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.mtkcompile 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.