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Research interests

I am a member of the Numerical Analysis Group. My interests are in the design and analysis of efficient and robust parallel numerical methods for engineering and physical problems with heterogeneous material properties that vary over multiple scales. This is typical in energy and environmental applications, but also in material science and manufacturing. My research spans the whole range from the regularity analysis of solutions to the efficient parallel implementation of novel methods and their industrial application. I am particularly interested in multilevel and multiscale methods for partial differential equations with strongly varying and high contrast coefficients, in particular domain decomposition and multigrid methods, preconditioners for systems of PDEs, iterative eigensolvers, and multiscale discretisation techniques with applications in oil reservoir simulation, radioactive waste disposal, numerical weather and climate prediction, novel optical materials or composite materials.

More recently my particular focus has been on the interface between computational mathematics and statistics/probability. In most applications with heterogeneous material properties the coefficients are not known exactly. In fact, they are usually highly uncertain. One of the most popular ways to deal with uncertainty is stochastic modelling. However, most of the statistical tools for uncertainty quantification are either very inaccurate or computationally infeasible for typical engineering applications. Similar things can be said for data assimilation, for example in numerical weather prediction. My current research focusses mainly on two promising variants of the classical Monte Carlo method, namely multilevel Monte Carlo and quasi-Monte Carlo, which can provide highly accurate and efficient tools for uncertainty quantification. More recently we have extended the technology also to Bayesian inference by developing a multilevel Markov chain Monte Carlo method. The new methods are also of interest in time dependent problems with random noise (SDEs), e.g. in mathematical finance or in atmospheric dispersion modelling.

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Multilevel Methods Mathematics
Elliptic PDE Mathematics
Monte Carlo methods Engineering & Materials Science
Monte Carlo method Mathematics
Coefficient Mathematics
Partial differential equations Engineering & Materials Science
Domain Decomposition Mathematics
Preconditioner Mathematics

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Projects 2009 2023

Re-shaping the Test Pyramid

Scheichl, R. & Anaya-Izquierdo, K.


Project: Research council

Numerical methods
Singular value decomposition


Scheichl, R.


Project: EU Commission

Sustainable development
Numerical methods
Fusion reactions

Multiscale Modelling of Aerospace Composites

Scheichl, R.


Project: Research council

Composite materials

Research Output 2002 2018

A hybrid Alternating Least Squares - TT Cross algorithm for parametric PDEs

Dolgov, S. & Scheichl, R., 26 Dec 2018, (Accepted/In press) In : SIAM/ASA Journal on Uncertainty Quantification. 28 p.

Research output: Contribution to journalArticle

Open Access
Maximum entropy methods
Probability density function
Mathematical operators
1 Citations (Scopus)

Analysis of circulant embedding methods for sampling stationary random fields

Graham, I. G., Kuo, F. Y., Nuyens, D., Scheichl, R. & Sloan, I. H., 2018, In : SIAM Journal on Numerical Analysis (SINUM). 56, 3, p. 1871-1895 27 p.

Research output: Contribution to journalArticle

Open Access
Random Field
Regular hexahedron
Positive Definiteness
Circulant Matrix
Correlation Length

A Stein variational Newton method

Detommaso, G., Cui, T., Marzouk, Y., Spantini, A. & Scheichl, R., 8 Jun 2018, (Accepted/In press) Advances in Neural Information Processing Systems (NIPS) 2018.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Citations (Scopus)

Circulant embedding with QMC: analysis for elliptic PDE with lognormal coefficients

Graham, I., Kuo, F. Y., Nuyens, D., Scheichl, R. & Sloan, I. H., 1 Oct 2018, In : Numerische Mathematik. 140, 2, p. 479-511 33 p.

Research output: Contribution to journalArticle

Open Access
Quasi-Monte Carlo
Elliptic PDE
Monte Carlo methods
Random Field

Continuous Level Monte Carlo and Sample-Adaptive Model Hierarchies

Detommaso, G., Dodwell, T. & Scheichl, R., 21 Feb 2018, (Accepted/In press) In : SIAM/ASA Journal on Uncertainty Quantification.

Research output: Contribution to journalArticle