Direct and Inverse Problems for Models with Uncertain Data
Modelling realworld physical systems always includes uncertainties (at least implicitly) regarding the measurement and determination of the data of the model. This for instance involves material properties, loading and also domains. We are concerned with infinite dimensional space dependent systems which are described by PDE. Typical applications are the simulation of mechanical systems and of groundwater flow in porous media. Since the solutions directly depend on the data, the measurement of this dependence in probabilistic terms is required in order to asses the reliability of the calculated solution. By this, it becomes feasible to deduce and quantify the solution uncertainty depending on data uncertainties.Illustration of forward propagation of uncertainties and inverse parameter identification.
 projection methods based on polynomial chaos (stochastic Galerkin FEM and, in wider sense, also collocation und regression)
 modern sampling methods (multilevel and Quasi Monte Carlo)
 methods based on stochastic differential equations (SDEs)
Adaptive Functional Representations and Modern Compression Techniques
The evaluation of forward or inverse stochastic problems often can be accellerated significantly (in terms of convergence rates) by employing functional representations. Moreover, such a representation allows for a numerical analysis, which resembles and extends concepts known from deterministic PDEs. With discretizations in generalized polynomial chaos as e.g. used for stochastic Galerkin methods, (reliable) a posteriori error estimates can be computed. These iteratively lead to problemadapted solution spaces with optimal convergence. Since the discrete algebraic systems exhibit a high dimensionality, model reduction techniques are often inevitable. Apart from the adaptation of the discrete space, we employ modern hierarchical tensor methods for the compression of the operators and the parametric solutions. This is tightly related to a possible representation as lowrank manifold. The tensor train (TT) format has proven advantageous for this. Applications for the developed methods can e.g. be found with samplingfree Bayesian inversion, topology optimization under uncertainties, and the determination of effective material models in case of media with multiscale properties.Mesh adaptivity with oscillating coefficient field.
SDE and Samplingbased Methods for Random PDEs
As an alternative to Monte Carlo methods, we investigate methods which exploit the equivalence of PDEs with random data and stochastic differential equations (FeynmanKac). Such highly parallelizable approaches typically require the reconstruction of a global solution representation. However, they allow for a complete separate and local control of all discretization parameters. To be more specific, pointwise solution realizations in the physical domain are determined by some appropriate numerical method (EulerMaruyama). Subsequently these are either used for a global or local polynomial regression or an interpolation on a mesh.Regression approach for the pointwise solution of an SDE equivalent to the random PDE.
Publications
Monographs

P. Deuflhard, M. Grötschel, D. Hömberg, U. Horst, J. Kramer, V. Mehrmann, K. Polthier, F. Schmidt, Ch. Schütte, M. Skutella, J. Sprekels, eds., MATHEON  Mathematics for Key Technologies, 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, 453 pages, (Collection Published).
Articles in Refereed Journals

M. Drieschner, M. Eigel, R. Gruhlke, D. Hömberg, Y. Petryna, Comparison of various uncertainty models with experimental investigations regarding the failure of plates with holes, Reliability Engineering and System Safety, 203 (2020), pp. 107106/1107106/12, DOI 10.1016/j.ress.2020.107106 .
Abstract
Unavoidable uncertainties due to natural variability, inaccuracies, imperfections or lack of knowledge are always present in real world problems. To take them into account within a numerical simulation, the probability, possibility or fuzzy set theory as well as a combination of these are potentially usable for the description and quantification of uncertainties. In this work, different monomorphic and polymorphic uncertainty models are applied on linear elastic structures with nonperiodic perforations in order to analyze the individual usefulness and expressiveness. The first principal stress is used as an indicator for structural failure which is evaluated and classified. In addition to classical sampling methods, a surrogate model based on artificial neural networks is presented. With regard to accuracy, efficiency and resulting numerical predictions, all methods are compared and assessed with respect to the added value. Real experiments of perforated plates under uniaxial tension are validated with the help of the different uncertainty models. 
CH. Bayer, D. Belomestny, M. Redmann, S. Riedel, J.G.M. Schoenmakers, Solving linear parabolic rough partial differential equations, Journal of Mathematical Analysis and Applications, 490 (2020), 124236, DOI 10.1016/j.jmaa.2020.124236 .
Abstract
We study linear rough partial differential equations in the setting of [Friz and Hairer, Springer, 2014, Chapter 12]. More precisely, we consider a linear parabolic partial differential equation driven by a deterministic rough path W of Hölder regularity α with ⅓ < α ≤ ½ . Based on a stochastic representation of the solution of the rough partial differential equation, we propose a regression Monte Carlo algorithm for spatiotemporal approximation of the solution. We provide a full convergence analysis of the proposed approximation method which essentially relies on the new bounds for the higher order derivatives of the solution in space. Finally, a comprehensive simulation study showing the applicability of the proposed algorithm is presented. 
G. Dong, H. Guo, Parametric polynomial preserving recovery on manifolds, SIAM Journal on Scientific Computing, 42 (2020), pp. A1885A1912, DOI 10.1137/18M1191336 .

