Abstract
Uncertainty quantification (UQ) is defined differently by different disciplines. Here, we first review an applied and computational mathematician’s definition of UQ for complex systems, especially in the context of partial differential equations (PDEs) with random inputs. We then discuss the types of stochastic noises that are used as inputs to the PDEs and, for the case of infinite stochastic processes, how those inputs are approximated so that they are amenable to computations. We then review methods that are used to obtain approximations of solutions of PDEs with random inputs, with special emphases given to stochastic Galerkin and stochastic sampling methods, including sparse-grid methods in the latter case. We close with a brief foray into where UQ in the PDE setting is going moving forwards.
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Notes
- 1.
Some attribute this quote as a slightly modification of a quote attributed to Albert Einstein: A theory is something nobody believes, except the person who made it. An experiment is something everybody believes, except the person who made it.
- 2.
Pink noise is “half-way” between white and brown noise; it is referred to as pink because in some circles brown noise is referred to as red noise.
- 3.
Polynomial chaos was a term coined by Norbert Weiner when he studied PDEs driven by white noise and whose solution displayed chaotic behaviors. He expressed white noise random fields by truncated expansions in terms of orthogonal polynomials. In the mathematical and engineering UQ communities, polynomial chaos is used as a substitute name for orthogonal polynomial approximation, even though very few of the problems addressed by those communities have solutions that display any chaotic tendencies. We ourselves eschew the use of “polynomial chaos” and instead call it by what it actually is, namely, orthogonal polynomial approximation.
- 4.
There is even a non-intrusive version of SGMs, but that version is better viewed as a sampling method.
- 5.
It is well known that in one dimension, evenly spaced points are “bad” interpolation points for general smooth function, bad because the interpolation error can get can get worse as the degree of the polynomial increase and the point spacing decreases. On the other hand, the unevenly spaced Chebyshev points are known to be ideal for the interpolation of smooth function in one dimension.
- 6.
In truth, any sampling method can be referred to as being a stochastic collocation methods because “stochastic” simply refers to the fact that we are dealing with random variables within some domain in parameter space and “collocation” simply means that we are evaluating the function, in our case the solution of the PDE, at points in the domain, ergo, we are sampling the solution at points in the parameter domain. However, stochastic collocation methods is now thought of as referring to a class of methods for which deterministic sampling is done on a structured set of points that are much fewer, e.g., much sparser, than, e.g., a tensor product of points, ergo, the synonymous moniker “sparse grids.”
- 7.
The quadrature rules that use Smolyak of sparse grids as the quadrature points are not always interpolatory quadrature rules, but they are all built using combinations such rules.
- 8.
MC and QMC points are useless for interpolation.
References
Ghanem R, Higdon D, Owhadi H (eds) (2017) Handbook of uncertainty quantification. Springer, Berlin
Gunzburger M, Webster C, Zhang G (2014) Stochastic finite element methods for partial differential equations with random input data. Acta Numer 23:521–650
Smith R (2014) Uncertainty quantification: theory, implementation, and applications. SIAM, Philadelphia
Soize C, (2017) Uncertainty quantification: an accelerated course with advanced applications in computational engineering. Springer, Berlin
Sullivan R (2015) Introduction to uncertainty quantification. Springer, Berlin
Acknowledgements
The author “Max Gunzburger” was supported in part by the US Department of Energy Office of Advanced Scientific Computing Research and the US Air Force Office of Scientific Research.
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Gunzburger, M. (2019). An Applied/Computational Mathematician’s View of Uncertainty Quantification for Complex Systems. In: Fillion, N., Corless, R., Kotsireas, I. (eds) Algorithms and Complexity in Mathematics, Epistemology, and Science. Fields Institute Communications, vol 82. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-9051-1_5
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