Sensitivity and Uncertainty

Collabotors: Jeff Borggaard (Virginia Tech), Vitor Nunes (UT Dallas)

When the quantification of uncertainty in distributed systems is important for making design decisions and managing uncertainty under operation, one may want to address questions such as: what is the `worst' distributed perturbation of this parameter?, and how likely is that perturbation? In \cite{borggaard2013suq} we propose a methodology based on Fréchet sensitivity analysis to interrogate stochastic descriptions of parameters. Let \(z(q)\) be the output of a PDE corresponding to the random parameter \(q\). For any given realization \(q_0\) of \(q\), the Fréchet derivative of \(z\) with respect to \(q\) is the unique bounded linear operator \(D_q[z(q0)]\) satisfying \[ z(q_0+h) = z(q_0) + D_q[z(q_0)]h+O(h^2), \] for small perturbations \(h\). The Fréechet derivative thus defines a local linear approximation of the model response \(z(q_0+h)\). Under certain regularity assumptions, the Fréchet operator is in fact Hilbert-Schmidt and therefore has a singular value decomposition \[ D_q[z(q_0)]h = \sum_{n=1}^\infty \sigma_n \langle h,v_n\rangle u_n, \] where \(\sigma_1 \geq \sigma_2 \geq ... \geq \sigma_n \rightarrow 0\), and \(\{v_n\}_{n=1}^\infty\) and \(\{u_n\}_{n=1}^\infty\) are orthonormal basis functions of the parameter- and output space respectively. This decomposition yields a hierarchical ordering of the local parametric variations to which the model output is most sensitive, namely the directions \(v_n\) corresponding to the dominant singular values \(\sigma_n\). The associated functions \(u_n\) represent the direction in which the model responds. By combining a statistical description of the model with this spectral decomposition we are able to study, as well as quantify, the interplay between parameter uncertainty and system sensitivity and hence obtain a more holistic understanding of the underlying mechanisms that cause a variation in the system response. Moreover, since the dominant parametric variations have the strongest influence on model response, they are also the easiest to identify from response measurements. We exploit this inherent regularization in solving the statistical inverse problem, by projecting the unknown parameter onto sets of these dominant variations through the use of a clustering algorithm.

Although it is not necessary to compute all singular vectors of the Fréchet operator, the computational cost associated with its spectral decomposition can be considerable. In the context of stochastic sampling, where the evaluation of each path involves the solution of a PDE, it nevertheless pays to compute this decomposition intermittently. We are currently investigating diagnostic tools, based on clustering, to determine when the decomposition is justified. The applicability of these strategies for more complex parameter estimation problems is also the subject of further study.