Kathrin Smetana

Unsaturated subsurface flow with surface water and nonlinear in- and outflow conditions

We analytically and numerically analyze groundwater flow in a homogeneous soil described by the Richards equation, coupled to surface water represented by a set of ordinary dierential equations (ODE’s) on parts of the domain boundary, and with nonlinear outflow conditions of Signorini’s type. The coupling of the partial dierential equation (PDE) and the ODE’s is given by nonlinear Robin boundary conditions. This article provides two major new contributions regarding these infiltration conditions. First, an existence result for the continuous coupled problem is established with the help of a regularization technique. Second, we analyze and validate a solver-friendly discretization of the coupled problem based on an implicit‐explicit time discretization and on finite elements in space. The discretized PDE leads to convex spatial minimization problems which can be solved eciently by monotone multigrid. Numerical experiments are provided using the Dune numerics framework.

Optimal local approximation spaces for component-based static condensation procedures

In this paper we introduce local approximation spaces for component-based static condensation (sc) procedures that are optimal in the sense of Kolmogorov. To facilitate simulations for large structures such as aircraft or ships, it is crucial to decrease the number of degrees of freedom on the interfaces, or “ports,” in order to reduce the size of the statically condensed system. To derive optimal port spaces we consider a (compact) transfer operator that acts on the space of harmonic extensions on a two-component system and maps the traces on the ports that lie on the boundary of these components to the trace of the shared port. Solving the eigenproblem for the composition of the transfer operator and its adjoint yields the optimal space. For a related work in the context of the generalized finite element method, we refer the reader to [I. Babuska and R. Lipton, Multiscale Model. Simul., 9 (2011), pp. 373--406]. We further introduce a spectral greedy algorithm to generalize the procedure to the parameter...

A Dimensional Reduction Approach Based on the Application of Reduced Basis Methods in the Framework of Hierarchical Model Reduction

In this article we introduce a new dimensional reduction approach which is based on the application of reduced basis (RB) techniques in the hierarchical model reduction (HMR) frame- work. Considering problems that exhibit a dominant spatial direction, the idea of HMR is to perform a Galerkin projection onto a reduced space, which combines the full solution space in the dominant direction with a reduction space in the transverse direction. The latter is spanned by modal or- thonormal basis functions. While so far the basis functions in the HMR approach have been chosen a priori (S. Perotto, A. Ern, and A. Veneziani, Multiscale Model. Simul., 8 (2010), pp. 1102-1127), for instance, as Legendre or trigonometric polynomials, in this work a highly nonlinear approximation is employed for the construction of the reduction space. To this end we first derive a lower dimen- sional parametrized problem in the transverse direction from the full problem where the parameters reflect the influence from the unknown solution in the dominant direction. Exploiting the good ap- proximation properties of RB methods, we then construct a reduction space by applying a proper orthogonal decomposition to a set of snapshots of the parametrized partial differential equation. For an efficient construction of the snapshot set we apply adaptive refinement in parameter space based on an a posteriori error estimate that is also derived in this article. We introduce our method for general elliptic problems such as advection-diffusion equations in two space dimensions. Numerical experiments demonstrate a fast convergence of the proposed dimensionally reduced approximation to the solution of the full dimensional problem and the computational efficiency of our new adaptive approach.

Hierarchical model reduction of nonlinear partial differential equations based on the adaptive empirical projection method and reduced basis techniques

