# Unique continuation for the wave equation: the stability landscape

Erik Burman, Lauri Oksanen, Janosch Preuss, Ziyao Zhao

► To cite this version:

Erik Burman, Lauri Oksanen, Janosch Preuss, Ziyao Zhao. Unique continuation for the wave equation: the stability landscape. PICOF 2025 - 11th International Conference on Inverse Problems, Control and Shape Optimization, Laboratory of Mathematical and Numerical Modeling in Engineering Sciences (LAMSIN), Oct 2025, Tunis, Tunisia. hal-05443659

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Submitted on 6 Jan 2026

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
Unique continuation for the wave equation
The stability landscape

Erik Burman¹, Lauri Oksanen², Janosch Preuss³ & Ziyao Zhao²

1. University College London, 2. University of Helsinki, 3. Inria Makutu

PICOF 2025, MS 7

Hammamet, October 30, 2025

<figure><img src="image_2.png" /></figure>

Engineering and
Physical Sciences
Research Council

<figure><img src="image_3.png" /></figure>

Funded by
the European Union

<figure><img src="image_4.png" /></figure>

European Research Council

<figure><img src="image_5.png" /></figure>

Research Council
of Finland
$$ \left\{ \begin{array}{l} \Box u := \partial_t^2 u - \Delta u = 0, \ (t,x) \in Q \\ u(0,x) = u_0, \quad \forall x \in \Omega, \\ \partial_t u(0,x) = u_1, \quad \forall x \in \Omega \end{array} \right. $$

<figure><img src="image_6.png" /></figure>
Unique continuation for the wave equation in $Q = (0, T) \times \Omega$

$$ egin{cases} 
\Box u := \partial_t^2 u - \Delta u = 0, (t,x) \in Q \\ 
u(0,x) = u_0, \quad \forall x \in \Omega, \\ 
\partial_t u(0,x) = u_1, \quad \forall x \in \Omega 
\end{cases} $$

Suppose we are given $u(t,x)|_{\omega_T} := u_\omega$ as data.

<figure><img src="image_7.png" /></figure>
$$ \left\{ \begin{array}{l} \square u := \partial_t^2 u - \Delta u = 0, (t,x) \in Q \\ \qquad \underline{u}(0,x) = u_0, \quad \forall x \in \Omega, \\ \partial_t u(0,x) = \underline{u}_1, \quad \forall x \in \Omega \end{array} \right. $$

Suppose we are given $u(t,x)|_{\omega_T} := u_\omega$ as data.

Can one extend the data to obtain $u$ in all of $Q$?

Yes, if ...

- $\omega_T \subset Q$ satisfies the geometric control condition,

- We have 'sufficient' information on $u|_{\Sigma}$

Actually only have access to perturbed data:

$$ u_{\omega}^{\delta} = u_{\omega} + \delta u \quad \text{in } \omega_{T}, \quad \delta u \in L^{2}(\omega_{T}). $$

<figure><img src="image_8.png" /></figure>
Stability estimates
# Some literature on control and observability of wave equations

**D. L. Russell**

Boundary value control theory of the higher-dimensional wave equation. II,
SIAM J. Control (1971)

**C. Bardos, G. Lebeau and J. Rauch,**

Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques,
Rend. Sem. Mat. Univ. Politec. Torino
(1988)

**N. Burq, P. Gérard**

Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes,
C. R. Acad. Sci. Paris Sér. I Math. (1997)

**J.-L. Lions**

Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 2: Perturbations,
Masson (1988)

**C. Bardos, G. Lebeau and J. Rauch,**

Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary,
SIAM J. Control Optim. (1992)

**L. Miller**

Escape function conditions for the observation, control, and stabilization of the wave equation,
SIAM J. Control Optim. (2002)

**L. Robbiano**

Fonction de coût et contrôle des solutions des équations hyperboliques,
Asymptotic Anal. (1995)

**J. Le Rousseau, G. Lebeau, P. Terpolilli & E. Trélat,**

"Geometric control condition for the wave equation with a time-dependent observation domain",
Analysis & PDE, 2017

**B. Dehman, S. Ervedoza and E. Zuazua**

Regional and partial observability and control of waves
arXiv:2504.18976 (2025)
Lipschitz stability estimate for $u|_{\Sigma}$ known

Theorem

Suppose that $\omega_T \subset Q$ satisfies the geometric control condition [Bardos, Lebeau & Rauch, '92]. Then for all $u \in H^1(Q)$

$$ \begin{aligned} & \|u\|_{L^{\infty}(0, T; L^2(\Omega))} + \|\partial_t u\|_{L^2(0, T; H^{-1}(\Omega))} \\ & \lesssim \|u\|_{L^2(\omega_T)} + \|u\|_{L^2(\Sigma)} + \|\Box u\|_{H^{-1}(Q)} \end{aligned} $$

E. Burman, A. Feizmohammadi, A. Münch & L. Oksanen, "Spacetime finite element methods for control problems subject to the wave equation.", ESAIM: COCV, 2023

But: What can be said if $u|_{\Sigma}$ is unknown?

<figure><img src="image_9.png" /></figure>
■ $H = \{(x^1, \dots, x^d) \in \mathbb{R}^d \mid x^1 < 0\}$,

■ $\Omega = H \cap B(0, R)$ and $\omega = \Omega \setminus \bar{B}(0, r)$.

■ Let $\beta > 0, \epsilon > 0, y = (\beta, 0, \dots, 0) \in \mathbb{R}^d$ and define

$$\psi(t, x) = |x - y|^2 - (1 - \epsilon)t^2.$$

■ Define $\rho_0 = r^2 + \beta^2$ and $\rho_1 = (r + \beta)^2$

■ Let $\rho \in (\rho_0, \rho_1)$ and $T > \sqrt{\frac{\rho_1 - \rho}{1 - \epsilon}}$

$$\mathcal{V}(s) := \{(t, x) \in (-T, T) \times \Omega \mid \psi(t, x) > s\}$$

<figure><img src="image_10.png" /></figure>

## Lemma

There exists $\alpha \in (0, 1)$ such that for $B = \mathcal{V}(\rho)$ and $\tilde{Q} = \mathcal{V}(\delta)$
$$\|u\|_{L^2(B)} \lesssim (\|u\|_{L^2(\omega_T)} + \|\Box u\|_{H^{-1}(\tilde{Q})})^{\alpha} \|u\|_{L^2(\tilde{Q})}^{1-\alpha}, \quad u \in H^1(\tilde{Q}).$$

E. Burman, L. Oksanen, J. Preuss & Z. Zhao, “Unique continuation for the wave equation: the stability landscape.”, arXiv:2510.20359, 2025
■ Suppose that $\mathcal{V}_M \subset L^2(\Sigma)$ is a finite dim. space (say $\dim(\mathcal{V}_M) = M$).

■ We denote $\mathcal{P}$ projection operator onto $\mathcal{V}_M$ and $Q = 1 - \mathcal{P}$.

■ Assume that the GCC is respected and that $\forall \varphi \in \mathcal{V}_M$, if $\varphi|_{\Gamma} = 0$, then $\varphi = 0$, where

$$ \Gamma := \{(t, x) \in \Sigma \mid \mathrm{dist}(x, \omega) \le \frac{T}{2} - |t - \frac{T}{2}|\} $$

**Theorem: Lipschitz stability estimate**

$$ \|u\|_{L^{\infty}(0,T;L^2(\Omega))} + \|\partial_t u\|_{L^2(0,T;H^{-1}(\Omega))} \lesssim \|u\|_{L^2(\omega_T)} + \|\Box u\|_{H^{-1}(Q)} + \|Qu\|_{L^2(\Sigma)} $$

E. Burman, L. Oksanen, J. Preuss & Z. Zhao, "Unique continuation for the wave equation: the stability landscape.", arXiv:2510.20359, 2025
## Lipschitz stab. knowing $u|_{\Sigma}$

$$ \begin{aligned} & \|u\|_{L^{\infty}(0, T; L^2(\Omega))} + \|\partial_t u\|_{L^2(0, T; H^{-1}(\Omega))} \\ & \lesssim \|u\|_{L^2(\omega_T)} + \|u\|_{L^2(\Sigma)} + \|\Box u\|_{H^{-1}(Q)} \end{aligned} $$

## Lipschitz stab. knowing $\mathcal{V}_M \ni u|_{\Sigma}$

$$ \begin{aligned} & \|u\|_{L^{\infty}(0, T; L^2(\Omega))} + \|\partial_t u\|_{L^2(0, T; H^{-1}(\Omega))} \\ & \lesssim \|u\|_{L^2(\omega_T)} + \|Qu\|_{L^2(\Sigma)} + \|\Box u\|_{H^{-1}(Q)} \end{aligned} $$

## Hölder stability

$$ \|u\|_{L^2(B)} \lesssim (\|u\|_{L^2(\omega_T)} + \|\Box u\|_{H^{-1}(\tilde{Q})})^{\alpha} \|u\|_{L^2(\tilde{Q})}^{1-\alpha}. $$

## Aim of the talk:

- Present numerical method that can be adapted easily to each regime of stability.