M. Eigel, M. Marschall, M. Multerer, An adaptive stochastic Galerkin tensor train discretization for randomly perturbed domains, SIAM/ASA Journal on Uncertainty Quantification, 8 (2020), pp. 11891214, DOI 10.1137/19M1246080 .
Abstract
A linear PDE problem for randomly perturbed domains is considered in an adaptive Galerkin framework. The perturbation of the domain's boundary is described by a vector valued random field depending on a countable number of random variables in an affine way. The corresponding KarhunenLoeve expansion is approximated by the pivoted Cholesky decomposition based on a prescribed covariance function. The examined highdimensional Galerkin system follows from the domain mapping approach, transferring the randomness from the domain to the diffusion coefficient and the forcing. In order to make this computationally feasible, the representation makes use of the modern tensor train format for the implicit compression of the problem. Moreover, an a posteriori error estimator is presented, which allows for the problemdependent iterative refinement of all discretization parameters and the assessment of the achieved error reduction. The proposed approach is demonstrated in numerical benchmark problems. 
M. Eigel, M. Marschall, M. Pfeffer, R. Schneider, Adaptive stochastic Galerkin FEM for lognormal coefficients in hierarchical tensor representations, Numerische Mathematik, 145 (2020), pp. 655692, DOI 10.1007/s00211020011231 .
Abstract
Stochastic Galerkin methods for nonaffine coefficient representations are known to cause major difficulties from theoretical and numerical points of view. In this work, an adaptive Galerkin FE method for linear parametric PDEs with lognormal coefficients discretized in Hermite chaos polynomials is derived. It employs problemadapted function spaces to ensure solvability of the variational formulation. The inherently high computational complexity of the parametric operator is made tractable by using hierarchical tensor representations. For this, a new tensor train format of the lognormal coefficient is derived and verified numerically. The central novelty is the derivation of a reliable residualbased a posteriori error estimator. This can be regarded as a unique feature of stochastic Galerkin methods. It allows for an adaptive algorithm to steer the refinements of the physical mesh and the anisotropic Wiener chaos polynomial degrees. For the evaluation of the error estimator to become feasible, a numerically efficient tensor format discretization is developed. Benchmark examples with unbounded lognormal coefficient fields illustrate the performance of the proposed Galerkin discretization and the fully adaptive algorithm. 
I. Papaioannou, M. Daub, M. Drieschner, F. Duddeck, M. Ehre, L. Eichner, M. Eigel, M. Götz, W. Graf, L. Grasedyck, R. Gruhlke, D. Hömberg, M. Kaliske, D. Moser, Y. Petryna, D. Straub, Assessment and design of an engineering structure with polymorphic uncertainty quantification, GAMMMitteilungen, 42 (2019), pp. e201900009/1e201900009/22, DOI 10.1002/gamm.201900009 .

D. Pivovarov, K. Willner, P. Steinmann, S. Brumme, M. Müller, T. Srisupattarawanit, G.P. Ostermeyer, C. Henning, T. Ricken, S. Kastian, S. Reese, D. Moser, L. Grasedyck, J. Biehler, M. Pfaller, W. Wall, Th. Kolsche, O. VON Estorff, R. Gruhlke, M. Eigel, M. Ehre, I. Papaioannou, D. Straub, S. Leyendecker, Challenges of order reduction techniques for problems involving polymorphic uncertainty, GAMMMitteilungen, 42 (2019), pp. e201900011/1e201900011/24.

