporates the newest
    available information, thereby sometimes
    converging faster than the Jacobi type
  • Maybe completely non-parallelizable since it is
    sequential in nature
  • When the dependency graph is sparse, it is
    possible that certain component updates can be
    parallelized
  • The degree of parallelism may depend on update
    ordering

e.g.) ordering 1?2?3?4
?The depth of the single iteration (sweep) is 3
12
Parallelization of Iterative Methods Using
Dependency Graph (contd)
  • the depth of the single iteration is 2
  • Finding an optimal update ordering that maximizes
    parallelisms in Gauss-Seidel algorithm is
    equivalent to an optimal coloring problem.

e.g.) ordering 1?3?4?2
13
Parallelization of Iterative Methods Using
Dependency Graph (contd)
  • Prop. 2.5
  • There exists an ordering such that a sweep
    of the Gauss-Seidel algorithm can be performed in
    K parallel steps if and only if there exists a
    coloring of the dependency graph that uses K
    colors and with the property that there exists no
    positive cycle with all nodes on the cycle having
    the same color.
  • Prop. 2.6
  • Suppose that if and only if
    . Then, there exists an ordering such
    that a sweep of the Gauss-Seidel algorithm can be
    performed in K parallel steps if and only if
    there exists a coloring of the dependency graph
    that uses at most K colors and such that adjacent
    nodes have different colors.
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