Adaptive High-dimensional Uncertainty Quantification


Dr. Martin Eigel

Dr. Elmar Zander (TU Braunschweig)

Manuel Marschall


stochastic coefficient field

ALEA is an open source library for research in new methods for Uncertainty Quantification (UQ). It's focus lies on functional spectral methods on the basis of polynomial chaos expansions and the treatment of high-dimensional discretizations. For this, adaptive sparse grid techniques and tensor based low-rank formats are incorporated. Apart from stochastic forward problems (PDEs with random data), methods for (sample-free) Bayesian inverse problems are available.


polynomial chaos sparsity ALEA has been employed for different WIAS projects and publications. It has been developed in cooperation with the TU Brunswick (group Prof. Matthies). The framework will be extended within the frame of ongoing research projects. Hence, the listed features may be in the test, development or planing phase. They will be added in due course.

  • systems of orthogonal polynomials
  • low-rank representations of PDEs with random data
  • adaptive stochastic Galerkin methods
  • classical and sample-free inverse Bayesian methods
  • adaptive sparse grid quadrature
  • domain decomposition techniques


  • ALEA is written in python and thus platform independent.
  • The (exchangeable) FEM backend is FEniCS by default. The WIAS library pdelib has also been used successfully.
  • The interoperability with the tensor library xerus is planned as part of joint research projects with the TU Berlin (research group Prof. Schneider).

Application areas

optimized structure

  • Adaptive stochastic Galerkin methods
  • Adaptive Galerkin methods in hierarchical tensor formats
  • Bayesian inverse problems in hierarchical tensor formats
  • Domain decomposition and localisation methods for stochastic PDEs