## Strategy to design the method:

- Apply stability estimates to $u - u_h$.

- Common denominator: Need to control $\|\Box(u - u_h)\|_{H^{-1}(Q)}$.

- Specific adaptations for each regime.
A stabilized primal dual finite element
method based on dG in time
# Selection of literature: UC for the wave equation (Lipschitz stable case)

## Backward differences in time + P1-FEM space

E. Burman, A. Feizmohammadi, and L. Oksanen, *A finite element data assimilation method for the wave equation*, Math. Comp. (2020)

<figure><img src="image_11.png" /></figure>

## PINNs

S. Mishra and R. Molinaro, *Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for PDEs.*, IMA J. Numer. Anal. (2021)

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

## C1-conforming space-time FEM

N. Cindea, A. Münch

*Inverse problems for linear hyperbolic equations using mixed formulations*, Inverse Problems (2015)

<figure><img src="image_14.png" /></figure>

## C0-conforming space-time FEM

E. Burman, A. Feizmohammadi, A. Münch, and L. Oksanen, *Space time stabilized finite element methods for a unique continuation problem subject to the wave equation*, ESAIM: Math. Model. Numer. Anal. (2021)

<figure><img src="image_15.png" /></figure>

## Least-squares FEM

W. Dahmen, H. Monsuur and R. Stevenson

*Least squares solvers for ill-posed PDEs that are conditionally stable*, ESAIM: Math. Model. Numer. Anal. (2023)

<figure><img src="image_16.png" /></figure>

## Hybrid DG

E. Burman, G. Delay and A. Ern, *The unique continuation problem for the wave equation discretized with a high-order space-time nonconforming method*, Numerische Mathematik (2025)

## DG in time + CG in space

E. Burman and J. Preuss, *Unique continuation for the wave equation based on a discontinuous Galerkin time discretization*, IMA J. Numer. Anal. (2025)
* Equidistant time discretization
  $\Delta t = |t_{n+1} - t_n| = 1/N$ for $n = 0, \dots, N-1$.

* $v_{\pm}^n(x) := \lim_{\tau \to 0^+} v(x, t_n \pm \tau)$, and $[[v^n]] := v_+^n - v_-^n$.

<figure><img src="image_17.png" /></figure>
* Equidistant time discretization

$$ \Delta t = |t_{n+1} - t_n| = 1/N \text{ for } n = 0, \dots, N-1. $$

* $v_{\pm}^n(x) := \lim_{\tau \to 0^+} v(x, t_n \pm \tau)$, and $[[v^n]] := v_+^n - v_-^n$.

* $H^1$-conforming FE space $V_h^k$ of order $k$ sur $\Omega$.

* $W_{h,\Delta t}^{q,k} := \otimes_{n=0}^{N-1} \mathcal{P}^q(I_n) \otimes V_h^k, q \in \mathbb{N}_0, k \in \mathbb{N}, h \sim \Delta t$.

* We will set $s = \min\{q, k\}$.

<figure><img src="image_18.png" /></figure>
* Equidistant time discretization

$$ \Delta t = |t_{n+1} - t_n| = 1/N \text{ for } n = 0, \dots, N-1. $$

* $v_{\pm}^n(x) := \lim_{\tau \to 0^+} v(x, t_n \pm \tau)$, and $[[v^n]] := v_+^n - v_-^n$.

* $H^1$-conforming FE space $V_h^k$ of order $k$ sur $\Omega$.

* $W_{h,\Delta t}^{q,k} := \otimes_{n=0}^{N-1} \mathcal{P}^q(I_n) \otimes V_h^k$, $q \in \mathbb{N}_0$, $k \in \mathbb{N}$, $h \sim \Delta t$.

* We will set $s = \min\{q, k\}$.

* First order system: $\underline{\mathbf{U}}_h = (\underline{u}_1, \underline{u}_2) \approx (u, \partial_t u) = \mathbf{U}$.

* $\underline{\mathbf{U}}_h \in W_{h,\Delta t}^{q,k} := W_{h,\Delta t}^{q,k} \times W_{h,\Delta t}^{q,k}$ for $q \in \mathbb{N}_0$.

$$ \underline{A}[\underline{\mathbf{U}}_h, \underline{\mathbf{Y}}_h] = \sum_{n=0}^{N-1} (\partial_t \underline{u}_2, \underline{y}_1)_{Q^n} + (\nabla \underline{u}_1, \nabla \underline{y}_1)_{Q^n} + (\partial_t \underline{u}_1 - \underline{u}_2, \underline{y}_2)_{Q^n} \\ - (\nabla \underline{u}_1 \cdot \mathbf{n}, \underline{y}_1)_{\Sigma^n} $$

<figure><img src="image_19.png" /></figure>
$$
\min_{u \in H^1(Q)} \frac{1}{2} \|u - u_{\omega}\|_{L^2(\omega_T)}^2 \quad \text{such that} \quad \Box u = 0, \text{ in } H^{-1}(Q),
$$
$$
\min_{u \in H^1(Q)} \frac{1}{2} \|u - u_{\omega}\|_{L^2(\omega_T)}^2 \quad \text{such that} \quad \Box u = 0, \text{ in } H^{-1}(Q),
$$

$$
\underline{\mathcal{L}}(\underline{\mathbf{U}}_h, \underline{\mathbf{Z}}_h) := \frac{1}{2} \underbrace{\|\underline{u}_1 - \underline{u}_{\omega}^{\delta}\|_{\omega_T}^2}_{\text{data fidelity}} + \underbrace{\underline{A}[\underline{\mathbf{U}}_h, \underline{\mathbf{Z}}_h]}_{\text{PDE constraint}} + \underbrace{\frac{1}{2}\underline{S}(\underline{\mathbf{U}}_h, \underline{\mathbf{U}}_h) - \frac{1}{2}\underline{S}^*(\underline{\mathbf{Z}}_h, \underline{\mathbf{Z}}_h)}_{\text{stabilization}}.
$$
$$
\min_{u \in H^1(Q)} \frac{1}{2} \|u - u_\omega\|_{L^2(\omega_T)}^2 \quad \text{such that} \quad \Box u = 0, \text{ in } H^{-1}(Q),
$$

$$
\underline{\mathcal{L}}(\underline{\mathbf{U}}_h, \underline{\mathbf{Z}}_h) := \frac{1}{2} \underbrace{\|u_1 - u_\omega^\delta\|_{\omega_T}^2}_{\text{data fidelity}} + \underbrace{\underline{A}[\underline{\mathbf{U}}_h, \underline{\mathbf{Z}}_h]}_{\text{PDE constraint}} + \underbrace{\frac{1}{2}\underline{S}(\underline{\mathbf{U}}_h, \underline{\mathbf{U}}_h) - \frac{1}{2}\underline{S}^*(\underline{\mathbf{Z}}_h, \underline{\mathbf{Z}}_h)}_{\text{stabilization}}.
$$

First order optimality conditions: Find $(\underline{\mathbf{U}}_h, \underline{\mathbf{Z}}_h) \in W_{h, \Delta t}^{q,k} \times W_{h, \Delta t}^{q_*, k_*}$ such that

$$
\begin{gather*}
(\underline{u}_1, \underline{w}_1)_{\omega_T} + \underline{A}[\underline{W}_h, \underline{Z}_h] + \underline{S}(\underline{U}_h, \underline{W}_h) = (u_\omega^\delta, \underline{w}_1)_{\omega_T}, \quad \forall \underline{W}_h \in W_{h, \Delta t}^{q,k}, \\
\underline{A}[\underline{U}_h, \underline{Y}_h] - \underline{S}^*(\underline{Y}_h, \underline{Z}_h) = 0, \quad \forall \underline{Y}_h \in W_{h, \Delta t}^{q_*, k_*}.
\end{gather*}
$$

$$
\rightsquigarrow B[(\underline{\mathbf{U}}_h, \underline{\mathbf{Z}}_h), (\underline{\mathbf{W}}_h, \underline{\mathbf{Y}}_h)] = (\underline{u}_1, \underline{w}_1)_{\omega_T} + \underline{A}[\underline{\mathbf{W}}_h, \underline{\mathbf{Z}}_h] + \underline{S}(\underline{\mathbf{U}}_h, \underline{\mathbf{W}}_h) + \underline{A}[\underline{\mathbf{U}}_h, \underline{\mathbf{Y}}_h] - \underline{S}^*(\underline{\mathbf{Y}}_h, \underline{\mathbf{Z}}_h)
$$
$$ \min_{u \in H^1(Q)} \frac{1}{2} \|u - u_{\omega}\|_{L^2(\omega_T)}^2 \quad \text{such that} \quad \Box u = 0, \text{ in } H^{-1}(Q), $$

$$ \mathcal{L}(\underline{\mathbf{U}}_h, \underline{\mathbf{Z}}_h) := \frac{1}{2} \underbrace{\|u_1 - u_{\omega}^{\delta}\|_{\omega_T}^2}_{\text{data fidelity}} + \underbrace{A[\underline{\mathbf{U}}_h, \underline{\mathbf{Z}}_h]}_{\text{PDE constraint}} + \underbrace{\frac{1}{2}S(\underline{\mathbf{U}}_h, \underline{\mathbf{U}}_h) - \frac{1}{2}S^*(\underline{\mathbf{Z}}_h, \underline{\mathbf{Z}}_h)}_{\text{stabilization}}. $$

