32 research outputs found
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. Babu\v{s}ka 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.Comment: 31 pages, 14 figures, 1 table, 1 algorith
(Parametrized) First Order Transport Equations: Realization of Optimally Stable Petrov-Galerkin Methods
We consider ultraweak variational formulations for (parametrized) linear
first order transport equations in time and/or space. Computationally feasible
pairs of optimally stable trial and test spaces are presented, starting with a
suitable test space and defining an optimal trial space by the application of
the adjoint operator. As a result, the inf-sup constant is one in the
continuous as well as in the discrete case and the computational realization is
therefore easy. In particular, regarding the latter, we avoid a stabilization
loop within the greedy algorithm when constructing reduced models within the
framework of reduced basis methods. Several numerical experiments demonstrate
the good performance of the new method
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 differential equations (ODE's) on parts of the domain
boundary, and with nonlinear outflow conditions of Signorini's type. The
coupling of the partial differential 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 efficiently by
monotone multigrid. Numerical experiments are provided using the DUNE numerics
framework.Comment: 34 pages, 5 figure
A fully algebraic and robust two-level Schwarz method based on optimal local approximation spaces
Two-level domain decomposition preconditioners lead to fast convergence and
scalability of iterative solvers. However, for highly heterogeneous problems,
where the coefficient function is varying rapidly on several possibly
non-separated scales, the condition number of the preconditioned system
generally depends on the contrast of the coefficient function leading to a
deterioration of convergence. Enhancing the methods by coarse spaces
constructed from suitable local eigenvalue problems, also denoted as adaptive
or spectral coarse spaces, restores robust, contrast-independent convergence.
However, these eigenvalue problems typically rely on non-algebraic information,
such that the adaptive coarse spaces cannot be constructed from the fully
assembled system matrix. In this paper, a novel algebraic adaptive coarse
space, which relies on the a-orthogonal decomposition of (local) finite element
(FE) spaces into functions that solve the partial differential equation (PDE)
with some trace and FE functions that are zero on the boundary, is proposed. In
particular, the basis is constructed from eigenmodes of two types of local
eigenvalue problems associated with the edges of the domain decomposition. To
approximate functions that solve the PDE locally, we employ a transfer
eigenvalue problem, which has originally been proposed for the construction of
optimal local approximation spaces for multiscale methods. In addition, we make
use of a Dirichlet eigenvalue problem that is a slight modification of the
Neumann eigenvalue problem used in the adaptive generalized Dryja-Smith-Widlund
(AGDSW) coarse space. Both eigenvalue problems rely solely on local Dirichlet
matrices, which can be extracted from the fully assembled system matrix. By
combining arguments from multiscale and domain decomposition methods we derive
a contrast-independent upper bound for the condition number
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. Babuška 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-dependent setting and to construct a quasi-optimal parameter-independent port space. Moreover, it is shown that, given a certain tolerance and an upper bound for the ports in the system, the spectral greedy constructs a port space that yields an sc approximation error on a system of arbitrary configuration which is smaller than this tolerance for all parameters in a rich train set. We present our approach for isotropic linear elasticity, although the idea may be readily applied to any linear coercive problem. Numerical experiments demonstrate the very rapid and exponential convergence both of the eigenvalues and of the sc approximation based on spectral modes for nonseparable and irregular geometries such as an I-beam with an internal crack.United States. Air Force Office of Scientific Research. Multidisciplinary University Research Initiative (Grant FA9550-09-1-0613)United States. Office of Naval Research (Grant N00014-11-1-0713
A dimensional reduction approach based on the application of reduced basis methods in the context of hierarchical model reduction
In dieser Dissertation wird vor dem Anwendungshintergrund von Grundwasserströmungen eine neue Dimensionsreduktionsmethode hergeleitet, welche reduzierte Basistechniken zur Generierung von Basisfunktionen innerhalb der hierarchischen Modellreduktionsmethode anwendet. Dabei wird entlang der dominanten Fließrichtung des betrachteten Phänomens ein Standarddiskretisierungsverfahren eingesetzt und mit optimal an das Problem angepassten hierarchischen Basisfunktionen in transversaler Richtung kombiniert. Die hierarchischen Basen werden hierbei mit reduzierte Basistechniken aus Lösungen von in der Arbeit hergeleiteten parameterabhängigen niederdimensionalen Problemen ausgewählt. In einem zweiten Schritt wird die vorgeschlagene Dimensionsreduktionsmethode weiterentwickelt um auch nichtlineare Differentialgleichungen effizient behandeln zu können. In numerischen Experimenten für lineare und nichtlineare Differentialgleichungen wird die schnelle Konvergenz und Effizienz der Methode nachgewiesen. <br/