Multivariate interpolation of large sets of scattered data
ACM Transactions on Mathematical Software (TOMS)
Nonlinear differential equations and dynamical systems
Nonlinear differential equations and dynamical systems
SIAM Journal on Numerical Analysis
Computational Differential Equations
Computational Differential Equations
Generalized Green's Functions and the Effective Domain of Influence
SIAM Journal on Scientific Computing
A Measure-Theoretic Computational Method for Inverse Sensitivity Problems I: Method and Analysis
SIAM Journal on Numerical Analysis
A method for solving stochastic equations by reduced order models and local approximations
Journal of Computational Physics
A Posteriori Error Analysis of Parameterized Linear Systems Using Spectral Methods
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Numerical Analysis
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A very common problem in science and engineering is the determination of the effects of uncertainty or variation in parameters and data on the output of a deterministic nonlinear operator. For example, such variations may describe the effect of experimental error or may arise as part of a sensitivity analysis of the model. The Monte-Carlo Method is a justifiably popular tool for analyzing such effects. It does, however, require many evaluations of the operator and it is difficult to extract precise information about the accuracy of any particular result. In this paper, we borrow techniques from a posteriori error analysis for finite element methods to compute information about the derivative of an operator with respect to its parameters. These techniques employ the generalized Green's function to describe how variation propagates into the solution around localized points in the parameter space. We show how this derivative information can be used either to create a higher order method or produce an error estimate for information computed from a given representation. In the latter case, this provides the basis for adaptive sampling according to the variation in the output values. Both the higher order method and the adaptive sampling method are generally orders of magnitude faster than Monte-Carlo methods in the case that the parameter space is not too high dimensional.