Efficient and certified solution of parametrized one-way coupled problems through DEIM-based data projection across non-conforming interfaces

Abstract

One of the major challenges of coupled problems is to manage nonconforming meshes at the interface between two models and/or domains, due to different numerical schemes or domain discretizations employed. Moreover, very often complex submodels depend on (e.g., physical or geometrical) parameters, thus making the repeated solutions of the coupled problem through high-fidelity, full-order models extremely expensive, if not unaffordable. In this paper, we propose a reduced order modeling (ROM) strategy to tackle parametrized one-way coupled problems made by a first, master model and a second, slave model; this latter depends on the former through Dirichlet interface conditions. We combine a reduced basis method, applied to each subproblem, with the discrete empirical interpolation method to efficiently interpolate or project Dirichlet data across either conforming or non-conforming meshes at the domains interface, building a low-dimensional representation of the overall coupled problem. The proposed technique is numerically verified by considering a series of test cases involving both steady and unsteady problems, after deriving a posteriori error estimates on the solution of the coupled problem in both cases. This work arises from the need to solve staggered cardiac electrophysiological models and represents the first step towards the setting of ROM techniques for the more general two-way Dirichlet-Neumann coupled problems solved with domain decomposition sub-structuring methods, when interface non-conformity is involved.

Appendix : A: A posteriori error estimator for unsteady reduced basis models

To find an a posteriori error estimate for the ROM approximation in a time-dependent case, according to [55], we can start by considering the following parametrized linear dynamical system for a vector $\mathbf {u}(t;\boldsymbol {\mu }) \in \mathbb {R}^{n}$:

$$ \begin{cases}\frac{d}{dt} \mathbf{u}(t;\boldsymbol{\mu}) = \mathbb{A}_{N}(t;\boldsymbol{\mu})\mathbf{u}(t;\boldsymbol{\mu}) + \mathbf{f}_{N}(t;\boldsymbol{\mu}), & \quad t \in (0,T) \\ \mathbf{u}(0;\boldsymbol{\mu}) = \mathbf{u}_{0}(\boldsymbol{\mu}) & \end{cases} $$

Here the matrix $\mathbb {A}_{N}(t;\boldsymbol {\mu }) \in \mathbb {R}^{N \times N}$ and the vector $\mathbf {f}_{N}(t;\boldsymbol {\mu }) \in \mathbb {R}^{N}$, where N denotes the dimension of the reference FOM space, are μ-dependent. Moreover, we define the projection matrix $\mathbb {V} \in \mathbb {R}^{N\times n}$ defined through RB methods, where n ≤ N is the ROM dimension. Then, the reduced dynamical system is:

$$ \begin{cases} \frac{d}{dt} \mathbf{u}_{n}(t;\boldsymbol{\mu}) = \mathbb{A}_{n}(t;\boldsymbol{\mu})\mathbf{u}_{n}(t;\boldsymbol{\mu}) + \mathbf{f}_{n}(t;\boldsymbol{\mu}), & t \in [0,T]\\ \mathbf{u}_{n}(0;\boldsymbol{\mu}) = \mathbf{u}_{n,0}(\boldsymbol{\mu}) \end{cases} $$

(A1)

where $\mathbb {A}_{n}(t;\boldsymbol {\mu }) = \mathbb {V}^{T}\mathbb {A}_{N}(t;\boldsymbol {\mu })\mathbb {V}$, $\mathbf {f}_{n}(t;\boldsymbol {\mu }) = \mathbb {V}^{T}\mathbf {f}_{N}(t;\boldsymbol {\mu }) \mathbb {V}$, u_n(t; μ) is the reduced approximation, i.e., $\mathbf {u}_{N}(t;\boldsymbol {\mu }) \approx \mathbb {V}\mathbf {u}_{n}(t;\boldsymbol {\mu })$, and u_n,0(μ) is the projection of u₀(μ) onto the reduced space.

Let us now denote the error and the residual as

$$ \begin{array}{l} \mathbf{e}(t;\boldsymbol{\mu}) := \mathbf{u}(t;\boldsymbol{\mu}) - \mathbb{V}\mathbf{u}_{n}(t;\boldsymbol{\mu}), \\ \mathbf{r}(t;\boldsymbol{\mu}) := \mathbb{A}_{N}(t;\boldsymbol{\mu})\mathbb{V}\mathbf{u}_{n}(t;\boldsymbol{\mu}) + \mathbf{f}_{N}(t;\boldsymbol{\mu}) - \mathbb{V} \frac{d}{dt}\mathbf{u}_{n}(t;\boldsymbol{\mu}), \end{array} $$

respectively; given a symmetric positive definite matrix $\mathbf {G} \in \mathbb {R}^{N \times N}$, let us denote by 〈⋅,⋅〉_G the induced inner product, and the induced norm as $\| \mathbf {u}\|_{\mathbf {G}} := \sqrt {\langle \mathbf {u},\mathbf {u}\rangle _{\mathbf {G}}}$ on R^N. Similarly, $\| \mathbb {A}\|_{\mathbf {G}} := \textup {sup}_{\| \mathbf {u}\|_{\mathbf {G}}}\| \mathbb {A}\mathbf {u}\|_{\mathbf {G}}$, for $\mathbb {A} \in \mathbb {R}^{N\times N}$. For example, if $\mathbf {G} = \mathbb {I}_{N \times N}$, i.e., it is the identity matrix, than we obtain the simple 2-norm used in this work. Then, the following a posteriori error estimate can be stated:

Proposition 1 (A posteriori error estimate)

Assuming that $\mathbb {A}_{N} (t;\boldsymbol {\mu }) = \mathbb {A}_{N}(\boldsymbol {\mu })$ is time-invariant and has eigenvalues with negative real part for all $\boldsymbol {\mu } \in {\mathscr{P}}$, than the solution is bounded by

$$ \sup_{t} \| \exp(\mathbb{A}_{N}(\boldsymbol{\mu})t)\|_{\mathbf{G}} \leq C_{1}(\boldsymbol{\mu}),$$

where C₁(μ) is a computable constant. Then, the following error estimates holds:

$$ \| \mathbf{u}(t;\boldsymbol{\mu}) - \mathbb{V}\mathbf{u}_{n}(t;\boldsymbol{\mu})\|_{\mathbf{G}} \leq C_{1}(\boldsymbol{\mu}) \left( \| \mathbf{e}(0;\boldsymbol{\mu})\|_{\mathbf{G}} + {{\int}_{0}^{T}} \| \mathbf{r}(\tau;\boldsymbol{\mu})\|_{\mathbf{G}} d \tau \right). $$

(A2)

Proof

From the residual definition, we obtain that

$$\mathbb{V}\frac{d}{dt}\mathbf{u}_{n}(t;\boldsymbol{\mu}) = \mathbb{A}_{N}(\boldsymbol{\mu})\mathbb{V}\mathbf{u}_{n}(t;\boldsymbol{\mu})+\mathbf{f}_{N}(t;\boldsymbol{\mu}) - \mathbf{r}(t;\boldsymbol{\mu}).$$

Subtracting this equation from the original system, we get the evolution system

$$ \begin{cases}\frac{d}{dt}\mathbf{e}(t;\boldsymbol{\mu}) = \mathbb{A}_{N}(\boldsymbol{\mu})\mathbf{e}(t;\boldsymbol{\mu}) + \mathbf{r}(t;\boldsymbol{\mu})\\ \mathbf{e}(0;\boldsymbol{\mu}) = \mathbf{u}_{0}(\boldsymbol{\mu}) - \mathbb{V} \mathbf{u}_{n,0}(\boldsymbol{\mu}). \end{cases} $$

(A3)

for the error, that admits the explicit solution

$$\mathbf{e}(t;\boldsymbol{\mu}) = \exp(\mathbb{A}_{N}(\boldsymbol{\mu})t)\mathbf{e}(0;\boldsymbol{\mu}) + {{\int}_{0}^{T}} \exp(\mathbb{A}_{N}(\boldsymbol{\mu})(T-\tau))\mathbf{r}(\tau;\boldsymbol{\mu})d\tau.$$

The thesis follows thanks to the assumption $\| \exp (\mathbb {A}_{N}(\boldsymbol {\mu })s)\|_{\mathbf {G}} \leq C_{1}(\boldsymbol {\mu })$ for $s \in \mathbb {R}^{+}$. □

Error relations similar to (A2) can also be found for time dependent systems, meaning when $\mathbb {A}_{N}(t;\boldsymbol {\mu })$ depends on time, by a suitable modification of C₁(μ). To do this, we first point out that the error evolution system (A3) holds also for time-variants systems. Then, integrating, we get

$$ \mathbf{e}(t;\boldsymbol{\mu}) = \mathbf{e}(0;\boldsymbol{\mu}) + {{\int}_{0}^{T}} \mathbb{A}_{N}(\tau;\boldsymbol{\mu})\mathbf{e}(\tau;\boldsymbol{\mu}) + \mathbf{r}(\tau;\boldsymbol{\mu})d\tau.$$

Denoting by Φ(t) := ∥e(t; μ)∥_G, $\boldsymbol {\alpha }(t):= \| \mathbf {e}(0;\boldsymbol {\mu })\|_{\mathbf {G}} + {{\int \limits }_{0}^{T}} \| \mathbf {r}(\tau ;\boldsymbol {\mu })\|_{\mathbf {G}}d\tau$ and $\boldsymbol {\beta }(t) := \| \mathbb {A}_{N}(\tau ;\boldsymbol {\mu })\|_{\mathbf {G}}$, we can obtain

$$\boldsymbol{\Phi}(t) \leq \boldsymbol{\alpha}(t) + {{\int}_{0}^{T}} \boldsymbol{\beta}(\tau)\boldsymbol{\Phi}(\tau)d\tau.$$

Moreover, assuming an upper bound $\| \mathbb {A}_{N}(t;\boldsymbol {\mu })\|_{\mathbf {G}} \leq C_{3}(\boldsymbol {\mu })$ for t ∈ [0,T], $\boldsymbol {\mu } \in {\mathscr{P}}$, using the Gronwall inequality, we can write

$$ \begin{array}{ll} \boldsymbol{\Phi}(t) &\leq \boldsymbol{\alpha}(t) + {{\int}_{0}^{T}} \boldsymbol{\alpha}(\tau)\boldsymbol{\beta}(\tau)\exp \left( {{\int}_{s}^{T}} \boldsymbol{\beta}(r)dr \right)d\tau \leq \boldsymbol{\alpha}(t)(1 + C_{3}(\boldsymbol{\mu})t \exp(C_{3} t)). \end{array} $$

Then, equation (A2) can be found denoting by $C_{1} := 1 + C_{3}(\boldsymbol {\mu })T \exp (C_{3} T)$.