deim(
    full_vars,
    linear_coeffs,
    constant_part,
    nonlinear_part,
    reduced_vars,
    linear_projection_matrix,
    nonlinear_projection_matrix;
    kwargs...
)

Compute the reduced model by applying the Discrete Empirical Interpolation Method (DEIM).

This method allows users to input the projection matrices of their choice.

Given the projection matrix $V\in\mathbb R^{n\times k}$ for the dependent variables $\mathbf y\in\mathbb R^n$ and the projection matrix $U\in\mathbb R^{n\times m}$ for the nonlinear function $\mathbf F\in\mathbb R^n$, the full-order model (FOM)

\[\frac{d}{dt}\mathbf y(t)=A\mathbf y(t)+\mathbf g(t)+\mathbf F(\mathbf y(t))\]

is transformed to the reduced-order model (ROM)

\[\frac{d}{dt}\hat{\mathbf y}(t)=\underbrace{V^TAV}_{k\times k}\hat{\mathbf y}(t)+V^T \mathbf g(t)+\underbrace{V^TU(P^TU)^{-1}}_{k\times m}\underbrace{P^T\mathbf F(V \hat{\mathbf y}(t))}_{m\times1}\]

where $P=[\mathbf e_{\rho_1},\dots,\mathbf e_{\rho_m}]\in\mathbb R^{n\times m}$, $\rho_1,\dots,\rho_m$ are interpolation indices from the DEIM point selection algorithm, and $\mathbf e_{\rho_i}=[0,\ldots,0,1,0,\ldots,0]^T\in\mathbb R^n$ is the $\rho_i$-th column of the identity matrix $I_n\in\mathbb R^{n\times n}$.

Arguments

• full_vars::AbstractVector: the dependent variables $\underset{n\times 1}{\mathbf y}$ in FOM.
• linear_coeffs::AbstractMatrix: the coefficient matrix $\underset{n\times n}A$ of linear terms in FOM.
• constant_part::AbstractVector: the constant terms $\underset{n\times 1}{\mathbf g}$ in FOM.
• nonlinear_part::AbstractVector: the nonlinear functions $\underset{n\times 1}{\mathbf F}$ in FOM.
• reduced_vars::AbstractVector: the dependent variables $\underset{k\times 1}{\hat{\mathbf y}}$ in the reduced-order model.
• linear_projection_matrix::AbstractMatrix: the projection matrix $\underset{n\times k}V$ for the dependent variables $\mathbf y$.
• nonlinear_projection_matrix::AbstractMatrix: the projection matrix $\underset{n\times m}U$ for the nonlinear functions $\mathbf F$.

Return

• reduced_rhss: the right-hand side of ROM.
• linear_projection_eqs: the linear projection mapping $\mathbf y=V\hat{\mathbf y}$.

deim(
    sys::ModelingToolkit.ODESystem,
    snapshot::AbstractMatrix,
    pod_dim::Integer;
    deim_dim::Integer = pod_dim,
    name::Symbol = Symbol(nameof(sys), :_deim),
    kwargs...
) -> ModelingToolkit.ODESystem

Reduce a ModelingToolkit.ODESystem using the Proper Orthogonal Decomposition (POD) with the Discrete Empirical Interpolation Method (DEIM).

snapshot should be a matrix with the data of each time instance as a column.

The LHS of equations in sys are all assumed to be 1st order derivatives. Use ModelingToolkit.ode_order_lowering to transform higher order ODEs before applying DEIM.

sys is assumed to have no internal systems. End users are encouraged to call ModelingToolkit.structural_simplify beforehand.

The POD basis used for DEIM interpolation is obtained from the snapshot matrix of the nonlinear terms, which is computed by executing the runtime-generated function for nonlinear expressions.