Multiscale problems and uncertainty quantification
Engineering applications that involves several different scales, multiscale problems (MS), and as well as uncertainty in the data, uncertainty quantification (UQ), are very computationally challenging. Because of the numerical difficulties in solving this kind of problems special numerical methods and algorithms has to be developed.
- Daniel Elfverson, Umeå University (MS and UQ)
- Donald Estep, Colorado State University (UQ)
- Emmanuil Georgoulis, Leicester University (MS)
- Fredrik Hellman, Uppsala University (MS and UQ)
- Patrick Henning, EPFL (MS)
- Mats Larson, Umeå University (MS)
- Axel Målqvist, University of Gothenburg and Chalmers (MS and UQ)
- Anna Persson, University of Gothenburg and Chalmers (MS)
- Daniel Peterseim, Universität Bonn (MS)
Multiscale problems are some of the greatest challenges in computational mathematics today. In all branches of the engineering sciences we encounter problems with features on several different scales. A typical example is simulations in a heterogenous media where material data such as module of elasticity, conductivity or permeability, varies in space over several different scales. In order to solve these problems efficiently we propose an adaptive multiscale method where the critical parameters of the method are chosen automatically through an adaptive algorithm.
In engineering applications it is very common that the data is given by experimental measurements. It is therefore associated with measurement errors. It is natural to model these errors using a probabilistic representation of the data. We develop efficient methods for forward sensitvity analysis of partial differential equations with uncertainty in the data. We present error estimates taking into account both the numerical and the statistical error e.g. when approximating the cumulative distribution function of a quantity of interest using numerical techniques.
Selected publications related to the project.
- Uncertainty quantification for approximate p-quantiles for physical models with stochastic inputs. In SIAM/ASA Journal on Uncertainty Quantification, volume 2, pp 826-850, 2014. (DOI).
- Localization of elliptic multiscale problems. In Mathematics of Computation, volume 83, number 290, pp 2583-2603, 2014. (DOI).
- A localized orthogonal decomposition method for semi-linear elliptic problems. In Mathematical Modelling and Numerical Analysis, volume 48, pp 1331-1349, 2014. (DOI).
- Convergence of a discontinuous Galerkin multiscale method. In SIAM Journal on Numerical Analysis, volume 51, pp 3351-3372, 2013. (DOI).
- Multiscale methods for elliptic problems. In Multiscale Modeling & simulation, volume 9, pp 1064-1086, 2011. (DOI).
- Nonparametric density estimation for randomly perturbed elliptic problems I: Computational methods, a posteriori analysis, and adaptive error control. In SIAM Journal on Scientific Computing, volume 31, pp 2935-2959, 2009. (DOI).
- Adaptive variational multiscale methods based on a posteriori error estimation: Energy norm estimates for elliptic problems. Mats G. Larson and Axel Målqvist. In Computer Methods in Applied Mechanics and Engineering, volume 196, pp 2313-2324, 2007.