M. Eigel, R. Schneider, P. Trunschke, S. Wolf, Variational Monte CarloBridging concepts of machine learning and high dimensional partial differential equations, Advances in Computational Mathematics, 45 (2019), pp. 25032532, DOI 10.1007/s10444019097238 .
Abstract
A statistical learning approach for parametric PDEs related to Uncertainty Quantification is derived. The method is based on the minimization of an empirical risk on a selected model class and it is shown to be applicable to a broad range of problems. A general unified convergence analysis is derived, which takes into account the approximation and the statistical errors. By this, a combination of theoretical results from numerical analysis and statistics is obtained. Numerical experiments illustrate the performance of the method with the model class of hierarchical tensors. 
M. Eigel, J. Neumann, R. Schneider, S. Wolf, Risk averse stochastic structural topology optimization, Computer Methods in Applied Mechanics and Engineering, 334 (2018), pp. 470482, DOI 10.1016/j.cma.2018.02.003 .
Abstract
A novel approach for riskaverse structural topology optimization under uncertainties is presented which takes into account random material properties and random forces. For the distribution of material, a phase field approach is employed which allows for arbitrary topological changes during optimization. The state equation is assumed to be a highdimensional PDE parametrized in a (finite) set of random variables. For the examined case, linearized elasticity with a parametric elasticity tensor is used. Instead of an optimization with respect to the expectation of the involved random fields, for practical purposes it is important to design structures which are also robust in case of events that are not the most frequent. As a common riskaware measure, the Conditional Value at Risk (CVaR) is used in the cost functional during the minimization procedure. Since the treatment of such highdimensional problems is a numerically challenging task, a representation in the modern hierarchical tensor train format is proposed. In order to obtain this highly efficient representation of the solution of the random state equation, a tensor completion algorithm is employed which only required the pointwise evaluation of solution realizations. The new method is illustrated with numerical examples and compared with a classical Monte Carlo sampling approach. 
M. Eigel, M. Marschall, R. Schneider, Samplingfree Bayesian inversion with adaptive hierarchical tensor representations, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 34 (2018), pp. 035010/1035010/29, DOI 10.1088/13616420/aaa998 .
Abstract
The statistical Bayesian approach is a natural setting to resolve the illposedness of inverse problems by assigning probability densities to the considered calibration parameters. Based on a parametric deterministic representation of the forward model, a samplingfree approach to Bayesian inversion with an explicit representation of the parameter densities is developed. The approximation of the involved randomness inevitably leads to several high dimensional expressions, which are often tackled with classical sampling methods such as MCMC. To speed up these methods, the use of a surrogate model is beneficial since it allows for faster evaluation with respect to calibration parameters. However, the inherently slow convergence can not be remedied by this. As an alternative, a complete functional treatment of the inverse problem is feasible as demonstrated in this work, with functional representations of the parametric forward solution as well as the probability densities of the calibration parameters, determined by Bayesian inversion. The proposed samplingfree approach is discussed in the context of hierarchical tensor representations, which are employed for the adaptive evaluation of a random PDE (the forward problem) in generalized chaos polynomials and the subsequent highdimensional quadrature of the loglikelihood. This modern compression technique alleviates the curse of dimensionality by hierarchical subspace approximations of the involved low rank (solution) manifolds. All required computations can be carried out efficiently in the lowrank format. A priori convergence is examined, considering all approximations that occur in the method. Numerical experiments demonstrate the performance and verify the theoretical results. 
L. Donati, M. Heida, M. Weber, B. Keller, Estimation of the infinitesimal generator by squareroot approximation, Journal of Physics: Condensed Matter, 30 (2018), pp. 425201/1425201/14, DOI 10.1088/1361648X/aadfc8 .
Abstract
For the analysis of molecular processes, the estimation of timescales, i.e., transition rates, is very important. Estimating the transition rates between molecular conformations is  from a mathematical point of view  an invariant subspace projection problem. A certain infinitesimal generator acting on function space is projected to a lowdimensional rate matrix. This projection can be performed in two steps. First, the infinitesimal generator is discretized, then the invariant subspace is approximated and used for the subspace projection. In our approach, the discretization will be based on a Voronoi tessellation of the conformational space. We will show that the discretized infinitesimal generator can simply be approximated by the geometric average of the Boltzmann weights of the Voronoi cells. Thus, there is a direct correlation between the potential energy surface of molecular structures and the transition rates of conformational changes. We present results for a 2ddiffusion process and Alanine dipeptide. 
M. Eigel, J. Neumann, R. Schneider, S. Wolf, Nonintrusive tensor reconstruction for high dimensional random PDEs, Computational Methods in Applied Mathematics, 19 (2019), pp. 3953 (published online on 25.07.2018), DOI 10.1515/cmam20180028 .
Abstract
This paper examines a completely nonintrusive, samplebased method for the computation of functional lowrank solutions of high dimensional parametric random PDEs which have become an area of intensive research in Uncertainty Quantification (UQ). In order to obtain a generalized polynomial chaos representation of the approximate stochastic solution, a novel blackbox rankadapted tensor reconstruction procedure is proposed. The performance of the described approach is illustrated with several numerical examples and compared to Monte Carlo sampling. 
F. Anker, Ch. Bayer, M. Eigel, M. Ladkau, J. Neumann, J.G.M. Schoenmakers, SDE based regression for random PDEs, SIAM Journal on Scientific Computing, 39 (2017), pp. A1168A1200.
Abstract
A simulation based method for the numerical solution of PDE with random coefficients is presented. By the FeynmanKac formula, the solution can be represented as conditional expectation of a functional of a corresponding stochastic differential equation driven by independent noise. A time discretization of the SDE for a set of points in the domain and a subsequent Monte Carlo regression lead to an approximation of the global solution of the random PDE. We provide an initial error and complexity analysis of the proposed method along with numerical examples illustrating its behaviour. 
F. Anker, Ch. Bayer, M. Eigel, J. Neumann, J.G.M. Schoenmakers, A fully adaptive interpolated stochastic sampling method for linear random PDEs, International Journal for Uncertainty Quantification, 7 (2017), pp. 189205, DOI 10.1615/Int.J.UncertaintyQuantification.2017019428 .
Abstract
