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The power of mathematical modeling comes from the ability to predict the be-havior of a system, which often is governed by differential equations, given some inputdata. Most commonly it is assumed that information about the input data, which couldrepresent initial conditions, boundary conditions, and/or material and system param-eters, is known exactly. But increasingly it is important to consider problems wherethere is significant uncertainty in the values of the input data, generally characterizedin some probabilistic manner. This uncertainty can arise from many different causes,including noise in the measurement of input data, or inherent uncertainty in physicalphenomena that characterize the problem.

The SIGEST paper in this issue contains seminal work on one such problem: thesolution of elliptic partial differential equations where both the model coefficients andthe forcing term may be subject to uncertainty which can be characterized by a finitenumber of random variables. A class of methods that has been used to solve such prob-lems is stochastic Galerkin methods. “A Stochastic Collocation Method for Elliptic Par-tial Differential Equations with Random Input Data,” by Ivo Babuska, Fabio Nobile, andRaul Tempone, originally published in the SIAM Journal on Numerical Analysis in 2007, in-troduces a new stochastic Galerkin method for these problems where the uncertaintyis treated through the use of a stochastic collocation method. The use of colloca-tion enables the method to treat a wider range of problems than noncollocation-basedstochastic Galerkin methods, including problems where the input data depend nonlin-early upon the random variables. The authors provide a rigorous convergence analysisfor their method and prove exponential convergence of the probability error. Part ofthe appeal of this approach is its practical applicability, as the method lends itself toworking directly with existing simulation models.

The authors have added a large amount of new material to the SIGEST versionof the paper. Most significantly, an extensive new section 6 describes further develop-ments related to the topic of this paper since its initial publication. This includes thedevelopment of sparse grid collocation methods, which allow the approach to be ex-tended efficiently to cases where the problem depends on a moderately large numberof random variables, whereas the original work was best suited to cases where theuncertainty could be characterized by a fairly small number of random variables.

This SIGEST paper provides a window onto a topic that currently is very im-portant in the applied mathematics and scientific community, as well as offering anexcellent combination of theoretical analysis and practical applicability. We hope it willoffer many SIAM readers a view into an exciting area.

The Editors

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