TY - JOUR
T1 - ANOVA Expansions and Efficient Sampling Methods for Parameter Dependent Nonlinear PDEs
AU - Y. Cao, Z. Chen & M. Gunzburger
JO - International Journal of Numerical Analysis and Modeling
VL - 2
SP - 256
EP - 273
PY - 2009
DA - 2009/06
SN - 6
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/766.html
KW - ANOVA expansions, nonlinear partial differential equations, surrogate optimization, parameter sampling methods.
AB - <p style="text-align: justify;">The impact of parameter dependent boundary conditions on solutions of a class of nonlinear partial differential equations and on optimization
problems constrained by such equations is considered. The tools used to gain
insights about these issues are the Analysis of Variance (ANOVA) expansion
of functions and the related notion of the effective dimension of a function;
both concepts are reviewed. The effective dimension is then used to study
the accuracy of truncated ANOVA expansions. Then, based on the ANOVA
expansions of functionals of the solutions, the effects of different parameter
sampling methods on the accuracy of surrogate optimization approaches to
constrained optimization problems are considered. Demonstrations are given
to show that whenever truncated ANOVA expansions of functionals provide accurate approximations, optimizers found through a simple surrogate optimization strategy are also relatively accurate. Although the results are presented
and discussed in the context of surrogate optimization problems, most also apply to other settings such as stochastic ensemble methods and reduced-order
modeling for nonlinear partial differential equations.</p>