A numerical method for the fully adaptive sampling and interpolation of PDE with random data is presented. It is based on the idea that the solution of the PDE with stochastic data can be represented as conditional expectation of a functional of a corresponding stochastic differential equation (SDE). The physical domain is decomposed subject to a nonuniform grid and a classical Euler scheme is employed to approximately solve the SDE at grid vertices. Interpolation with a conforming finite element basis is employed to reconstruct a global solution of the problem. An a posteriori error estimator is introduced which provides a measure of the different error contributions. This facilitates the formulation of an adaptive algorithm to control the overall error by either reducing the stochastic error by locally evaluating more samples, or the approximation error by locally refining the underlying mesh. Numerical examples illustrate the performance of the presented novel method. 
M. Eigel, M. Pfeffer, R. Schneider, Adaptive stochastic Galerkin FEM with hierarchical tensor representations, Numerische Mathematik, 136 (2017), pp. 765803.
Abstract
The solution of PDE with stochastic data commonly leads to very highdimensional algebraic problems, e.g. when multiplicative noise is present. The Stochastic Galerkin FEM considered in this paper then suffers from the curse of dimensionality. This is directly related to the number of random variables required for an adequate representation of the random fields included in the PDE. With the presented new approach, we circumvent this major complexity obstacle by combining two highly efficient model reduction strategies, namely a modern lowrank tensor representation in the tensor train format of the problem and a refinement algorithm on the basis of a posteriori error estimates to adaptively adjust the different employed discretizations. The adaptive adjustment includes the refinement of the FE mesh based on a residual estimator, the problemadapted stochastic discretization in anisotropic Legendre Wiener chaos and the successive increase of the tensor rank. Computable a posteriori error estimators are derived for all error terms emanating from the discretizations and the iterative solution with a preconditioned ALS scheme of the problem. Strikingly, it is possible to exploit the tensor structure of the problem to evaluate all error terms very efficiently. A set of benchmark problems illustrates the performance of the adaptive algorithm with higherorder FE. Moreover, the influence of the tensor rank on the approximation quality is investigated. 
F. Lanzara, V. Maz'ya, G. Schmidt, A fast solution method for time dependent multidimensional Schrödinger equations, Applicable Analysis. An International Journal, published online on 08.08.2017, urlhttps://doi.org/10.1080/00036811.2017.1359571, DOI 10.1080/00036811.2017.1359571 .
Abstract
In this paper we propose fast solution methods for the Cauchy problem for the multidimensional Schrödinger equation. Our approach is based on the approximation of the data by the basis functions introduced in the theory of approximate approximations. We obtain high order approximations also in higher dimensions up to a small saturation error, which is negligible in computations, and we prove error estimates in mixed Lebesgue spaces for the inhomogeneous equation. The proposed method is very efficient in high dimensions if the densities allow separated representations. We illustrate the efficiency of the procedure on different examples, up to approximation order 6 and space dimension 200. 
M.H. Farshbaf Shaker, R. Henrion, D. Hömberg, Properties of chance constraints in infinite dimensions with an application to PDE constrained optimization, SetValued and Variational Analysis. Theory and Applications. Springer, Dordrecht. English., 26 (2018), pp. 821841 (published online on 11.10.2017), DOI 10.1007/s1122801704525 .
Abstract
Chance constraints represent a popular tool for finding decisions that enforce a robust satisfaction of random inequality systems in terms of probability. They are widely used in optimization problems subject to uncertain parameters as they arise in many engineering applications. Most structural results of chance constraints (e.g., closedness, convexity, Lipschitz continuity, differentiability etc.) have been formulated in a finitedimensional setting. The aim of this paper is to generalize some of these wellknown semicontinuity and convexity properties to a setting of control problems subject to (uniform) state chance constraints. 
M. Eigel, Ch. Merdon, J. Neumann, An adaptive multilevel MonteCarlo method with stochastic bounds for quantities of interest in groundwater flow with uncertain data, SIAM/ASA Journal on Uncertainty Quantification, 4 (2016), pp. 12191245.
Abstract
The focus of this work is the introduction of some computable a posteriori error control to the popular multilevel Monte Carlo sampling for PDE with stochastic data. We are especially interested in applications in the geosciences such as groundwater flow with rather rough stochastic fields for the conductive permeability. With a spatial discretisation based on finite elements, a goal functional is defined which encodes the quantity of interest. The devised goaloriented error estimator enables to determine guaranteed a posteriori error bounds for this quantity. In particular, it allows for the adaptive refinement of the mesh hierarchy used in the multilevel Monte Carlo simulation. In addition to controlling the deterministic error, we also suggest how to treat the stochastic error in probability. Numerical experiments illustrate the performance of the presented adaptive algorithm for a posteriori error control in multilevel Monte Carlo methods. These include a localised goal with problemadapted meshes and a slit domain example. The latter demonstrates the refinement of regions with low solution regularity based on an inexpensive explicit error estimator in the multilevel algorithm. 
M. Eigel, Ch. Merdon, Local equilibration error estimators for guaranteed error control in adaptive stochastic higherorder Galerkin finite element methods, SIAM/ASA Journal on Uncertainty Quantification, 4 (2016), pp. 13721397.
Abstract
Equilibration error estimators have been shown to commonly lead to very accurate guaranteed error bounds in the a posteriori error control of finite element methods for second order elliptic equations. Here, we extend previous results by the design of equilibrated fluxes for higherorder finite element methods with nonconstant coefficients and illustrate the favourable performance of different variants of the error estimator within two deterministic benchmark settings. After the introduction of the respective parametric problem with stochastic coefficients and the stochastic Galerkin FEM discretisation, a novel a posteriori error estimator for the stochastic error in the energy norm is devised. The error estimation is based on the stochastic residual and its decomposition into approximation residuals and a truncation error of the stochastic discretisation. Importantly, by using the derived deterministic equilibration techniques for the approximation residuals, the computable error bound is guaranteed for the considered class of problems. An adaptive algorithm allows the simultaneous refinement of the deterministic mesh and the stochastic discretisation in anisotropic Legendre polynomial chaos. Several stochastic benchmark problems illustrate the efficiency of the adaptive process. 
F. Lanzara, V. Maz'ya, G. Schmidt, Approximation of solutions to multidimensional parabolic equations by approximate approximations, Applied and Computational Harmonic Analysis. TimeFrequency and TimeScale Analysis, Wavelets, Numerical Algorithms, and Applications, 41 (2016), pp. 749767.