First order optimality conditions: Find $(\underline{\mathbf{U}}_h, \underline{\mathbf{Z}}_h) \in W_{h, \Delta t}^{q, k} \times W_{h, \Delta t}^{q_*, k_*}$ such that

$$ (\underline{u}_1, \underline{w}_1)_{\omega_T} + A[\underline{\mathbf{W}}_h, \underline{\mathbf{Z}}_h] + S(\underline{\mathbf{U}}_h, \underline{\mathbf{W}}_h) = (u_{\omega}^{\delta}, \underline{w}_1)_{\omega_T}, \quad \forall \underline{\mathbf{W}}_h \in W_{h, \Delta t}^{q, k}, \\ A[\underline{\mathbf{U}}_h, \underline{\mathbf{Y}}_h] - S^*(\underline{\mathbf{Y}}_h, \underline{\mathbf{Z}}_h) = 0, \quad \forall \underline{\mathbf{Y}}_h \in W_{h, \Delta t}^{q_*, k_*}. $$

$$ \rightsquigarrow B[(\underline{\mathbf{U}}_h, \underline{\mathbf{Z}}_h), (\underline{\mathbf{W}}_h, \underline{\mathbf{Y}}_h)] = (\underline{u}_1, \underline{w}_1)_{\omega_T} + A[\underline{\mathbf{W}}_h, \underline{\mathbf{Z}}_h] + S(\underline{\mathbf{U}}_h, \underline{\mathbf{W}}_h) + A[\underline{\mathbf{U}}_h, \underline{\mathbf{Y}}_h] - S^*(\underline{\mathbf{Y}}_h, \underline{\mathbf{Z}}_h) $$

■ Taking $\underline{\mathbf{W}}_h = \underline{\mathbf{U}}_h$ et $\underline{\mathbf{Y}}_h = -\underline{\mathbf{Z}}_h$ yields

$$ B[(\underline{\mathbf{U}}_h, \underline{\mathbf{Z}}_h), (\underline{\mathbf{U}}_h, -\underline{\mathbf{Z}}_h)] = \|\underline{u}_1\|_{\omega_T}^2 + S(\underline{\mathbf{U}}_h, \underline{\mathbf{U}}_h) + S^*(\underline{\mathbf{Z}}_h, \underline{\mathbf{Z}}_h) := \|\| (\underline{\mathbf{U}}_h, \underline{\mathbf{Z}}_h) \|\|^2 \rightsquigarrow \text{stability.} $$
Stabilization $\underline{S}(\cdot, \cdot)$

Sufficiently strong
such that $\|\|\cdot, \cdot\|$ is a norm on
$\mathcal{W}_{h, \Delta t}^{q, k} \times \mathcal{W}_{h, \Delta t}^{q^*, k_*}$.

Sufficiently
weak such that
$$ \|\|(\mathbf{U}_h - \mathbf{U}, \mathbf{Z}_h)\|\| \lesssim h^s \|\mathbf{u}\|_{\diamond} + \|\delta \mathbf{u}\|_{L^2(\omega_T)} $$

Consistency (ideally)
$\underline{S}(\mathbf{U}, \mathbf{U}) = 0$
$\mathbf{U} = (u, \partial_t u)$
exact sol.

■ Boost convergence in weak norm by applying stability estimate to the error $e_h$, i.e.

$$ \begin{aligned} & \|e_h\|_{L^\infty(0, T; L^2(\Omega))} + \|\partial_t e_h\|_{L^2(0, T; H^{-1}(\Omega))} \\ & \lesssim \|e_h\|_{\omega_T} + \|e_h\|_{\Sigma} + \|\Box e_h\|_{H^{-1}(Q)}. \end{aligned} $$

■ Note: not possible to apply directly to $e_h = u - \underline{u}_1 \notin H^1(Q)$.

■ Ignoring this for the moment, the key is to control

$$ \begin{aligned} \|\Box e_h\|_{H^{-1}(Q)} &= \|\Box \underline{u}_1\|_{H^{-1}(Q)} \\ &:= \sup_{\substack{y \in H_0^1(Q), \\ \|y\|_{H^1(Q)} = 1}} \int \{-(\partial_t \underline{u}_1)\partial_t y + \nabla \underline{u}_1 \nabla y\} \\ &\stackrel{!}{\lesssim} \|\|(\underline{U}_h - \mathbf{U}, \underline{Z}_h)\|\| + \mathcal{O}(h^s) \end{aligned} $$

■ Recall $\|\|(\underline{U}_h - \mathbf{U}, \cdot)\|\| = \|\underline{u}_1 - u\|_{\omega_T}^2 + \underline{S}(\underline{U}_h - \mathbf{U}, \underline{U}_h - \mathbf{U})$.
$$ \|\Box \underline{u}_1\|_{H^{-1}(Q)} := \sup_{\substack{y \in H_0^1(Q), \\ \|y\|_{H^1(Q)}=1}} \int_Q \{-(\partial_t \underline{u}_1)\partial_t y + \nabla \underline{u}_1 \nabla y\} $$

$$ = \sup_{\substack{y \in H_0^1(Q), \\ \|y\|_{H^1(Q)}=1}} \left\{ \sum_{n=0}^{N-1} \int_{I_n} \sum_{K \in \mathcal{T}_h} (\partial_t \underline{u}_2 - \Delta \underline{u}_1, y)_K dt + \sum_{n=0}^{N-1} (\underline{u}_2 - \partial_t \underline{u}_1, \partial_t y)_{Q^n} \right. \\ \left. + \sum_{n=0}^{N-1} \int_{I_n} \sum_{F \in \mathcal{F}_i} (\llbracket \nabla \underline{u}_1 \cdot \mathbf{n} \rrbracket_F, y)_F dt + \sum_{n=1}^{N-1} (\llbracket \underline{u}_2^n \rrbracket, y)_{\Omega} \right\} $$

<figure><img src="image_20.png" /></figure>

t

$$ \begin{array}{c} t_{n+1} \\ \hline t_n \end{array} \begin{array}{c} \underset{\underline{u}_2 \downarrow}{u_n^\mathrm{n}} \\ \underset{\underline{u}_1 \uparrow}{u_n^\mathrm{n}} \end{array} \quad \begin{array}{c} Q_n \\ \Sigma_n \end{array} $$

$$ \rightsquigarrow \underline{S}(\underline{u}_h, \underline{w}_h) = \sum_{n=0}^{N-1} \int_{I_n} \sum_{F \in \mathcal{F}_i} h(\llbracket \nabla \underline{u}_1 \rrbracket, \llbracket \nabla \underline{w}_1 \rrbracket)_F dt + \sum_{n=0}^{N-1} (\underline{u}_2 - \partial_t \underline{u}_1, \underline{w}_2 - \partial_t \underline{w}_1)_{Q^n} $$

$$ + \sum_{n=0}^{N-1} \int_{I_n} \sum_{K \in \mathcal{T}_h} h^2(\partial_t \underline{u}_2 - \Delta \underline{u}_1, \partial_t \underline{w}_2 - \Delta \underline{w}_1)_K dt + \sum_{n=1}^{N-1} \frac{1}{\Delta t} (\llbracket \underline{u}_2^n \rrbracket, \llbracket \underline{w}_2^n \rrbracket)_{\Omega} + \dots $$
■ Argument from above not valid since $\underline{u}_1 \notin H^1(Q)$.

■ Replace $\underline{u}_1$ by $\tilde{u}_1 := L_h \underline{u}_1$ where lifting operator $L_h: W_{h,\Delta t}^{k,q} \to C^0((0,T); V_h^k)$ defined as:

■ Let $\vartheta_n(t) := (t_{n+1} - t)/(t_{n+1} - t_n)$ for $t \in I_n$ and set

$$L_h \underline{w}(t) = \underline{w}(t) - [[\underline{w}^n]] \vartheta_n(t), \quad t \in I_n, n \ge 1, \quad L_h \underline{w}(t) = \underline{w}(t), \quad t \in I_0.$$

Now $L_h \underline{w}$ is continuous in time since we have

$$\lim_{t \downarrow t_n} L_h \underline{w}(t) = \underline{w}_+^n - [[\underline{w}^n]] = \underline{w}_-^n = \lim_{t \uparrow t_n} L_h \underline{w}(t).$$

■ To apply argument to $e_h = u - \tilde{u}_1$, need to strengthen stabilization to control jumps in time:

$$\underline{s}(\underline{U}_h, \underline{W}_h) = + \sum_{n=1}^{N-1} \left\{ \frac{1}{\Delta t} ([[\underline{u}_1^n], [[\underline{w}_1^n]]]_\Omega + \Delta t ([[\nabla \underline{u}_1^n], [[\nabla \underline{w}_1^n]]]_\Omega)) \right\}$$
Lipschitz stab. knowing $u|_{\Sigma}$

$$ \begin{aligned} & \|u\|_{L^{\infty}(0,T;L^2(\Omega))} + \|\partial_t u\|_{L^2(0,T;H^{-1}(\Omega))} \\ & \lesssim \|u\|_{L^2(\omega_T)} + \|u\|_{L^2(\Sigma)} + \|\Box u\|_{H^{-1}(Q)} \end{aligned} $$

