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{\large \bf
%Please insert here the title of your contribution!
Back-tracking $Ax=b$
%Forward Dynamics, Elimination Methods, and Formulation Stiffness in
%Robot Simulation
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%Please insert here the address(es) of the author(s)!
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{\bf Uri M. Ascher}
%Collaborators: D. Pai, B. Cloutier, P. Lin, D. Moulton, J. Morel
Dept. of Computer Science, Univ. of British Columbia
Vancouver, B.C., V6T 1Z4, Canada
ascher@cs.ubc.ca
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%Please insert here the abstract!
A physical problem begets a mathematical model.
A mathematical model begets a numerical method for
an approximate, computable solution. A numerical method
begets large, sparse systems of
linear algebraic equations $Ax = b$.
Ideally, one would like to design a solver that automatically
and efficiently computes the solution $x$ for any such system.
But this ideal may require too much to be feasible in such generality.
In some situations, specific features of the underlying discretized
problem may be crucial for the design of efficient algorithms
for the
linear algebra.
Furthermore, there are instances where the best approach
to design fast algorithms to find $x$ is not to
solve $Ax = b$ at all.
Rather, start by {\em back-tracking} away from the algebraic system,
to the numerical method that gives rise to it.
A reinterpretation of the numerical method may then yield
an embedding of $Ax = b$ in a larger linear algebra system
which is easier to solve.
We will consider two instances of this process: One is
the numerical simulation of multibody systems,
such as robot manipulators,
where the composite
rigid body method is replaced by the articulated body
method, yielding a solution algorithm with optimal
complexity in the number of bodies. The other instance involves
the numerical solution of nodal and mixed
finite element methods, where a reformulation in terms
of hybrid methods allows the design of highly efficient
multilevel preconditioners for the larger system.
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