# Brief Overview of Sensitivity Analysis Research

Computer-modeled 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 first-order 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, differential-algebraic 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 matrix-vector 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 reverse-mode 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.

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