Nitsche stability term:

$$ \underline{S} = + \sum_{n=0}^{N-1} h^{-1}(\underline{u}_1, \underline{w}_1) \Sigma^n $$
Lipschitz stab. knowing $u|_{\Sigma}$

$$ \begin{aligned} & \|u\|_{L^{\infty}(0, T; L^2(\Omega))} + \|\partial_t u\|_{L^2(0, T; H^{-1}(\Omega))} \\ & \lesssim \|u\|_{L^2(\omega_T)} + \|u\|_{L^2(\Sigma)} + \|\Box u\|_{H^{-1}(Q)} \end{aligned} $$

Lipschitz stab. knowing $V_M \ni u|_{\Sigma}$

$$ \begin{aligned} & \|u\|_{L^{\infty}(0, T; L^2(\Omega))} + \|\partial_t u\|_{L^2(0, T; H^{-1}(\Omega))} \\ & \lesssim \|u\|_{L^2(\omega_T)} + \|Qu\|_{L^2(\Sigma)} + \|\Box u\|_{H^{-1}(Q)} \end{aligned} $$

Nitsche stability term:

$$ \underline{S} = + \sum_{n=0}^{N-1} h^{-1}(\underline{u}_1, \underline{w}_1) \Sigma^n $$

Control distance to $V_M$:

$$ \underline{S} = + (Qu_1, Qw_1)_{\Sigma} $$
## Lipschitz stab. knowing $u|_{\Sigma}$

$$ \begin{aligned} & \|u\|_{L^{\infty}(0, T; L^2(\Omega))} + \|\partial_t u\|_{L^2(0, T; H^{-1}(\Omega))} \\ & \lesssim \|u\|_{L^2(\omega_T)} + \|u\|_{L^2(\Sigma)} + \|\Box u\|_{H^{-1}(Q)} \end{aligned} $$

## Lipschitz stab. knowing $\mathcal{V}_M \ni u|_{\Sigma}$

$$ \begin{aligned} & \|u\|_{L^{\infty}(0, T; L^2(\Omega))} + \|\partial_t u\|_{L^2(0, T; H^{-1}(\Omega))} \\ & \lesssim \|u\|_{L^2(\omega_T)} + \|Qu\|_{L^2(\Sigma)} + \|\Box u\|_{H^{-1}(Q)} \end{aligned} $$

## Hölder stability

$$ \begin{aligned} \|u\|_{L^2(B)} & \lesssim (\|u\|_{L^2(\omega_T)} + \|\Box u\|_{H^{-1}(\tilde{Q})})^{\alpha} \\ & \quad \times \|u\|_{L^2(\tilde{Q})}^{1-\alpha}. \end{aligned} $$

Nitsche stability term:

$$ \underline{S} += \sum_{n=0}^{N-1} h^{-1}(\underline{u}_1, \underline{w}_1)^{\Sigma^n} $$

Control distance to $\mathcal{V}_M$:

$$ \underline{S} += (Qu_1, Qw_1)^{\Sigma} $$

Tikhonov term (weakly inconsistent):

$$ \underline{S} += \gamma \sum_{n=0}^{N-1} h^{2\epsilon}(\underline{u}_1, \underline{w}_1)^{Q^n} $$
■ There is freedom in choosing the dual stabilization as $\underline{\mathbf{Z}}_h$ approximates zero.

■ It needs to provide a norm on $W_{h, \Delta t}^{q_*, k_*}$ and give some control on $A[\cdot, \cdot]$, i.e.

$$A[\mathbf{U}, \underline{\mathbf{Y}}_h] \lesssim \left( \sum_{n=0}^{N-1} \left\{ \|\partial_t u_2\|_{Q^n}^2 + \|\nabla u_1\|_{Q^n}^2 + \|\partial_t u_1\|_{Q^n}^2 + \|u_2\|_{Q^n}^2 \right. \right. \\ \left. \left. + \int_{I_n} \sum_{K \in \mathcal{T}_h} h^2 \|u_1\|_{H^2(K)}^2 \, \mathbf{d}t \right) \right)^{1/2} \underline{S}^*(\underline{\mathbf{Y}}_h, \underline{\mathbf{Y}}_h)^{1/2}.$$

■ We choose

$$\underline{S}^*(\underline{\mathbf{Y}}_h, \underline{\mathbf{Z}}_h) := \sum_{n=0}^{N-1} \{ (\underline{y}_1, \underline{z}_1)_{Q^n} + (\nabla \underline{y}_1, \nabla \underline{z}_1)_{Q^n} + (\underline{y}_2, \underline{z}_2)_{Q^n} + h^{-1}(\underline{y}_1, \underline{z}_1)_{\Sigma^n} \}.$$
Lipschitz stab. knowing $u|_{\Sigma}$

$$ \begin{aligned} & \|u\|_{L^{\infty}(0,T;L^2(\Omega))} + \|\partial_t u\|_{L^2(0,T;H^{-1}(\Omega))} \\ & \lesssim \|u\|_{L^2(\omega_T)} + \|u\|_{L^2(\Sigma)} + \|\Box u\|_{H^{-1}(Q)} \end{aligned} $$

$$ \begin{aligned} & \|u - \tilde{u}_1\|_{L^{\infty}([0,T];L^2(\Omega))} \\ & \quad + \|\partial_t(u - \tilde{u}_1)\|_{L^2([0,T];H^{-1}(\Omega))} \\ & \lesssim h^s \|u\|_{H^{s+2}(Q)} + \|\delta u\|_{L^2(\omega_T)}. \end{aligned} $$
Lipschitz stab. knowing $u|_{\Sigma}$

$$ \begin{aligned} & \|u\|_{L^{\infty}(0,T;L^2(\Omega))} + \|\partial_t u\|_{L^2(0,T;H^{-1}(\Omega))} \\ & \lesssim \|u\|_{L^2(\omega_T)} + \|u\|_{L^2(\Sigma)} + \|\Box u\|_{H^{-1}(Q)} \end{aligned} $$

$$ \begin{aligned} & \|u - \tilde{u}_1\|_{L^{\infty}([0,T];L^2(\Omega))} \\ & \quad + \|\partial_t(u - \tilde{u}_1)\|_{L^2([0,T];H^{-1}(\Omega))} \\ & \lesssim h^s \|u\|_{H^{s+2}(Q)} + \|\delta u\|_{L^2(\omega_T)}. \end{aligned} $$

Lipschitz stab. knowing $V_M \ni u|_{\Sigma}$

$$ \begin{aligned} & \|u\|_{L^{\infty}(0,T;L^2(\Omega))} + \|\partial_t u\|_{L^2(0,T;H^{-1}(\Omega))} \\ & \lesssim \|u\|_{L^2(\omega_T)} + \|Qu\|_{L^2(\Sigma)} + \|\Box u\|_{H^{-1}(Q)} \end{aligned} $$

$$ \begin{aligned} & \|u - \tilde{u}_1\|_{L^{\infty}([0,T];L^2(\Omega))} \\ & \quad + \|\partial_t(u - \tilde{u}_1)\|_{L^2([0,T];H^{-1}(\Omega))} \\ & \lesssim h^s \|u\|_{H^{s+2}(Q)} + \|\delta u\|_{L^2(\omega_T)}. \end{aligned} $$
## Lipschitz stab. knowing $u|_{\Sigma}$

$$ \begin{aligned} & \|u\|_{L^{\infty}(0, T; L^2(\Omega))} + \|\partial_t u\|_{L^2(0, T; H^{-1}(\Omega))} \\ & \lesssim \|u\|_{L^2(\omega_T)} + \|u\|_{L^2(\Sigma)} + \|\Box u\|_{H^{-1}(Q)} \end{aligned} $$

$$ \begin{aligned} & \|u - \tilde{u}_1\|_{L^{\infty}([0, T]; L^2(\Omega))} \\ & \quad + \|\partial_t(u - \tilde{u}_1)\|_{L^2([0, T]; H^{-1}(\Omega))} \\ & \lesssim h^s \|u\|_{H^{s+2}(Q)} + \|\delta u\|_{L^2(\omega_T)}. \end{aligned} $$

## Lipschitz stab. knowing $\mathcal{V}_M \ni u|_{\Sigma}$

$$ \begin{aligned} & \|u\|_{L^{\infty}(0, T; L^2(\Omega))} + \|\partial_t u\|_{L^2(0, T; H^{-1}(\Omega))} \\ & \lesssim \|u\|_{L^2(\omega_T)} + \|Qu\|_{L^2(\Sigma)} + \|\Box u\|_{H^{-1}(Q)} \end{aligned} $$

$$ \begin{aligned} & \|u - \tilde{u}_1\|_{L^{\infty}([0, T]; L^2(\Omega))} \\ & \quad + \|\partial_t(u - \tilde{u}_1)\|_{L^2([0, T]; H^{-1}(\Omega))} \\ & \lesssim h^s \|u\|_{H^{s+2}(Q)} + \|\delta u\|_{L^2(\omega_T)}. \end{aligned} $$