In this paper we extend the hierarchical model reduction framework based on reduced basis techniques recently introduced in [M. Ohlberger and K. Smetana, SIAM J. Sci. Comput. 36 (2014) A714–A736] for the application to nonlinear partial differential equations. The major new ingredient to accomplish this goal is the introduction of the adaptive empirical projection method, which is an adaptive integration algorithm based on the (generalized) empirical interpolation method [M. Barrault, et al. , C. R. Math. Acad. Sci. Paris Series I 339 (2004) 667–672; Y. Maday and O. Mula, A generalized empirical interpolation method: Application of reduced basis techniques to data assimilation. In Analysis and Numerics of Partial Differential Equations . Vol. 4 of Springer INdAM Series . Springer Milan (2013) 221–235]. Different from other partitioning concepts for the empirical interpolation method we perform an adaptive decomposition of the spatial domain. We project both the variational formulation and the range of the nonlinear operator onto reduced spaces. Those reduced spaces combine the full dimensional (finite element) space in an identified dominant spatial direction and a reduction space or collateral basis space spanned by modal orthonormal basis functions in the transverse direction. Both the reduction and the collateral basis space are constructed in a highly nonlinear fashion by introducing a parametrized problem in the transverse direction and associated parametrized operator evaluations, and by applying reduced basis methods to select the bases from the corresponding snapshots. Rigorous a priori and a posteriori error estimators, which do not require additional regularity of the nonlinear operator are proven for the adaptive empirical projection method and then used to derive a rigorous a posteriori error estimator for the resulting hierarchical model reduction approach. Numerical experiments for an elliptic nonlinear diffusion equation demonstrate a fast convergence of the proposed dimensionally reduced approximation to the solution of the full-dimensional problem. Runtime experiments verify a close to linear scaling of the reduction method in the number of degrees of freedom used for the computations in the dominant direction.

Approximation of skewed interfaces with tensor-based model reduction procedures

In this article we introduce a procedure, which allows to recover the potentially very good approximation properties of tensor-based model reduction procedures for the solution of partial differential equations in the presence of interfaces or strong gradients in the solution which are skewed with respect to the coordinate axes. The two key ideas are the location of the interface either by solving a lower-dimensional partial differential equation or by using data functions and the subsequent removal of the interface of the solution by choosing the determined interface as the lifting function of the Dirichlet boundary conditions. We demonstrate in numerical experiments for linear elliptic equations and the reduced basis-hierarchical model reduction approach that the proposed procedure locates the interface well and yields a significantly improved convergence behavior even in the case when we only consider an approximation of the interface.

Adaptive modelling of coupled hydrological processes with application in water management

This paper presents recent results of a network project aiming at the modelling and simulation of coupled surface and subsurface flows. In particular, a discontinuous Galerkin method for the shallow water equations has been developed which includes a special treatment of wetting and drying. A robust solver for saturated–unsaturated groundwater flow in homogeneous soil is at hand, which, by domain decomposition techniques, can be reused as a subdomain solver for flow in heterogeneous soil. Coupling of surface and subsurface processes is implemented based on a heterogeneous nonlinear Dirichlet–Neumann method, using the dune-grid-glue module in the numerics software DUNE.

Randomized local model order reduction

In this paper we propose local approximation spaces for localized model order reduction procedures such as domain decomposition and multiscale methods. Those spaces are constructed from local solutions of the partial differential equation (PDE) with random boundary conditions, yield an approximation that converges provably at a nearly optimal rate, and can be generated at close to optimal computational complexity. In many localized model order reduction approaches like the generalized finite element method, static condensation procedures, and the multiscale finite element method local approximation spaces can be constructed by approximating the range of a suitably defined transfer operator that acts on the space of local solutions of the PDE. Optimal local approximation spaces that yield in general an exponentially convergent approximation are given by the left singular vectors of this transfer operator [I. Babuska and R. Lipton 2011, K. Smetana and A. T. Patera 2016]. However, the direct calculation of these singular vectors is computationally very expensive. In this paper, we propose an adaptive randomized algorithm based on methods from randomized linear algebra [N. Halko et al. 2011], which constructs a local reduced space approximating the range of the transfer operator and thus the optimal local approximation spaces. The adaptive algorithm relies on a probabilistic a posteriori error estimator for which we prove that it is both efficient and reliable with high probability. Several numerical experiments confirm the theoretical findings.