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In this paper, we propose a Static Condensation Reduced Basis Element (SCRBE) approach for the Reynolds Lubrication Equation (RLE). The SCRBE method is a computational tool that allows to efficiently analyze parametrized structures which can be decomposed into a large number of similar components. Here, we extend the methodology to allow for a more general domain decomposition, a typical example being a checkerboard-pattern assembled from similar components. To this end, we extend the formulation and associated a posteriori error bound procedure. Our motivation comes from the analysis of the pressure distribution in plain journal bearings governed by the RLE. However, the SCRBE approach presented is not limited to bearings and the RLE, but directly extends to other component-based systems. We show numerical results for plain bearings to demonstrate the validity of the proposed approach.
We consider the efficient and reliable solution of linear-quadratic optimal control
problems governed by parametrized parabolic partial differential equations. To this end,
we employ the reduced basis method as a low-dimensional surrogate model to solve the
optimal control problem and develop a posteriori error estimation
procedures that provide rigorous bounds for the error in the optimal control and the
associated cost functional. We show that our approach can be applied to problems involving
control constraints and that, even in the presence of control constraints, the reduced
order optimal control problem and the proposed bounds can be efficiently evaluated in an
offline-online computational procedure. We also propose two greedy sampling procedures to
construct the reduced basis space. Numerical results are presented to confirm the validity
of our approach.
In this paper, we employ the reduced basis method as a surrogate model for the solution
of linear-quadratic optimal control problems governed by parametrized elliptic partial
differential equations. We present a posteriori error estimation and dual
procedures that provide rigorous bounds for the error in several quantities of interest:
the optimal control, the cost functional, and general linear output functionals of the
control, state, and adjoint variables. We show that, based on the assumption of affine
parameter dependence, the reduced order optimal control problem and the proposed bounds
can be efficiently evaluated in an offline-online computational procedure. Numerical
results are presented to confirm the validity of our approach.
In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter
dependence to problems involving (a) nonaffine dependence on the
parameter, and (b) nonlinear dependence on the field variable.
The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computational
decomposition. We first review the coefficient function approximation procedure: the essential ingredients are (i) a good collateral
reduced-basis approximation space, and (ii) a stable and inexpensive
interpolation procedure. We then apply this approach to linear nonaffine and nonlinear elliptic and parabolic equations; in each
instance, we discuss the reduced-basis approximation and the associated offline-online computational
procedures. Numerical results are presented to assess our approach.
In this paper, we extend the reduced-basis methods and associated a
posteriori error estimators developed earlier for elliptic partial
differential equations to parabolic problems with affine parameter
dependence. The essential new ingredient is the presence of time in the
formulation and solution of the problem – we shall “simply” treat
time as an additional, albeit special, parameter. First, we introduce
the reduced-basis recipe – Galerkin projection onto a space WN
spanned by solutions of the governing partial differential equation at
N selected points in parameter-time space – and develop a new greedy
adaptive procedure to “optimally” construct the parameter-time sample
set. Second, we propose error estimation and adjoint procedures that
provide rigorous and sharp bounds for the error in specific outputs of
interest: the estimates serve a priori to construct our samples,
and a posteriori to confirm fidelity. Third, based on the
assumption of affine parameter dependence, we develop offline-online
computational procedures: in the offline stage, we generate the
reduced-basis space; in the online stage, given a new parameter value, we
calculate the reduced-basis output and associated error bound. The
operation count for the online stage depends only on N (typically
small) and the parametric complexity of the problem; the method is thus
ideally suited for repeated, rapid, reliable evaluation of input-output
relationships in the many-query or real-time contexts.
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