## Hölder stability

$$ \begin{aligned} & \|u\|_{L^2(B)} \lesssim (\|u\|_{L^2(\omega_T)} + \|\Box u\|_{H^{-1}(\tilde{Q})})^{\alpha} \\ & \times \|u\|_{L^2(\tilde{Q})}^{1-\alpha}. \end{aligned} $$

$$ \begin{aligned} & \|u - \underline{u}_1\|_{L^2(B)} \\ & \lesssim h^{\alpha s} \|u\|_{H^{s+2}(Q)} + h^{-(1-\alpha)s} \|\delta u\|_{L^2(\omega_T)}. \end{aligned} $$
Numerical experiments
<figure><img src="image_21.png" /></figure>

<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /></figure>

<figure><img src="image_25.png" /></figure>

<figure><img src="image_26.png" /></figure>

<figure><img src="image_27.png" /></figure>

■ Optimal convergence is observed as predicted by theory.
Hölder stable case for clean data: $||u - u_1||_{L^2(B)} \lesssim h^{\alpha s} ||u||_{H^{s+2}(Q)}$

- For clean data actually observe the same (optimal) convergence in $B$ as for the Lipschitz stable case.

<figure><img src="image_28.png" /></figure>

- Up to the boundary observe logarithmic convergence (not covered by our theory).
Convergence of $||u - u_1||_{L^2(B_\kappa)}$ in the subdomains $B_\kappa = \mathcal{V}(\kappa\rho)$

<figure><img src="image_29.png" /></figure>

<table><tr><td>■ Recall that V(κρ) := {((t, x) ∈ (-T, T) × Ω | ψ(t, x) > κρ)}. ■ Note that B = B<sub>1</sub>.</td></tr></table>
■ Consider exact solution $u = 5 \cos(\sqrt{2}\frac{\pi}{2}t) \cos(\frac{\pi}{2}x) \cos(\frac{\pi}{2}y)$.

■ Noise $\delta u \sim \Pi_h(u \cos(2\pi x))$ such that $\|\delta u\|_{L^2(\omega_T)} \sim h^\theta$ for $\theta \in [1, 2]$.

■ According to error estimate $\|u - \underline{u}_1\|_{L^2(B)} \lesssim h^{\alpha s} \|u\|_{H^m(Q)} + h^{\theta-s(1-\alpha)}$.

<figure><img src="image_30.png" /></figure>

■ We observe $\alpha = 1$. How to make the reduced stability appear?
Let's look in the kernel of the system matrix.

Find the smallest $\lambda$ and corresponding mode ($\mathbf{U}_h^\lambda, \mathbf{Z}_h^\lambda$) such that

$$ \begin{bmatrix} \tilde{\mathbf{S}} & \mathbf{A} \\ \mathbf{A}^T & \mathbf{S}^* \end{bmatrix} \begin{bmatrix} \mathbf{U}_h^\lambda \\ \mathbf{Z}_h^\lambda \end{bmatrix} = \lambda \begin{bmatrix} \mathbf{M}_u & 0 \\ 0 & \mathbf{M}_z \end{bmatrix} \begin{bmatrix} \mathbf{U}_h^\lambda \\ \mathbf{Z}_h^\lambda \end{bmatrix}. $$

For $N = 64$ and $s = q = k = 1$ we find $\lambda \sim 6 \cdot 10^{-4}$.

A small perturbation in the data domain $\omega_T$ creates huge perturbation outside.

Evolution of the mode follows the geometry of the stability region.

<figure><img src="image_31.png" /></figure>

<figure><img src="image_32.png" /></figure>
Convergence in $B$ for noisy data, $s = q = k = 1$

- Noise $\delta u \sim \Pi_h(\underline{u}_1^\lambda)$ such that $\|\delta u\|_{L^2(\omega_T)} \sim h^\theta$ for an $\theta \in [1, 2]$.

- According to error estimate $\|u - \underline{u}_1\|_{L^2(B)} \lesssim h^{\alpha s} \|u\|_{H^m(Q)} + h^{\theta-s(1-\alpha)}$.

<figure><img src="image_33.png" /></figure>

- Reduced stability $\alpha \sim 0.85$ is indeed observed.
<figure><img src="image_34.png" /></figure>

■ Let $\mathcal{V}_M := \operatorname{span}\{\varphi_m \mid m \in \{1, \dots, M\}\}$,

■ Consider the Fourier modes

$$ \varphi_m = \cos\left(\sqrt{2}\frac{m\pi}{4}t\right) \cos\left(\frac{m\pi}{4}x\right) \cos\left(\frac{m\pi}{4}y\right). $$

■ And the exact solution $u = 5\varphi_2$. Clearly, $u|_{\Sigma} \in \mathcal{V}_2$.

$$ \frac{\|u - \underline{u}_1\|_{Q \setminus B}}{\|u - \underline{u}_1\|_B} = \frac{q = k = 2, q_* = 0, k_* = 1}{\|u - \underline{u}_1\|_B} $$

<figure><img src="image_35.png" /></figure>
And if solution can only be approximated in $V_M$ with limited precision ?

■ Consider exact solution $u_η = 5 \cdot ϕ_2 + ηϕ_3$ for $η > 0$ and $V_2$.

■ Note that $ϕ_2 \in V_2$ but $ϕ_3 \notin V_2$.

<figure><img src="image_36.png" /></figure>
- Assume no information about $u|_{\partial\Omega}$ known.

<figure><img src="image_37.png" /></figure>

- Let $v_0 := (2\pi + 2\epsilon - a)/2\epsilon$

$\|u - u_h\|_{L^2(Q_T)}$ for $T = 2 - 2\epsilon$

<figure><img src="image_38.png" /></figure>
- Assume no information about $u|_{\partial\Omega}$ known.

- Let $v_0 := (2\pi + 2\epsilon - a)/2\epsilon$

<figure><img src="image_39.png" /></figure>

Observability property*

- For every $v > v_0$ there exists $T_0 < \infty$ such that $\cup_{t \in (0, T)} \omega(t)$ satisfies the GCC for $T > T_0$.

- We have $T_0 \sim 2 - 2\epsilon$ as $v \to \infty$.

$\|u - u_h\|_{L^2(Q_T)}$ for $T = 2 - 2\epsilon$

<figure><img src="image_40.png" /></figure>

<footnote>(*): J. Le Rousseau, G. Lebeau, P. Terpolilli & E. Trélat, "Geometric control for the wave equation with a time-dependent observation domain", Analysis & PDE, 2017</footnote>
- Assume no information about $u|_{\partial\Omega}$ known.

- Let $v_0 := (2\pi + 2\epsilon - a)/2\epsilon$

<figure><img src="image_41.png" /></figure>

Observability property*

- For every $v > v_0$ there exists $T_0 < \infty$ such that $\cup_{t \in (0, T)} \omega(t)$ satisfies the GCC for $T > T_0$.

- We have $T_0 \sim 2 - 2\epsilon$ as $\nu \to \infty$.

$$ \|u - u_h\|_{L^2(Q_T)} \text{ for } T = 2 - 2\epsilon $$

<figure><img src="image_42.png" /></figure>

<footnote>(*): J. Le Rousseau, G. Lebeau, P. Terpolilli & E. Trélat, "Geometric control for the wave equation with a time-dependent observation domain", Analysis & PDE, 2017</footnote>
- Assume no information about $u|_{\partial\Omega}$ known.

- Let $v_0 := (2\pi + 2\epsilon - a)/2\epsilon$

<figure><img src="image_43.png" /></figure>

Observability property*

- For every $v > v_0$ there exists $T_0 < \infty$ such that $\cup_{t \in (0, T)} \omega(t)$ satisfies the GCC for $T > T_0$.

- We have $T_0 \sim 2 - 2\epsilon$ as $v \to \infty$.

$\|u - u_h\|_{L^2(Q_T)}$ for $T = 2 - 2\epsilon$

<figure><img src="image_44.png" /></figure>

<footnote>(*): J. Le Rousseau, G. Lebeau, P. Terpolilli & E. Trélat, "Geometric control for the wave equation with a time-dependent observation domain", Analysis & PDE, 2017</footnote>
$$ \min_{u \in H^1(Q)} \frac{1}{2} \|u - u_{\omega}\|^2_{L^2(\omega_T)} \text{ such that } \Box u = 0, \text{ in } H^{-1}(Q), $$

■ Suppose that the geometric control condition holds.

<table><thead><tr><th>$u|_{\Sigma}$ known</th><th>$u|_{\Sigma} = ?$</th><th>Knowing $\mathcal{V}_M \ni u|_{\Sigma}$</th></tr></thead><tbody><tr><td>Lipschitz stability<br>in $Q$</td><td>Hölder stability in $B$<br>logarithmic in $Q \setminus B$</td><td>Recover Lipschitz stability<br>if $(\dim(\mathcal{V}_M)$ small)</td></tr></tbody></table>

■ Discretization by stabilized primal dual finite element method.