M. Eigel, C.J. Gittelson, Ch. Schwab, E. Zander, A convergent adaptive stochastic Galerkin finite element method with quasioptimal spatial meshes, ESAIM: Mathematical Modelling and Numerical Analysis, 49 (2015), pp. 13671398.
Abstract
We analyze aposteriori error estimation and adaptive refinement algorithms for stochastic Galerkin Finite Element methods for countablyparametric elliptic boundary value problems. A residual error estimator which separates the effects of gpcGalerkin discretization in parameter space and of the Finite Element discretization in physical space in energy norm is established. It is proved that the adaptive algorithm converges, and to this end we establish a contraction property satisfied by its iterates. It is shown that the sequences of triangulations which are produced by the algorithm in the FE discretization of the active gpc coefficients are asymptotically optimal. Numerical experiments illustrate the theoretical results. 
F. Lanzara, G. Schmidt, On the computation of highdimensional potentials of advectiondiffusion operators, Mathematika. A Journal of Pure and Applied Mathematics, 61 (2015), pp. 309327.

W. Giese, M. Eigel, S. Westerheide, Ch. Engwer, E. Klipp, Influence of cell shape, inhomogeneities and diffusion barriers in cell polarization models, Physical Biology, 12 (2015), pp. 066014/1066014/18.
Abstract
In silico experiments bear the potential to further the understanding of biological transport processes by allowing a systematic modification of any spatial property and providing immediate simulation results for the chosen models. We consider cell polarization and spatial reorganization of membrane proteins which are fundamental for cell division, chemotaxis and morphogenesis. Our computational study is motivated by mating and budding processes of S. cerevisiae. In these processes a key player during the initial phase of polarization is the GTPase Cdc42 which occurs in an active membranebound form and an inactive cytosolic form. We use partial differential equations to describe the membranecytosol shuttling of Cdc42 during budding as well as mating of yeast. The membrane is modeled as a thin layer that only allows lateral diffusion and the cytosol is modeled as a volume. We investigate how cell shape and diffusion barriers like septin structures or bud scars influence Cdc42 cluster formation and subsequent polarization of the yeast cell. Since the details of the binding kinetics of cytosolic proteins to the membrane are still controversial, we employ two conceptual models which assume different binding kinetics. An extensive set of in silico experiments with different modeling hypotheses illustrate the qualitative dependence of cell polarization on local membrane curvature, cell size and inhomogeneities on the membrane and in the cytosol. We examine that spatial inhomogenities essentially determine the location of Cdc42 cluster formation and spatial properties are crucial for the realistic description of the polarization process in cells. In particular, our computer simulations suggest that diffusion barriers are essential for the yeast cell to grow a protrusion. 
TH. Arnold, A. Rathsfeld, Reflection of plane waves by rough surfaces in the sense of Born approximation, Mathematical Methods in the Applied Sciences, 37 (2014), pp. 20912111.
Abstract
The topic of the present paper is the reflection of electromagnetic plane waves by rough surfaces, i.e., by smooth and bounded perturbations of planar faces. Moreover, the contrast between the cover material and the substrate beneath the rough surface is supposed to be low. In this case, a modification of Stearns' formula based on Born approximation and Fourier techniques is derived for a special class of surfaces. This class contains the graphs of functions if the interface function is a radially modulated almost periodic function. For the Born formula to converge, a sufficient and almost necessary condition is given. A further technical condition is defined, which guarantees the existence of the corresponding far field of the Born approximation. This far field contains plane waves, farfield terms like those for bounded scatterers, and, additionally, a new type of terms. The derived formulas can be used for the fast numerical computations of far fields and for the statistics of random rough surfaces. 
M. Eigel, C. Gittelson, Ch. Schwab, E. Zander, Adaptive stochastic Galerkin FEM, Computer Methods in Applied Mechanics and Engineering, 270 (2014), pp. 247269.

F. Lanzara, V. Maz'ya, G. Schmidt, Fast cubature of volume potentials over rectangular domains by approximate approximations, Applied and Computational Harmonic Analysis. TimeFrequency and TimeScale Analysis, Wavelets, Numerical Algorithms, and Applications, 36 (2014), pp. 167182.
Abstract
In the present paper we study highorder cubature formulas for the computation of advectiondiffusion potentials over boxes. By using the basis functions introduced in the theory of approximate approximations, the cubature of a potential is reduced to the quadrature of one dimensional integrals. For densities with separated approximation, we derive a tensor product representation of the integral operator which admits efficient cubature procedures in very high dimensions. Numerical tests show that these formulas are accurate and provide approximation of order O(h^{6}) up to dimension 10^{8}.
Contributions to Collected Editions

CH. Bayer, H. Oberhauser, Splitting methods for SPDEs: From robustness to financial engineering, optimal control and nonlinear filtering, in: Splitting Methods in Communication, Imaging, Science, and Engineering, R. Glowinski, S.J. Osher, W. Yin, eds., Scientific Computation, Springer International Publishing Switzerland, Cham, 2016, pp. 499539.
Abstract
In this survey chapter we give an overview of recent applications of the splitting method to stochastic (partial) differential equations, that is, differential equations that evolve under the influence of noise. We discuss weak and strong approximations schemes. The applications range from the management of risk, financial engineering, optimal control and nonlinear filtering to the viscosity theory of nonlinear SPDEs.
Preprints, Reports, Technical Reports

