Professor of Scientific Computing & Deputy Head of Department

4 West 2.20

Dept of Mathematical Sciences

Email: r.scheichl@bath.ac.uk

# Robert Scheichl

## Profile

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.

### Publications

Peterseim, D. and Scheichl, R., 2016. Robust numerical upscaling of elliptic multiscale problems at high contrast. *Computational Methods in Applied Mathematics*, 16 (4), pp. 579-603.

Fletcher, T. A., Kim, T., Dodwell, T. J., Butler, R., Scheichl, R. and Newley, R., 2016. Resin treatment of free edges to aid certification of through thickness laminate strength. *Composite Structures*, 146, pp. 26-33.

Ferreiro-Castilla, A., Kyprianou, A. and Scheichl, R., 2016. An euler-poisson scheme for Lévy driven SDEs. *Journal of Applied Probability*, 53 (1), pp. 262-278.

Graham, I., Scheichl, R. and Ullmann, E., 2016. Mixed Finite Element Analysis of Lognormal Diffusion and Multilevel Monte Carlo Methods. *Stochastics and Partial Differential Equations : Analysis and Computations*, 4 (1), pp. 41-75.

Dedner, A., Mueller, E. and Scheichl, R., 2016. Efficient multigrid preconditioners for atmospheric flow simulations at high aspect ratio. *International Journal for Numerical Methods in Fluids*, 80 (1), pp. 76-102.

Fletcher, T., Reinarz, A., Dodwell, T., Butler, R., Scheichl, R. and Newley, R., 2016. Efficient Modelling and Accurate Certification of Curved Aerospace Laminates. *In*: *ECCM17 - 17th European Conference on Composite Materials*, 2016-06-26 - 2016-06-30.

Mueller, E., Scheichl, R. and Vainikko, E., 2015. Petascale solvers for anisotropic PDEs in atmospheric modelling on GPU clusters. *Parallel Computing*, 50, pp. 53-69.

Graham, I. G., Kuo, F. Y., Nichols, J. A., Scheichl, R., Schwab, C. and Sloan, I. H., 2015. Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients. *Numerische Mathematik*, 131 (2), pp. 329-368.

Mueller, E., Scheichl, R. and Shardlow, T., 2015. Improving multilevel Monte Carlo for stochastic differential equations with application to the Langevin equation. *Proceedings of the Royal Society A*, 471 (2176).

Dauge, M., Norton, R. A. and Scheichl, R., 2015. Regularity for Maxwell eigenproblems in photonic crystal fibre modelling. *BIT Numerical Mathematics*, 55 (1), pp. 59-80.

Dodwell, T., Ketelsen, C., Scheichl, R. and Teckentrup, A. L., 2015. A hierarchical multilevel Markov chain Monte Carlo algorithm with applications to uncertainty quantification in subsurface flow. *SIAM/ASA Journal on Uncertainty Quantification*, 3 (1), pp. 1075-1108.

Loisel, S., Nguyen, H. and Scheichl, R., 2015. Optimized Schwarz and 2-Lagrange multiplier methods for multiscale elliptic PDEs. *SIAM Journal on Scientific Computing*, 37 (6), A2896-A2923.

Kim, T., Fletcher, T., Dodwell, T., Butler, R., Scheichl, R., Ankersen, J. and Newley, R., 2015. The effect of free edges on inter-laminar performance of curved laminates. *In*: *56th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference 2015*, 2015-01-05 - 2015-01-09. American Institute of Aeronautics and Astronautics Inc..

Butler, R., Dodwell, T. J., Kim, T., Kynaston, S., Scheichl, R., Haftka, R. T. and Kim, N. H., 2015. Uncertainty quantification of composite structures with defects using multilevel monte carlo simulations. *In*: *17th AIAA Non-Deterministic Approaches Conference 2015*, 2015-01-05 - 2015-01-09. American Institute of Aeronautics and Astronautics Inc..

Mueller, E. H. and Scheichl, R., 2014. Massively parallel solvers for elliptic partial differential equations in numerical weather and climate prediction:scalability of elliptic solvers in NWP. *Quarterly Journal of the Royal Meteorological Society*, 140 (685), pp. 2608-2624.

Spillane, N., Dolean, V., Hauret, P., Nataf, F., Pechstein, C. and Scheichl, R., 2014. Achieving robustness through coarse space enrichment in the two level Schwarz framework. *In*: Erhel, J., Gander, M. J., Halpern, L., Pichot, G., Sassi, T. and Widlund, O., eds. *Domain Decomposition Methods in Science and Engineering XXI.Vol. 98.* Springer, pp. 447-455.

Spillane, N., Dolean, V., Hauret, P., Nataf, F., Pechstein, C. and Scheichl, R., 2014. Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps. *Numerische Mathematik*, 126 (4), pp. 741-770.

Ferreiro-Castilla, A., Kyprianou, A.E., Scheichl, R. and Suryanarayana, G., 2014. Multilevel Monte Carlo simulation for Lévy processes based on the Wiener–Hopf factorisation. *Stochastic Processes and their Applications*, 124 (2), pp. 985-1010.

