We present a model reduction formulation for parametrized nonlinear partial differential equations (PDEs). Our approach builds on two ingredients: reduced basis (RB) spaces which provide rapidly ...convergent approximations to the parametric manifold; sparse empirical quadrature rules which provide rapid evaluation of the nonlinear residual and output forms associated with the RB spaces. We identify both the RB spaces and the sparse quadrature rules in the offline stage through a greedy training procedure over the parameter domain; the procedure requires the dual norm of the finite element (FE) residual at many training points in the parameter domain, but only very few FE solutions—the snapshots retained in the RB space. The quadrature rules are identified by a linear program (LP) empirical quadrature procedure (EQP) which (i) admits efficient solution by a simplex method, and (ii) directly controls the solution error induced by the approximate quadrature. We demonstrate the formulation for a parametrized neo-Hookean beam: the dimension of the approximation space and the number of quadrature points are both reduced by two orders of magnitude relative to FE treatment, with commensurate savings in computational cost.
•A reduced basis method for parametrized nonlinear PDEs is developed.•Empirical quadrature procedure (EQP) finds sparse quadrature for parametric integrands.•Greedy training strategy simultaneously identifies reduced basis and EQP.•Significant computational savings are achieved for a parametrized neo-Hookean beam.
error bounds for reduced basis approximations, in sharp contrast to classical (pessimistic) exponentially growing energy estimates. The paper contains a full analysis and various extensions for the ...formulation introduced briefly by Urban and Patera (2012) as well as numerical results for a model reaction-convection-diffusion equation.>
We present a reduced basis technique for long-time integration of parametrized incompressible turbulent flows. The new contributions are threefold. First, we propose a constrained Galerkin ...formulation that corrects the standard Galerkin statement by incorporating prior information about the long-time attractor. For explicit and semi-implicit time discretizations, our statement reads as a constrained quadratic programming problem where the objective function is the Euclidean norm of the error in the reduced Galerkin (algebraic) formulation, while the constraints correspond to bounds for the maximum and minimum value of the coefficients of the N-term expansion. Second, we propose an a posteriori error indicator, which corresponds to the dual norm of the residual associated with the time-averaged momentum equation. We demonstrate that the error indicator is highly-correlated with the error in mean flow prediction, and can be efficiently computed through an offline/online strategy. Third, we propose a Greedy algorithm for the construction of an approximation space/procedure valid over a range of parameters; the Greedy is informed by the a posteriori error indicator developed in this paper. We illustrate our approach and we demonstrate its effectiveness by studying the dependence of a two-dimensional turbulent lid-driven cavity flow on the Reynolds number.
We present a two-level parameterized Model Order Reduction (pMOR) technique for the linear hyperbolic Partial Differential Equation (PDE) of time-domain elastodynamics. In order to approximate the ...frequency-domain PDE, we take advantage of the Port-Reduced Reduced-Basis Component (PR-RBC) method to develop (in the offline stage) reduced bases for subdomains; the latter are then assembled (in the online stage) to form the global domains of interest. The PR-RBC approach reduces the effective dimensionality of the parameter space and also provides flexibility in topology and geometry. In the online stage, for each query, we consider a given parameter value and associated global domain. In the first level of reduction, the PR-RBC reduced bases are used to approximate the frequency-domain solution at selected frequencies. In the second level of reduction, these instantiated PR-RBC approximations are used as surrogate truth solutions in a Strong Greedy approach to identify a reduced basis space; the PDE of time-domain elastodynamics is then projected on this reduced space. We provide a numerical example to demonstrate the computational capability and assess the performance of the proposed two-level approach.
•Two-level model order reduction approach for parameterized elastodynamics PDE.•Methodology demonstration for 2-D bridge with 63 global parameters.•Cost comparison: model order reduction 28× faster than finite element.
The convergence and efficiency of the reduced basis method used for the approximation of the solutions to a class of problems written as a parametrized PDE depends heavily on the choice of the ...elements that constitute the “reduced basis”. The purpose of this paper is to analyze the a priori convergence for one of the approaches used for the selection of these elements, the greedy algorithm. Under natural hypothesis on the set of all solutions to the problem obtained when the parameter varies, we prove that three greedy algorithms converge; the last algorithm, based on the use of an a posteriori estimator, is the approach actually employed in the calculations.
A reduced basis method for the wave equation Glas, Silke; Patera, Anthony T.; Urban, Karsten
International journal of computational fluid dynamics,
02/2020, Letnik:
34, Številka:
2
Journal Article
Recenzirano
Odprti dostop
In this contribution, we derive an a posteriori error estimator for the second-order wave equation motivated by energy-based a priori estimates by Bernardi and Süli "Time and Space Adaptivity for the ...Second-order Wave Equation." Mathematical Models and Methods in Applied Sciences 15 (2): 199-225. This estimate (which is valid for general discretisations) is then used to derive a POD-Greedy reduced basis approach for the parameterised wave equation. The quantitative performance of the online-efficient error estimator is shown for an illustrative example, keeping in mind that model reduction of parametrised hyperbolic problems is a challenge.
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