■ Stability estimates drive design of regularization / stabilization.

■ Error estimates that reflect the stability properties of the continuum problem.

E. Burman, L. Oksanen, J. Preuss & Z. Zhao, "Unique continuation for the wave equation: the stability landscape.", arXiv:2510.20359, 2025
Thank you for your attention!
# Appendix
■ Let $B^0[\cdot, \cdot]$ denote the bilinear form with the term $(Qu_1, Qw_1)_\Sigma$ removed.

■ We introduce an additional variable on $\Sigma$ and solve the augmented variational formulation:

Find $(\underline{U}_h, \underline{\mu}_h, \underline{Z}_h) \in W_{h,\Delta t}^{k,q} \times V_M \times W_{h,\Delta t}^{k_*,q_*}$ such that

$$B^#[(\underline{U}_h, \underline{\mu}_h, \underline{Z}_h), (\underline{W}_h, \underline{\eta}_h, \underline{Y}_h)] = (\tilde{u}_\omega, \underline{w}_1)_{\omega_T}$$

for all $(\underline{W}_h, \underline{\eta}_h, \underline{Y}_h) \in W_{h,\Delta t}^{k,q} \times V_M \times W_{h,\Delta t}^{k_*,q_*}$, where

$$B^#[(\underline{U}_h, \underline{\mu}_h, \underline{Z}_h), (\underline{W}_h, \underline{\eta}_h, \underline{Y}_h)] := B^0[(\underline{U}_h, \underline{Z}_h), (\underline{W}_h, \underline{Y}_h)] + (\underline{u}_1 - \underline{\mu}_h, \underline{w}_1 - \underline{\eta}_h)_\Sigma.$$
Is there a penalty for choosing $\mathcal{V}$ too large?

■ Let $u = 5\varphi_8$.

■ Compare convergence using optimal space $\mathcal{V} = \operatorname{span}\{\varphi_8\}$ with convergence using $\mathcal{V} = \operatorname{span}\{\varphi_1, \dots, \varphi_8\}$

■ In both cases a convergence

$$ \|u - \underline{u}_1\|_{L^2(Q)} \le C_\nu h^{s+1} \|u\|_{H^{s+1}(Q)}. $$

is observed, but the constant $C_\nu$ depends on $\mathcal{V}$!

<figure><img src="image_45.png" /></figure>
■ Let $u_M = 5\varphi_M$.

■ Calculate the constant $C_{V_M}$ in

$$C_{V_M} = \frac{\|u_M - \underline{u}_1\|_{L^2(Q)}}{h^{s+1} \|u_M\|_{H^{s+1}(Q)}}$$

for $s = k = q = 2$

■ It seems that $C_{V_M}$ grows exponentially with $\dim(V_M) = M$.

We have to keep $V_M$ small!

$C_{V_M}$ for $q = k = 2$

<figure><img src="image_46.png" /></figure>

<figure><img src="image_47.png" /></figure>

■ How to find $V_M$: For example using POD [Boulakia, James & Lombardi, '24] or ML.
We consider spatial dimension $d = 2$ and choose the values for the parameters as follows:

$$R = 1, \quad r = 3/4, \quad \rho = \rho_0 + \frac{(\rho_1 - \rho_0)}{10}, \quad \epsilon = 5 \times 10^{-2}.$$

We compute up to

$$T = \sqrt{\frac{\rho_1 - \rho}{1 - \epsilon}} \approx 0.843.$$

* The asymptotic convergence rates appear to be insensitive to the specific choice of these parameters as long as one respects the relations:

$$\rho \in (\rho_0, \rho_1) \text{ and } T > \sqrt{\frac{\rho_1 - \rho}{1 - \epsilon}}.$$
$$ \|u - \underline{u}_1\|_{L^2(B)} \lesssim h^{\alpha s} \|u\|_{H^{s+2}(Q)} + h^{-(1-\alpha)s} \|\delta u\|_{L^2(\omega_T)}. $$

Two regimes separated by

$$ h_{\min} := \left( \frac{\|\delta u\|_{L^2(\omega_T)}}{\|u\|_{H^{s+2}(Q)}} \right)^{1/s}. $$

* For $h > h_{\min}$ first term dominates and mesh refinement will improve accuracy.

* For $h < h_{\min}$ second term dominates and mesh refinement will lead to poorer accuracy.

∴ Stop refinement at $h = h_{\min}$ (requires estimate of $\|u\|_{H^{s+2}(Q)}$ and the noise level).
Using $B[(\mathbf{U}_h, \mathbf{Z}_h), (\mathbf{W}_h, \mathbf{Y}_h)] = (\tilde{u}_\omega, \underline{w}_1)_{\omega_T}$, consistency and that $\mathbf{U}$ solves the wave equation:

$$
\begin{align*}
B[(\mathbf{U}_h - \Pi_h \mathbf{U}, \mathbf{Z}_h), (\mathbf{W}_h, \mathbf{Y}_h)]
&= (u - \Pi_h u, \underline{w}_1)_{\omega_T} - \underline{S}(\Pi_h \mathbf{U}, \mathbf{W}_h) - A[\Pi_h \mathbf{U}, \mathbf{Y}_h] + (\delta u, \underline{w}_1)_{\omega_T} \\
&= (u - \Pi_h u, \underline{w}_1)_{\omega_T} - \underline{S}(\Pi_h \mathbf{U} - \mathbf{U}, \mathbf{W}_h) - A[\Pi_h \mathbf{U} - \mathbf{U}, \mathbf{Y}_h] + (\delta u, \underline{w}_1)_{\omega_T} \\
&\lesssim (\mathbf{h}^s \|u\|_{H^{s+2}(Q)} + \|\delta u\|_{L^2(\omega_T)}) \|\underline{(\mathbf{W}_h, \mathbf{Y}_h)}\|.
\end{align*}
$$

The inf-sup condition

$$
\sup_{(\underline{\mathbf{W}}_h, \underline{\mathbf{Y}}_h) \in W_{h, \Delta t}^{k,q} \times W_{h, \Delta t}^{k*,q*}} \frac{B[(\underline{\mathbf{U}}_h, \underline{\mathbf{Z}}_h), (\underline{\mathbf{W}}_h, \underline{\mathbf{Y}}_h)]}{\|\underline{(\underline{\mathbf{W}}_h, \underline{\mathbf{Y}}_h)}\|} \gtrsim \|(\underline{\mathbf{U}}_h, \underline{\mathbf{Z}}_h)\|
$$

now implies that

$$
\|\underline{(\underline{\mathbf{U}}_h - \Pi_h \mathbf{U}, \underline{\mathbf{Z}}_h)}\| \lesssim \mathbf{h}^s \|\underline{u}\|_{\diamond} + \|\delta \underline{u}\|_{L^2(\omega_T)}.
$$
Stabilisation $S(\cdot, \cdot)$

Strong enough s.t.
$$ |||\cdot, \cdot||| $$ is norm on
$$ W_{h, \Delta t}^{q,k} \times W_{h, \Delta t}^{q_*, k_*} $$

Weak enough s.t.
$$ \|\|(\mathbf{U}_h - \mathbf{U}, \mathbf{Z}_h)\|\| \\ \lesssim h^{\min\{q,k\}} \|\mathbf{u}\|_{\diamond} $$

Strong enough to
control residual
$$ \|\Box e_h\|_{H^{-1}(Q)} \\ \lesssim \|\|(\mathbf{U}_h - \mathbf{U}, \mathbf{Z}_h)\|\| $$

- Provides regularization.

- Desirable: consistency. For sufficiently smooth exact solution $\mathbf{U} = (u, \partial_t u)$:

$S(\mathbf{U}, \mathbf{U}) = 0$.
Stabilisation $S(\cdot, \cdot)$

Strong enough s.t.
$|||(\cdot, \cdot)|||$ is norm on
$W_{h, \Delta t}^{q,k} \times W_{h, \Delta t}^{q_*, k_*}$

Weak enough s.t.
$$ |||(\mathbf{U}_h - \mathbf{U}, \mathbf{Z}_h)||| \\ \lesssim h^{\min\{q,k\}} ||u||_\bullet $$

Strong enough to
control residual
$$ \begin{aligned} & \|\Box e_h\|_{H^{-1}(Q)} \\ \lesssim & |||(\underline{\mathbf{U}}_h - \mathbf{U}, \underline{\mathbf{Z}}_h)||| \end{aligned} $$

$$ |||(\underline{\mathbf{U}}_h, \underline{\mathbf{Z}}_h)|||^2 := |\underline{\mathbf{U}}_h|^2_{\underline{\mathcal{S}}} + ||\underline{\mathbf{u}}_1||^2_{\omega_T} + ||\underline{\mathbf{Z}}_h||^2_{\underline{\mathcal{S}}^*} = 0. $$

■ Need to show: $\underline{\mathbf{U}}_h = 0$.
Stabilisation $S(\cdot, \cdot)$

Strong enough s.t.
$$ \|\|\cdot\|\|_S $$ is norm on
$$ W_{h, \Delta t}^{q,k} \times W_{h, \Delta t}^{q*,k*}. $$

Weak enough s.t.
$$ \|\| (\mathbf{U}_h - \mathbf{U}, \mathbf{Z}_h) \|\| \le h^{\min\{q,k\}} \|\mathbf{u}\|_{\diamond} $$

Strong enough to
control residual
$$ \|\Box e_h\|_{H^{-1}(Q)} \le \|\| (\mathbf{U}_h - \mathbf{U}, \mathbf{Z}_h) \|\| $$

$$ \|\| (\mathbf{U}_h, \mathbf{Z}_h) \|\|^2 := |\mathbf{U}_h|_S^2 + \|\underline{u}_1\|_{\omega_T}^2 + \|\mathbf{Z}_h\|_{S^*}^2 = 0. $$

■ Need to show: $\underline{\mathbf{U}}_h = 0$.

