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dynamical systems are typically stiff) and then solving the resulting
matrix problem at each time step.  This approach can be
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
disadvantageous for a parallel implementation, especially for MIMD
parallel computers having a high communication latency, since the
processors will have to synchronize repeatedly for each timestep.  A
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
effective approach to solving the IVP with a parallel computer is
to decompose the problem at the ODE level.  That is, the large system
is decomposed into smaller subsystems, each of which is assigned to a
single processor.  The IVP is solved iteratively by solving the
smaller IVP's for each subsystem, using fixed values from previous
iterations for the variables from other subsystems.  This dynamic
iteration process is known as waveform relaxation (WR) or sometimes as
the Picard-Lindelof iteration.  In this section, conjugate-direction
methods are considered for accelerating the classical dynamic
iteration methods.  The approach is to first convert the IVP to a
system of second-kind Volterra integral equations by using a ``dynamic
preconditioner.''  Next, it is shown that the classical dynamic
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iteration methods are obtained by applying the Richardson iteration to
the integral equation system.  Finally, conjugate-direction methods
are developed for accelerating the classical dynamic iteration
methods.  This development is approached by considering
conjugate-direction methods as Galerkin methods.  Remark: Theorem only
provides a description of the asymptotic behavior of the dynamic
iteration.  In Miekkala and Nevanlinna examine the dynamic iteration
on the infinite interval.  In that case, intuition about the
convergence rate of the dynamic iteration can be obtained, although
the extent to which that intuition applies for a given dynamic
iteration on a finite interval depends on the stiffness of the
problem.  Another approach to solving is to apply a Galerkin method to
solving a variational formulation of the problem.  This approach leads
directly to the conjugate-direction methods.
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
Consider the problem of numerically solving the linear time-varying
initial value problem (IVP), is a given right-hand side, and computed
over the simulation interval. There are several approaches to solving
the IVP.  The traditional numerical approach is to begin by
discretizing a in time with an implicit integration rule (since large
dynamical systems are typically stiff) and then solving the resulting
matrix problem at each time step.  This approach can be
disadvantageous for a parallel implementation, especially for MIMD
parallel computers having a high communication latency, since the
processors will have to synchronize repeatedly for each timestep.  A
effective approach to solving the IVP with a parallel computer is
to decompose the problem at the ODE level.  That is, the large system
is decomposed into smaller subsystems, each of which is assigned to a
single processor.  The IVP is solved iteratively by solving the
smaller IVP's for each subsystem, using fixed values from previous
iterations for the variables from other subsystems.  This dynamic
iteration process is known as waveform relaxation (WR) or sometimes as
the Picard-Lindelof iteration.  In this section, conjugate-direction
methods are considered for accelerating the classical dynamic
iteration methods.  The approach is to first convert the IVP to a
system of second-kind Volterra integral equations by using a ``dynamic
preconditioner.''  Next, it is shown that the classical dynamic
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this this this this this this 
this this this this this this 
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iteration methods are obtained by applying the Richardson iteration to
the integral equation system.  Finally, conjugate-direction methods
are developed for accelerating the classical dynamic iteration
methods.  This development is approached by considering
conjugate-direction methods as Galerkin methods.  Remark: Theorem only
provides a description of the asymptotic behavior of the dynamic
iteration.  In Miekkala and Nevanlinna examine the dynamic iteration
on the infinite interval.  In that case, intuition about the
convergence rate of the dynamic iteration can be obtained, although
the extent to which that intuition applies for a given dynamic
iteration on a finite interval depends on the stiffness of the
problem.  Another approach to solving is to apply a Galerkin method to
solving a variational formulation of the problem.  This approach leads
directly to the conjugate-direction methods.
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
ivp ivP iVp Ivp iVP IVp IVP
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