M. Eigel, O. Ernst, B. Sprungk, L. Tamellini, On the convergence of adaptive stochastic collocation for elliptic partial differential equations with affine diffusion, Preprint no. 2753, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2753 .
Abstract, PDF (325 kByte)
Convergence of an adaptive collocation method for the stationary parametric diffusion equation with finitedimensional affine coefficient is shown. The adaptive algorithm relies on a recently introduced residualbased reliable a posteriori error estimator. For the convergence proof, a strategy recently used for a stochastic Galerkin method with an hierarchical error estimator is transferred to the collocation setting. 
S. Riedel, Semiimplicit Taylor schemes for stiff rough differential equations, Preprint no. 2734, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2734 .
Abstract, PDF (538 kByte)
We study a class of semiimplicit Taylortype numerical methods that are easy to implement and designed to solve multidimensional stochastic differential equations driven by a general rough noise, e.g. a fractional Brownian motion. In the multiplicative noise case, the equation is understood as a rough differential equation in the sense of T. Lyons. We focus on equations for which the drift coefficient may be unbounded and satisfies a onesided Lipschitz condition only. We prove wellposedness of the methods, provide a full analysis, and deduce their convergence rate. Numerical experiments show that our schemes are particularly useful in the case of stiff rough stochastic differential equations driven by a fractional Brownian motion. 
M. Redmann, S. Riedel, RungeKutta methods for rough differential equations, Preprint no. 2708, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2708 .
Abstract, PDF (393 kByte)
We study RungeKutta methods for rough differential equations which can be used to calculate solutions to stochastic differential equations driven by processes that are rougher than a Brownian motion. We use a Taylor series representation (Bseries) for both the numerical scheme and the solution of the rough differential equation in order to determine conditions that guarantee the desired order of the local error for the underlying RungeKutta method. Subsequently, we prove the order of the global error given the local rate. In addition, we simplify the numerical approximation by introducing a RungeKutta scheme that is based on the increments of the driver of the rough differential equation. This simplified method can be easily implemented and is computational cheap since it is derivativefree. We provide a full characterization of this implementable RungeKutta method meaning that we provide necessary and sufficient algebraic conditions for an optimal order of convergence in case that the driver, e.g., is a fractional Brownian motion with Hurst index 1/4 < H ≤ 1/2. We conclude this paper by conducting numerical experiments verifying the theoretical rate of convergence. 
M. Hintermüller, S.M. Stengl, Th.M. Surowiec, Uncertainty quantification in image segmentation using the AmbrosioTortorelli approximation of the MumfordShah energy, Preprint no. 2703, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2703 .
Abstract, PDF (930 kByte)
The quantification of uncertainties in image segmentation based on the MumfordShah model is studied. The aim is to address the error propagation of noise and other error types in the original image to the restoration result and especially the reconstructed edges (sharp image contrasts). Analytically, we rely on the AmbrosioTortorelli approximation and discuss the existence of measurable selections of its solutions as well as samplingbased methods and the limitations of other popular methods. Numerical examples illustrate the theoretical findings. 
D. Gahururu, M. Hintermüller, S.M. Stengl, Th.M. Surowiec, Generalized Nash equilibrium problems with partial differential operators: Theory, algorithms, and risk aversion, Preprint no. 2654, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2654 .
Abstract, PDF (337 kByte)
PDEconstrained (generalized) Nash equilibrium problems (GNEPs) are considered in a deterministic setting as well as under uncertainty. This includes a study of deterministic GNEPs with nonlinear and/or multivalued operator equations as forward problems and PDEconstrained GNEPs with uncertain data. The deterministic nonlinear problems are analyzed using the theory of generalized convexity for setvalued operators, and a variational approximation approach is proposed. The stochastic setting includes a detailed overview of the recently developed theory and algorithms for riskaverse PDEconstrained optimization problems. These new results open the way to a rigorous study of stochastic PDEconstrained GNEPs. 
M. Eigel, L. Grasedyck, R. Gruhlke, D. Moser, Low rank surrogates for polymorphic fields with application to fuzzystochastic partial differential equations, Preprint no. 2580, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2580 .
Abstract, PDF (1235 kByte)
We consider a general form of fuzzystochastic PDEs depending on the interaction of probabilistic and nonprobabilistic ("possibilistic") influences. Such a combined modelling of aleatoric and epistemic uncertainties for instance can be applied beneficially in an engineering context for realworld applications, where probabilistic modelling and expert knowledge has to be accounted for. We examine existence and welldefinedness of polymorphic PDEs in appropriate function spaces. The fuzzystochastic dependence is described in a highdimensional parameter space, thus easily leading to an exponential complexity in practical computations. To aleviate this severe obstacle in practise, a compressed lowrank approximation of the problem formulation and the solution is derived. This is based on the Hierarchical Tucker format which is constructed with solution samples by a nonintrusive tensor reconstruction algorithm. The performance of the proposed model order reduction approach is demonstrated with two examples. One of these is the ubiquitous groundwater flow model with KarhunenLoeve coefficient field which is generalized by a fuzzy correlation length.
Talks, Poster

S. Riedel, RungeKutta methods for rough differential equations (online talk), The DNA Seminar (spring 2020), Norwegian University of Science and Technology, Department of Mathematical Sciences, Trondheim, Norway, June 24, 2020.

S.M. Stengl, Uncertainty quantification of the AmbrosioTortorelli approximation in image segmentation, Workshop on PDE Constrained Optimization under Uncertainty and Mean Field Games, January 28  30, 2020, WIAS, Berlin, January 30, 2020.

R. Gruhlke, Bayesian upscaling with application to failure analysis of adhesive bonds in rotor blades, 3rd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECOMP 2019), Minisymposium 6II ``Uncertainty Computations with Reduced Order Models and LowRank Representations'', June 24  26, 2019, Crete, Greece, June 24, 2019.