■ If $|\underline{\mathbf{U}}_h|_S = 0$ guarantees $[\underline{u}_1^n] = 0$ for $n = 1, \dots, N-1$ then $\underline{u}_1 \in H^1(Q)$ and can apply continuum stability estimate

$$ \|\underline{u}_1\|_{L^{\infty}(0,T;L^2(\Omega))} \lesssim \|\underline{u}_1\|_{\omega_T} + \|\Box \underline{u}_1\|_{H^{-1}(Q)} + \|\underline{u}_1\|_{S^*}. $$
Stabilisation $S(\cdot, \cdot)$

Strong enough s.t.
$$ \|\|\cdot\|\|_S $$ is norm on
$$ W_{h, \Delta t}^{q,k} \times W_{h, \Delta t}^{q_*, k_*} $$

Weak enough s.t.
$$ \|\| (\mathbf{U}_h - \mathbf{U}, \mathbf{Z}_h) \|\| \lesssim h^{\min\{q,k\}} \|\mathbf{u}\|_{\odot} $$

Strong enough to
control residual

$$ \begin{aligned} & \|\Box e_h\|_{H^{-1}(Q)} \\ & \lesssim \|\| (\mathbf{U}_h - \mathbf{U}, \mathbf{Z}_h) \|\| \end{aligned} $$

$$ \|\| (\mathbf{U}_h, \mathbf{Z}_h) \|\|^2 := |\mathbf{U}_h|_S^2 + \|\underline{u}_1\|_{\omega_T}^2 + \|\mathbf{Z}_h\|_{S^*}^2 = 0. $$

■ Need to show: $\mathbf{U}_h = 0$.

■ If $|\mathbf{U}_h|_S = 0$ guarantees $[\underline{u}_1^n] = 0$ for $n = 1, \dots, N-1$ then $\underline{u}_1 \in H^1(Q)$ and can apply continuum stability estimate
$$ \|\underline{u}_1\|_{L^\infty(0,T;L^2(\Omega))} \lesssim \|\underline{u}_1\|_{\omega_T} + \|\Box \underline{u}_1\|_{H^{-1}(Q)} + \|\underline{u}_1\|_{\Sigma}. $$

■ If $|\mathbf{U}_h|_S = 0$ guarantees additionally:

$$ \begin{aligned} [\underline{u}_2^n] &= 0, n = 1, \dots, N-1, & [\nabla \underline{u}_1 \cdot \mathbf{n}]_F &= 0, \forall \mathbf{F} \in \mathcal{F}_i \\ \underline{u}_2 &= \partial_t \underline{u}_1, \text{ on } Q_n, & \partial_t \underline{u}_2 - \Delta \underline{u}_1 &= 0, \forall \mathbf{K} \in \mathcal{T}_h, \end{aligned} $$

$$ \rightsquigarrow \|\Box \underline{u}_1\|_{H^{-1}(Q)} = \sup_{\|y\|_{H_0^1(Q)}=1} \int_0^T \int_\Omega (-\partial_t \underline{u}_1 \partial_t y + \nabla \underline{u}_1 \nabla y) \, dx \, dt = 0. $$
Stabilisation $S(\cdot, \cdot)$

Strong enough s.t.
$\|\|\cdot, \cdot\|$ is norm on
$W_{h, \Delta t}^{q, k} \times W_{h, \Delta t}^{q_*, k_*}$.

Weak enough s.t.
$$ \|\|(\mathbf{U}_h - \mathbf{U}, \mathbf{Z}_h)\|\| \lesssim h^{\min\{q,k\}} \|\mathbf{u}\|_{\diamond} $$

Strong enough to
control residual
$$ \begin{aligned} & \|\Box e_h\|_{H^{-1}(Q)} \\ & \lesssim \|\|(\mathbf{U}_h - \mathbf{U}, \mathbf{Z}_h)\|\| $$

■ Corresponding stabilization:

$$
\begin{aligned}
\underline{S}(\underline{\mathbf{U}}_h, \underline{\mathbf{W}}_h) = & \sum_{n=0}^{N-1} \int_{I_n} \sum_{F \in \mathcal{F}_i} h(\|\nabla \underline{\mathbf{u}}_1\|, \|\nabla \underline{\mathbf{w}}_1\|)_{\mathcal{F}} \, dt \\
& + \sum_{n=0}^{N-1} \int_{I_n} \sum_{K \in \mathcal{T}_h} h^2(\partial_t \underline{\mathbf{u}}_2 - \Delta \underline{\mathbf{u}}_1, \partial_t \underline{\mathbf{w}}_2 - \Delta \underline{\mathbf{w}}_1)_{\mathcal{K}} \, dt \\
& + \sum_{n=1}^{N-1} \frac{1}{\Delta t} (\|\underline{\mathbf{u}}_1^n\|, \|\underline{\mathbf{w}}_1^n\|)_{\Omega} + \sum_{n=1}^{N-1} \frac{1}{\Delta t} (\|\underline{\mathbf{u}}_2^n\|, \|\underline{\mathbf{w}}_2^n\|)_{\Omega} \\
& + \dots
\end{aligned}
$$
Stabilisation $S(\cdot, \cdot)$

Strong enough s.t.
$$ \|\cdot\|_{\infty} $$ is norm on
$$ W_{h, \Delta t}^{q,k} \times W_{h, \Delta t}^{q_*, k_*} $$

Weak enough s.t.
$$ \| (\mathbf{U}_h - \mathbf{U}, \mathbf{Z}_h) \| \le h^{\min\{q,k\}} \|\mathbf{U}\|_{\bullet} $$

Strong enough to
control residual
$$ \begin{aligned} & \|\Box e_h\|_{H^{-1}(Q)} \\ & \lesssim \|(\mathbf{U}_h - \mathbf{U}, \mathbf{Z}_h)\| \end{aligned} $$

■ Split into discretization and approximation error

$$ \|\|(\mathbf{U}_h - \mathbf{U}, \mathbf{Z}_h)\| \le \underbrace{\|\|(\mathbf{U}_h - \mathbf{\Pi}_{q,k}^{\text{dc}} \mathbf{U}, \mathbf{Z}_h)\|}_{\text{inf-sup} + \text{var. formulation}} + \underbrace{\|\|(\mathbf{U} - \mathbf{\Pi}_{q,k}^{\text{dc}} \mathbf{U}, \mathbf{0})\|}_{\text{interpolation}} $$

■ Discretization error treated using standard procedures see e.g. [Burman, Feizmohammadi, Münch & Oksanen, 21']

■ To compensate for inverse powers of $\Delta t$ we assume $\Delta t \sim h$.
Stabilisation $S(\cdot, \cdot)$

Strong enough s.t.
$$ \|\|\cdot\|\| \text{ is norm on} $$
$$ W_{h, \Delta t}^{q,k} \times W_{h, \Delta t}^{q_*, k_*} $$

Weak enough s.t.
$$ \|\| (\mathbf{U}_h - \mathbf{U}, \mathbf{Z}_h) \|\| \le h^{\min\{q,k\}} \|u\|_{\diamond} $$

Strong enough to
control residual

$$ \|\Box e_h\|_{H^{-1}(Q)} \le \|\| (\mathbf{U}_h - \mathbf{U}, \mathbf{Z}_h) \|\| $$

■ Contrôlez l'error $e_h$ en appliquant l'estimation de la stabilité.

$$ \begin{aligned} & \|e_h\|_{L^\infty(0, T; L^2(\Omega))} + \|\partial_t e_h\|_{L^2(0, T; H^{-1}(\Omega))} \\ & \lesssim \|e_h\|_{\omega_T} + \|e_h\|_{\Sigma} + \|\Box e_h\|_{H^{-1}(Q)}. \end{aligned} $$

■ Note: Cannot take $e_h = u - \underline{u}_1 \notin H^1(Q)$.
Stabilisation $S(\cdot, \cdot)$

Strong enough s.t.
$$ \|\|\cdot\|\| \text{ is norm on} $$
$$ W_{h, \Delta t}^{q,k} \times W_{h, \Delta t}^{q_*, k_*} $$

Weak enough s.t.
$$ \|\| (\mathbf{U}_h - \mathbf{U}, \mathbf{Z}_h) \|\| $$
$$ \lesssim h^{\min\{q,k\}} \|u\|_{\diamond} $$

Strong enough to
control residual

$$
\begin{aligned}
& \|\Box e_h\|_{H^{-1}(Q)} \\
& \lesssim \|\| (\mathbf{U}_h - \mathbf{U}, \mathbf{Z}_h) \|\|
\end{aligned}
$$

■ Contrôlez l'error $e_h$ en appliquant l'estimation de la stabilité.

$$
\begin{aligned}
& \|e_h\|_{L^{\infty}(0, T; L^2(\Omega))} + \|\partial_t e_h\|_{L^2(0, T; H^{-1}(\Omega))} \\
& \lesssim \|e_h\|_{\omega_T} + \|e_h\|_{\Sigma} + \|\Box e_h\|_{H^{-1}(Q)}.
\end{aligned}
$$

■ Note: Cannot take $e_h = u - \underline{u}_1 \notin H^1(Q)$.

