Brief Overview of Sensitivity Analysis Research
Computermodeled simulations are used widely in the investigation of complex
physical systems. These models typically contain parameters, and the numerical
results can be highly sensitive to small changes in the parameter values. NSDE
investigators evaluate two main approaches for estimating the firstorder
sensitivity of simulations with respect to model parameters: the forward
sensitivity or tangent linear model method, and the backward or adjoint method.
For models comprised of ordinary differential equations and/or nonlinear
algebraic equations, the project is developing forward sensitivity variants of
the CASC parallel solvers PVODE, IDA and KINSOL. Forward sensitivity methods
can be formulated in terms of an ODE, differentialalgebraic equation or
algebraic equation (for steady
problems) for the solution sensitivities with respect to a fixed set of
parameters. These new equations can be appended to the original system, and the
resulting augmented system can be solved by the respective solver code. The
additional sensitivity equations contain terms involving Jacobian matrixvector
products and partial derivatives. Finite differences can be used to estimate
these terms. For some problems, however, this technique does not work. In
particular, difficulties can arise in applications where the solution
components are badly scaled. Such shortcomings motivate researchers to use
automatic differentiation (AD) as an efficient and exact technique for
evaluating various terms in the sensitivity equations. AD can also be used to
compute the forward sensitivities in a general simulation code by
differentiating the entire code with respect to a specified set of
parameters. This more general approach, however, is usually less efficient
because it does not exploit the user's knowledge of the computational
requirements of the simulation.
In cases where the sensitivity of the solution is needed with respect to a
large number of parameters, adjoint methods may prove more useful. For the
PVODE solver suite, part of this approach involves formulating and solving
adjoint sensitivity equations for the respective solver. Analogous to the
approach for the forward sensitivities, finite differences or AD can be used to
evaluate the adjoint sensitivity equations within the solver. Alternatively, a
reversemode of AD can be applied to differentiate a simulation code with
respect to the model parameters. Both approaches rely on the ability to
efficiently store or recompute the simulation in the forward direction (e.g.,
using some type of interpolation scheme). In particular, it may be possible to
store only a subset of the simulation results (a reduced basis) and to develop
interpolation schemes that express the interpolated solution as a member of
this reduced basis.