M. Eigel, A machine learning approach for explicit Bayesian inversion, Workshop 3 within the Special Semester on Optimization ``Optimization and Inversion under Uncertainty'', November 11  15, 2019, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Linz, Austria, November 12, 2019.

M. Eigel, A statistical learning approach for highdimensional PDEs, 3rd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECOMP 2019), Minisymposium 6IV ``Uncertainty Computations with Reduced Order Models and LowRank Representations'', June 24  26, 2019, Crete, Greece, June 25, 2019.

M. Eigel, A statistical learning approach for parametric PDEs, Workshop ``Scientific Computation using MachineLearning Algorithms'', April 25  26, 2019, University of Nottingham, UK, April 26, 2019.

M. Eigel, A statistical learning approach for parametric PDEs, École Polytechnique Fédérale de Lausanne (EPFL), Scientific Computing and Uncertainty Quantification, Lausanne, Switzerland, May 14, 2019.

M. Eigel, Some thoughts on adaptive stochastic Galerkin FEM, Sixteenth Conference on the Mathematics of Finite Elements and Applications (MAFELAP 2019), Minisymposium 17 ``Finite Element Methods for Efficient Uncertainty Quantification'', June 18  21, 2019, Brunel University London, Uxbridge, UK, June 18, 2019.

M. Marschall, Adaptive lowrank approximation in Bayesian inverse problems, 3rd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECOMP 2019), Minisymposium 6IV ``Uncertainty Computations with Reduced Order Models and LowRank Representations'', June 24  26, 2019, Crete, Greece, June 25, 2019.

M. Marschall, Lowrank surrogates in Bayesian inverse problems, 19th FrenchGermanSwiss Conference on Optimization (FGS'2019), Minisymposium 1 ``Recent Trends in Nonlinear Optimization 1'', September 17  20, 2019, Nice, France, September 17, 2019.

M. Marschall, Random domains in PDE problems with lowrank surrogates. Forward and backward, PhysikalischTechnische Bundesanstalt, Arbeitsgruppe 8.41 ``Mathematische Modellierung und Datenanalyse'', Berlin, April 10, 2019.

M. Eigel, Aspects of adaptive Galerkin FE for stochastic direct and inverse problems, Workshop ``Surrogate Models for UQ in Complex Systems'' (UNQW02), February 5  9, 2018, Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, February 7, 2018.

S.M. Stengl, Uncertainty quantification of the AmbrosioTortorelli approximation in image segmentation, MIA 2018  Mathematics and Image Analysis, HumboldtUniversität zu Berlin, January 15  17, 2018.

M. Eigel, R. Gruhlke, Domain decomposition for random highdimensional PDEs, Workshop ``Reducing Dimensions and Cost for UQ in Complex Systems'', Cambridge, UK, March 5  9, 2018.

M. Eigel, Adaptive Galerkin FEM for stochastic forward and inverse problems, Optimisation and Numerical Analysis Seminars, University of Birmingham, School of Mathematics, UK, February 15, 2018.

M. Eigel, Adaptive tensor methods for forward and inverse problems, SIAM Conference on Uncertainty Quantification (UQ18), Minisymposium 122 ``LowRank Approximations for the Forward and the Inverse Problems III'', April 16  19, 2018, Garden Grove, USA, April 19, 2018.

M. Marschall, Bayesian inversion with adaptive lowrank approximation, Analysis, Control and Inverse Problems for PDEs  Workshop of the FrenchGermanItalian LIA (Laboratoire International Associe) COPDESC on Applied Analysis, November 26  30, 2018, University of Naples Federico II and Accademia Pontaniana, Italy, November 29, 2018.

M. Eigel, A samplingfree adaptive Bayesian inversion with hierarchical tensor representations, European Conference on Numerical Mathematics and Advanced Applications (ENUMATH 2017), Minisymposium 15 ``Uncertainty Propagation'', September 25  29, 2017, Voss, Norway, September 27, 2017.

M. Eigel, Adaptive stochastic FE for explicit Bayesian inversion with hierarchical tensor representations, Institut National de Recherche en Informatique et en Automatique (INRIA), SERENA (Simulation for the Environment: Reliable and Efficient Numerical Algorithms) research team, Paris, France, June 1, 2017.

M. Eigel, Adaptive stochastic Galerkin FE and tensor compression for random PDEs, sc Matheon Workshop ``Reliable Methods of Mathematical Modeling'' (RMMM8), July 31  August 4, 2017, HumboldtUniversität zu Berlin, August 3, 2017.

M. Eigel, Aspects of stochastic Galerkin FEM, Universität Basel, Mathematisches Institut, Switzerland, November 10, 2017.

M. Eigel, Efficient Bayesian inversion with hierarchical tensor representation, 2nd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECOMP 2017), June 15  17, 2017, Rhodos, Greece, June 16, 2017.

M. Eigel, Explicit Bayesian inversion in hierarchical tensor representations, 4th GAMM Junior's and 1st GRK2075 Summer School 2017 ``Bayesian Inference: Probabilistic Way of Learning from Data'', July 10  14, 2017, Braunschweig, July 14, 2017.

M. Eigel, Stochastic topology optimization with hierarchical tensor reconstruction, Frontiers of Uncertainty Quantification in Engineering (FrontUQ 2017), September 6  8, 2017, München, September 7, 2017.

R. Gruhlke, Multiscale failure analysis with polymorphic uncertainties for optimal design of rotor blades, Frontiers of Uncertainty Quantification in Engineering (FrontUQ 2017), September 6  8, 2017, München, September 6, 2017.