Teckentrup, A.L., Scheichl, R., Giles, M.B. and Ullmann, E., 2013. Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients. *Numerische Mathematik*, 125 (3), pp. 569-600.

Freitag, M. A., 2013. *Large Scale Inverse Problems:Computational Methods and Applications in the Earth Sciences.* Walter de Gruyter.

Mueller, E., Guo, X., Scheichl, R. and Shi, S., 2013. Matrix-free GPU implementation of a preconditioned conjugate gradient solver for anisotropic elliptic PDEs. *Computing and Visualization in Science*, 16 (2), pp. 41-58.

Norton, R.A. and Scheichl, R., 2013. Planewave expansion methods for photonic crystal fibres. *Applied Numerical Mathematics*, 63, pp. 88-104.

Dolean, V., Nataf, F., Spillane, N. and Scheichl, R., 2013. A two-level Schwarz preconditioner for heterogeneous problems. Berlin: Springer, pp. 87-94.

Charrier, J., Scheichl, R. and Teckentrup, A. L., 2013. Finite element error analysis of elliptic PDEs with random coefficients and its application to multilevel Monte Carlo methods. *SIAM Journal on Numerical Analysis (SINUM)*, 51 (1), pp. 322-352.

Pechstein, C., Sarkis, M. and Scheichl, R., 2013. New theoretical coefficient robustness results for FETI-DP. *In*: Bank, R., Holst, M., Widlund, O. and Xu, J., eds. *Domain Decomposition Methods in Science and Engineering XX.* Berlin: Springer, pp. 313-320.

Scheichl, R., 2013. Robust coarsening in multiscale PDEs. Berlin: Springer, pp. 51-62.

Pechstein, C. and Scheichl, R., 2013. Weighted Poincaré inequalities. *IMA Journal of Numerical Analysis*, 33 (2), pp. 652-686.

Dolean, V., Nataf, F., Spillane, N. and Scheichl, R., 2012. Analysis of a two-level Schwarz method with coarse spaces based on local Dirichlet-to-Neumann maps. *Computational Methods in Applied Mathematics*, 12 (4), pp. 391-414.

Bastian, P., Blatt, M. and Scheichl, R., 2012. Algebraic multigrid for discontinuous Galerkin discretizations of heterogeneous elliptic problems. *Numerical Linear Algebra with Applications*, 19 (2), pp. 367-388.

Scheichl, R., Vassilevski, P. S. and Zikatanov, L. T., 2012. Multilevel methods for elliptic problems with highly varying coefficients on nonaligned coarse grids. *SIAM Journal on Numerical Analysis (SINUM)*, 50 (3), pp. 1675-1694.

Spillane, N., Dolean, V., Hauret, P., Nataf, F., Pechstein, C. and Scheichl, R., 2011. A robust two-level domain decomposition preconditioner for systems of PDEs. *Comptes Rendus Mathematique*, 349 (23-24), pp. 1255-1259.

Pechstein, C. and Scheichl, R., 2011. Analysis of FETI methods for multiscale PDEs. Part II: interface variation. *Numerische Mathematik*, 118 (3), pp. 485-529.

Graham, I. G., Kuo, F. Y., Nuyens, D., Scheichl, R. and Sloan, I. H., 2011. Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications. *Journal of Computational Physics*, 230 (10), pp. 3668-3694.

Buckeridge, S., Cullen, M. J. P., Scheichl, R. and Wlasak, M., 2011. A robust numerical method for the potential vorticity based control variable transform in variational data assimilation. *Quarterly Journal of the Royal Meteorological Society*, 137 (657), pp. 1083-1094.

Cliffe, K. A., Giles, M. B., Scheichl, R. and Teckentrup, A. L., 2011. Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients. *Computing and Visualization in Science*, 14 (1), pp. 3-15.

Scheichl, R., Vassilevski, P. S. and Zikatanov, L. T., 2011. Weak approximation properties of elliptic projections with functional constraints. *Multiscale Modeling and Simulation*, 9 (4), pp. 1677-1699.

Buckeridge, S. and Scheichl, R., 2010. Parallel geometric multigrid for global weather prediction. *Numerical Linear Algebra with Applications*, 17 (2-3), pp. 325-342.

Norton, R. and Scheichl, R., 2010. Convergence analysis of planewave expansion methods for 2D Schrodinger operators with discontinuous periodic potentials. *SIAM Journal on Numerical Analysis (SINUM)*, 47 (6), pp. 4356-4380.

Pechstein, C. and Scheichl, R., 2010. Weighted Poincaré inequalities and applications in domain decomposition. *In*: *Domain Decomposition Methods in Science and Engineering XIX.Vol. 78.* Heidelberg: Springer, pp. 197-204.

Van Lent, J., Scheichl, R. and Graham, I. G., 2009. Energy-minimizing coarse spaces for two-level Schwarz methods for multiscale PDEs. *Numerical Linear Algebra with Applications*, 16 (10), pp. 775-799.

Pechstein, C. and Scheichl, R., 2009. Scaling up through domain decomposition. *Applicable Analysis*, 88 (10-11), pp. 1589-1608.

Pechstein, C. and Scheichl, R., 2008. Analysis of FETI methods for multiscale PDEs. *Numerische Mathematik*, 111 (2), pp. 293-333.