Title: No'1
1Applications 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
2Applications 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.
3Applications Poissons equation (contd)
- JOR algorithm
- where fi,j(t)fi,j are known, whenever i or j is
equal to 0 or N. -
4Applications 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
.
5Applications 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.
6Applications 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.
7Applications 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.
8Applications Power Control of CDMA Uplink
(contd)
- JOR algorithm
- For each user i,
- where ß, Gi, gi, N0 and W are given.
9Parallelization 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.
10Parallelization 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
11Parallelization 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
12Parallelization 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
13Parallelization 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.