M. Marschall, Bayesian inversion using hierarchical tensors, 88th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2017), Section S15 ``Uncertainty Quantification'', March 6  10, 2017, Bauhaus Universität Weimar/Technische Universität Ilmenau, Weimar, March 8, 2017.

M. Marschall, Samplingfree Bayesian inversion with adaptive hierarchical tensor representation, Frontiers of Uncertainty Quantification in Engineering (FrontUQ 2017), September 6  8, 2017, München, September 7, 2017.

M. Marschall, Samplingfree Bayesian inversion with adaptive hierarchical tensor representation, International Conference on Scientific Computation and Differential Equations (SciCADE2017), MS21 ``Tensor Approximations of MultiDimensional PDEs'', September 11  15, 2017, University of Bath, UK, September 14, 2017.

M. Eigel, Adaptive stochastic Galerkin FEM with hierarchical tensor representations, Advances in Uncertainty Quantification Methods, Algorithms and Applications (UQAW 2016), January 5  10, 2016, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia, January 8, 2016.

M. Eigel, Adaptive stochastic Galerkin FEM with hierarchical tensor representations, 15th Conference on the Mathematics of Finite Elements and Applications (Brunel MAFELAP 2016), Minisymposium ``Uncertainty Quantification Using Stochastic PDEs and Finite Elements'', June 14  17, 2016, Brunel University London, Uxbridge, UK, June 14, 2016.

M. Eigel, Adaptive stochastic Galerkin FEM with hierarchical tensor representations, Joint Annual Meeting of DMV and GAMM, Section 18 ``Numerical Methods of Differential Equations'', March 7  11, 2016, Technische Universität Braunschweig, March 10, 2016.

M. Eigel, Bayesian inversion using hierarchical tensor approximations, SIAM Conference on Uncertainty Quantification, Minisymposium 67 ``Bayesian Inversion and Lowrank Approximation (Part II)'', April 5  8, 2016, Lausanne, Switzerland, April 6, 2016.

M. Eigel, Some aspects of adaptive random PDEs, Oberseminar, RheinischWestfälische Technische Hochschule Aachen, Institut für Geometrie und Praktische Mathematik, July 21, 2016.

J. Neumann, Adaptive SDE based sampling for random PDE, SIAM Conference on Uncertainty Quantification, Minisymposium 142 ``Error Estimation and Adaptive Methods for Uncertainty Quantification in Computational Sciences  Part II'', April 5  8, 2016, Lausanne, Switzerland, April 8, 2016.

J. Neumann, The phase field approach for topology optimization under uncertainties, ZIB Computational Medicine and Numerical Mathematics Seminar, KonradZuseZentrum für Informationstechnik Berlin, August 25, 2016.

J. Pellerin, RINGMesh: A programming library for geological model meshes, The 17th annual conference of the International Association for Mathematical Geosciences, September 5  13, 2015, Freiberg, September 8, 2015.

M. Eigel, Adaptive stochastic Galerkin FEM with hierarchical tensor representations, 2nd GAMM AGUQ Workshop on Uncertainty Quantification, September 10  11, 2015, Chemnitz, September 10, 2015.

M. Eigel, Fully adaptive higherorder stochastic Galerkin FEM in lowrank tensor representation, International Conference on Scientific Computation And Differential Equations (SciCADE 2015), September 14  18, 2015, Universität Potsdam, September 15, 2015.

M. Eigel, Guaranteed error bounds for adaptive stochastic Galerkin FEM, Technische Universität Braunschweig, Institut für Wissenschaftliches Rechnen, April 1, 2015.

M. Eigel, Stochastic adaptive FEM, Forschungsseminar Numerische Mathematik, HumboldtUniversität zu Berlin, Institut für Mathematik, January 28, 2015.

CH. Bayer, SDE based regression for random PDEs, Direct and Inverse Problems for PDEs with Random Coefficients, WIAS Berlin, November 13, 2015.

M. Eigel, A posteriori error control in stochastic FEM and MLMC, 27th Chemnitz FEM Symposium 2014, September 22  24, 2014, September 24, 2014.

M. Eigel, Adaptive spectral methods for stochastic optimisation problems, Technische Universität Berlin, Institut für Mathematik, May 22, 2014.

M. Eigel, Adaptive stochastic FEM, Universität Heidelberg, Interdisziplinäres Zentrum für Wissenschaftliches Rechnen (IWR), June 5, 2014.

M. Eigel, Guaranteed a posteriori error control with adaptive stochastic Galerkin FEM, SIAM Conference on Uncertainty Quantification (UQ14), March 31  April 3, 2014, Savannah, USA, April 1, 2014.

M. Ladkau, Brownian motion approach for spatial PDEs with stochastic data, International Workshop ``Advances in Optimization and Statistics'', May 15  16, 2014, Russian Academy of Sciences, Institute of Information Transmission Problems (Kharkevich Institute), Moscow, May 16, 2014.

J. Neumann, A posteriori error estimators for problems with uncertain data, Norddeutsches Kolloquium über Angewandte Analysis und Numerische Mathematik (NoKo), ChristianAlbrechtsUniversität zu Kiel, May 10, 2014.

J. Neumann, Stochastic bounds for quantities of interest in groundwater flow with uncertain data, Université ParisSud, Laboratoire d'Analyse Numérique, Orsay, France, October 9, 2014.