Toolboxmor

For the configuration of the case, and the usage in a C++ application, refer to ToolboxMor documentation. More details on the functions of the toolbox can be found in this page.

1. Model

We have a model defining a parameter-dependant problem to solve \(a(u, v; \mu) = f(v; \mu)\), and we are foccussing on the following outputs \(s_i(\mu) = L_i(\mu)^T u(\mu)\) for \(i\in [|1, n_\text{outputs}|]\), where \(n_\text{outputs}\) is the number of CRBOutputs described in the JSON file of the case.

We assume that we have the following decompositions :

\[\begin{align} a(u, v; \mu) &= \displaystyle\sum_{q=1}^{Q_a} \beta_A^q(\mu) a^q(u, v) \\ f(v; \mu) &= \displaystyle\sum_{p=1}^{Q_f} \beta_F^p(\mu) f^p(v)\\ l_i(v; \mu) &= \displaystyle\sum_{p=1}^{Q_{l_i}} \beta_{s_i}^p(\mu) l_i^p(v) \end{align}\]

In the term of matrices and vectors, it translates by :

\[\begin{align} A(\mu) &= \displaystyle\sum_{q=1}^{Q_a} \beta_A^q(\mu) A^q \\ F(\mu) &= \displaystyle\sum_{p=1}^{Q_f} \beta_F^p(\mu) F^p\\ L_i(\mu) &= \displaystyle\sum_{p=1}^{Q_{l_i}} \beta_{L_i}^p(\mu) L_i^p \end{align}\]

2. Get affine decomposition

Offline computation
[Aq, Fq] = model.getAffineDecomposition()

  • Aq is a list of lenght 1, and Aq[0] contains the list of the matrices of the affine decomposition of the bilinear form : Aq[0] = [ \(A^1\),…​, \(A^{Q_a}\)]

  • Fq is a list a lenght \(1 + n_\text{outputs}\). Fq[0] contains the affine decomposition of the right-hand side Fq[0] = [ \(F^1\),…​, \(F^{Q_f}\)], and for \(i\in [|1,n_\text{outputs}|]\), Fq[i] contains the affine decomposition of the \(i\)-th output Fq[i] = [ \(L_i^1\),…​, \(L_i^{Q_{l_i}}\)].

Online computation
[betaA, betaF] = model.computeBetaQm(mu)

where mu is a ParameterSpaceElement (see Parameters).

  • betaA is a list of length 1, and betaA[0] contains the coefficients of the affine decomposition betaA[0] = [ \(\beta_A^1\),…​, \(\beta_A^{Q_a}\) ].

  • betaF is a list of length \(1 + n_\text{outputs}\) contains the coefficients of the affine decompositions of \(f\), and \(s_i\) :

    • betaF[0][0] = [ \(\beta_F^1\),…​, \(\beta_F^{Q_f}\) ],

    • betaF[i][0] = [ \(\beta_{L_i}^1\),…​, \(\beta_{L_i}^{Q_{l_i}}\) ] for \(i\in [|1,n_\text{outputs}|]\).