GridapROMs.RBTransient

struct TransientIntegrationDomain <: IntegrationDomain
  domain_space::IntegrationDomain
  indices_time::Vector{Int32}
end

Integration domain for a projection operator in a transient problem

struct TransientMDEIMReduction{A,R<:Reduction{A,EuclideanNorm}} <: AbstractMDEIMReduction{A}
  reduction::R
  combine::Function
end

MDEIM struct employed in transient problems. The field combine is a function used to group the reductions relative to the various Jacobians(in general, more than one in transient problems) in a smart way. We consider, for example, the ODE

$\tfrac{du}{dt} - \nu \Delta u = f \ \ \text{in} \ \ Ω \times [0,T]$

subject to initial/boundary conditions. Upon applying a FE discretization in space, and a θ-method in time, one gets the space-time system

$A_{\theta} u_{\theta} = f_{\theta}$

where

\[A_{\theta} = \begin{bmatrix} A_1 + M / (\theta \Delta t) & & & & & \\ - M / (\theta \Delta t) & A_2 + M / (\theta \Delta t) & & & & \\ & - M / (\theta \Delta t) & A_3 + M / (\theta \Delta t) & & & \\ & & \ddots & \ddots & & \\ & & & & - M / (\theta \Delta t) & A_n + M / (\theta \Delta t) \end{bmatrix};\]

\[u_{\theta} = \left[(1-\theta)u_0 + \theta u_1, \hdots, (1-\theta)u_{n-1} + \theta u_n\right]^T;\]

\[f_{\theta} = \left[f_1, \hdots, f_n\right]^T;\]

\[A_k = A(t_{k-1} + \theta \Delta t);\]

\[f_k = f(t_{k-1} + \theta \Delta t).\]

Note: instead of multiplying $A_{\theta}$ by $u_{\theta}$, we multiply $\tilde{A}_{\theta}$ by $u$, where

\[\tilde{A}_{\theta} = tridiag((1-\theta)A_{k-1} - M / \Delta t, \theta A_k + M / \Delta t, 0).\]

We now denote with $\Phi$ and $\Psi$ the spatial and temporal basis obtained by reducing the snapshots associated to the state variable $u$. The Galerkin projection of the space-time system is equal to $\hat{A}_{\theta}\hat{u} = \hat{f}_{\theta}$, where $\hat{u}$ is the unknown, and

\[\begin{align*} \hat{A}_{\theta} &= \sum\limits_{k=1}^{n-1} ( (1-θ) \Phi^T A_k \Phi - \Phi^T M \Phi / \Delta t) \otimes \Psi[k-1,:]^T \Psi[k,:] + \sum\limits_{k=1}^n (\theta \Phi^T A_k \Phi + \Phi^T M \Phi / \Delta t) \otimes \Psi[k,:]^T \Psi[k,:] \\ &= \theta A_{backwards} + (1-\theta)A_{forwards} + (M_{backwards} + M_{forwards}) / \Delta t \\ \hat{f}_{\theta} &= \sum\limits_{k=1}^n \Phi^T f_k \otimes \Psi[k,:] \end{align*}\]

We notice that the expression of $\hat{A}_{\theta}$ can be written in a more general form as

\[\hat{A}_{\theta} = combine_A(A_{backwards},A_{forwards}) + combine_M(M_{backwards},M_{forwards}),\]

where combineA and combineM are two function specific to A and M:

\[\begin{align*} combine_A(x,y) &= \theta y + (1-\theta)y \\ combine_M(x,y) &= (x - y) / \Delta t \end{align*}\]

The same can be said of any time marching scheme. This is the meaning of the function combine. Note that for a time marching with $p$ interpolation points (e.g. for $\theta$ method, $p = 2$) the combine functions will have to accept $p$ arguments.

struct TransientProjection <: Projection
  projection_space::Projection
  projection_time::Projection
end

Projection operator for transient problems, containing a spatial projection and a temporal one. The space-time projection operator is equal to

projection_time ⊗ projection_space

which, for efficiency reasons, is never explicitly computed

struct TransientReduction{A,B,RS<:Reduction{A,B},RT<:Reduction{A,EuclideanNorm}} <: Reduction{A,B}
  reduction_space::RS
  reduction_time::RT
end

Wrapper for reduction methods in transient problems. The fields reduction_space and reduction_time respectively represent the spatial reduction method, and the temporal one

time_enrichment(red::SupremizerReduction,a_primal::Projection,basis_dual) -> AbstractMatrix

Temporal supremizer enrichment. (Approximate) Procedure:

1. for every b_dual ∈ Col(basis_dual)
2. compute Φ_primal_dual = get_basis(a_primal)'*get_basis(b_dual)
3. compute v = kernel(Φ_primal_dual)
4. compute v′ = orth_complement(v,a_primal)
5. enrich a_primal = [a_primal,v′]