Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods
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Citations
Nonlinear Model Reduction via Discrete Empirical Interpolation
An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations
A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems
Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations
Certified Reduced Basis Methods for Parametrized Partial Differential Equations
References
A general output bound result: application to discretization and iteration error estimation and control
Reliable Real-Time Solution of Parametrized Elliptic Partial Differential Equations: Application to Elasticity
Related Papers (5)
Frequently Asked Questions (9)
Q2. What is the problem with the inf-sup parameter?
Loss of stability can, in turn, lead to poor approximations—the inf-sup parameter enters in the denominator of the a priori convergence result.
Q3. What are the advantages of a nonintegrated approach?
there are significant computational and conditioning advantages associated with a ‘‘nonintegrated’’ approach, in which the authors introduce separate primal (u(mn)) and dual (c(mn)) approximation spaces for u(m) and c~m!, respectively.
Q4. What is the second numerical difficulty of the inf-sup parameter?
The second numerical difficulty is estimation of the inf-sup parameter, which for noncoercive problems plays the role of g(m) in Method The authora posteriori error estimation techniques.
Q5. What is the effectivity of the two?
If the primal and dual errors are a-orthogonal, or become increasingly orthogonal as N increases, then the effectivity will not, in fact, be bounded as N→` .
Q6. What is the on-line procedure for calculating the bounds for a given new ?
In the on-line stage, for any given new m, the authors first form AI N(m), FI N and AI 2N(m), FI 2N , then solve for uI N(m) and uI 2N(m), and finally evaluate sN ,2N6 (m): this requires O(4QN2)1O(16/3 N3) operations and O(4QN2) storage.
Q7. What is the method to estimate error?
Note that WN has good approximation properties both for the first and second lowest eigenfunctions, and hence eigenvalues; this is required by the Method The authorerror estimator to be presented below.
Q8. What is the cost of evaluating sN(m)?
In the on-line stage, for any given new m, the authors first form AI N from ~15!, then solve ~14! for uI N(m), and finally evaluate sN(m)5FI NT uI N(m): this requires O(QN2)1O(2/3 N3) operations and O(QN2) storage.
Q9. What is the new ingredient in the reduced-basis approximation and error estimator?
The essential new ingredient is the presence of the time variable, t.The reduced-basis approximation and error estimator procedures are similar to those for noncompliant nonsymmetric problems, except that the authors now include the time variable as an additional parameter.