Lifting of the jumps [Ern & Schieweck, 16']

Pick polynomial $\vartheta_n$ such that $\vartheta_n(t_n) = 1$, $\vartheta_n(t_{n+1}) = 0$ and

$$ \|\vartheta_n\|_{L^2(t_n)} \lesssim (\Delta t)^{1/2} \text{ and } \|\vartheta'_n\|_{L^2(t_n)} \lesssim (\Delta t)^{-1/2}. $$

$$ \mathcal{L}_{\Delta t} \underline{w}(t) = \underline{w}(t) - [\underline{w}^n] \vartheta_n(t), \quad t \in I_n, n \ge 1. $$

$$ \rightsquigarrow \text{ Define } e_h := u - \mathcal{L}_{\Delta t} \underline{u}_1 \in H^1(Q). $$
Stabilisation $S(\cdot, \cdot)$

Strong enough s.t.
$\|\|\cdot\|\|$ is norm on
$W_{h,\Delta t}^{q,k} \times W_{h,\Delta t}^{q_*,k_*}$.

Weak enough s.t.
$$ \|\|(\mathbf{U}_h - \mathbf{U}, \mathbf{Z}_h)\|\| \\ \lesssim h^{\min\{q,k\}} \|u\|_{\diamond} $$

Strong enough to
control residual
$$ \|\Box e_h\|_{H^{-1}(Q)} \\ \lesssim \|(\mathbf{U}_h - \mathbf{U}, \mathbf{Z}_h)\|\| $$

■ Contrôlez l'error $e_h$ en appliquant l'estimation de la stabilité.

$$ \|e_h\|_{L^\infty(0,T;L^2(\Omega))} + \|\partial_t e_h\|_{L^2(0,T;H^{-1}(\Omega))} \\ \lesssim \|e_h\|_{\omega_T} + \|e_h\|_{\Sigma} + \|\Box e_h\|_{H^{-1}(Q)} $$

■ Note: Cannot take $e_h = u - \underline{u}_1 \notin H^1(Q)$.

Lifting of the jumps [Ern & Schieweck, 16']

Pick polynomial $\vartheta_n$ such that $\vartheta_n(t_n) = 1$, $\vartheta_n(t_{n+1}) = 0$ and

$$ \|\vartheta_n\|_{L^2(t_n)} \lesssim (\Delta t)^{1/2} \quad \text{and} \quad \|\vartheta'_n\|_{L^2(t_n)} \lesssim (\Delta t)^{-1/2}. $$

$$ \mathcal{L}_{\Delta t} \underline{w}(t) = \underline{w}(t) - [\underline{w}^n]\vartheta_n(t), \quad t \in I_n, n \ge 1. $$

$$ \rightsquigarrow \text{ Define } e_h := u - \mathcal{L}_{\Delta t} \underline{u}_1 \in H^1(Q). $$

■ To control jumps in $\Box e_h$ need to stabilize $[\nabla \underline{u}_1^n]$ additionally!
What happens when GCC is violated

<figure><img src="image_48.png" /></figure>

<figure><img src="image_49.png" /></figure>

<figure><img src="image_50.png" /></figure>
$q = k = 2$

<table><thead><tr><th rowspan="2">N</th><th colspan="3">DFB</th><th colspan="3">M-I</th></tr><tr><th>ndof-tot</th><th>ndof-inv</th><th>iter</th><th>ndof-tot</th><th>ndof-inv</th><th>iter</th></tr></thead><tbody><tr><td>1</td><td>1,500</td><td>750</td><td>8</td><td>804</td><td>804</td><td>1</td></tr><tr><td>2</td><td>17,496</td><td>4,374</td><td>31</td><td>9,248</td><td>4,624</td><td>19</td></tr><tr><td>4</td><td>2.36 · 10<sup>5</sup></td><td>29,478</td><td>81</td><td>1.24 · 10<sup>5</sup></td><td>30,936</td><td>57</td></tr><tr><td>8</td><td>3.45 · 10<sup>6</sup></td><td>2.16 · 10<sup>5</sup></td><td>187</td><td>1.8 · 10<sup>6</sup></td><td>2.25 · 10<sup>5</sup></td><td>211</td></tr></tbody></table>
- $\Omega = [0, 1]$ with data domain $\omega = [0, 1/4]$ and final time $T = 1/2$

<figure><img src="image_51.png" /></figure>

- $B = \bigcup_{t \in [0, T]} B_t$, $B_t := \{x \le 1/4 + t \text{ for } t \le 1/4 \mid x \le 3/4 - t \text{ for } t \ge 1/4\}$
Need to solve

$$ \tilde{A}[\mathbf{W}_h, \mathbf{Z}_h] = (u_\omega - \underline{u}_1, \underline{w}_1)_{\omega_T} - S_h(\mathbf{U}_h, \mathbf{W}_h), \quad \forall \mathbf{W}_h \in \mathcal{W}_{h, \Delta t}^{k,q}. $$

$$ \tilde{A}[\underline{\mathbf{W}}_h, \underline{\mathbf{Z}}_h] = A[\underline{\mathbf{W}}_h, \underline{\mathbf{Z}}_h] + (\underline{w}_1, \underline{z}_1)_{\omega_T} + \frac{\lambda}{h} \sum_{n=0}^{N-1} (\underline{w}_1, \underline{z}_1)^{\Sigma^n} + \sum_{n=1}^{N-1} \{(\llbracket \underline{w}_1^n \rrbracket, \underline{z}_{2,+}^n)_{\Omega} + (\llbracket \underline{w}_2^n \rrbracket, \underline{z}_{1,+}^n)_{\Omega}\} $$

Integration by parts in time:

$$ \begin{aligned} \tilde{A}[\underline{\mathbf{W}}_h, \underline{\mathbf{Z}}_h] = & \sum_{n=0}^{N-1} \left\{ -(\underline{w}_2, \partial_t \underline{z}_1 + \underline{z}_2)_{Q^n} + a(\underline{w}_1, \underline{z}_1)_{Q^n} - (\underline{w}_1, \partial_t \underline{z}_2)_{Q^n} \right. \\ & \left. - (\nabla \underline{w}_1 \cdot \mathbf{n}, \underline{z}_1)_{\Sigma^n} + \frac{\lambda}{h} (\underline{w}_1, \underline{z}_1)_{\omega_T} - (\underline{w}_2^0, \underline{z}_{1,+}^0)_{\Omega} - (\underline{w}_{1,+}^0, \underline{z}_{2,+}^0)_{\Omega} \right. \\ & \left. + (\underline{w}_{2,-}^N, \underline{z}_{1,-}^N)_{\Omega} + (\underline{w}_{1,-}^N, \underline{z}_{2,-}^N)_{\Omega} - \sum_{n=1}^{N-2} \left\{ (\underline{w}_{2,-}^{n+1}, \llbracket \underline{z}_1^{n+1} \rrbracket)_{\Omega} + (\underline{w}_{1,-}^{n+1}, \llbracket \underline{z}_2^{n+1} \rrbracket)_{\Omega} \right\} \right\} \end{aligned} $$
For $\omega \subset \Omega$ want to find $u$ s.t.

$$ \begin{cases} \Delta u = 0, & \text{in } \Omega \\ u|_{\omega} = u_{\omega}, & \forall x \text{ in } \omega \end{cases} $$

for given $u_{\omega} \in L^2(\omega)$.

B

$\omega$

Conditional stability estimate

Let $B \subset \Omega$ **not touching the boundary** of $\Omega \setminus \omega$.
Then there exists $0 < \alpha < 1$ such that

$$ \|u\|_{H^1(B)} \lesssim \|u\|_{L^2(\Omega)}^{1-\alpha} (\|u\|_{L^2(\omega)} + \|-\Delta u\|_{H^{-1}(\Omega)})^{\alpha} $$

$k=2$

<figure><img src="image_52.png" /></figure>
E. Burman and L. Oksanen, Finite element approximation of unique continuation of functions with finite dimensional trace, Math. Models Methods Appl. Sci. (2024)

Example:

$$ \mathcal{V}_M = \operatorname{span}\{y \sin(m\pi x) \mid m = 1, \dots, M\} $$

$$ u = y \sin(\pi x) + \frac{1}{100} y \sin(2\pi x) $$

## Lipschitz stability

Assume we know in addition a finite dimensional subspace $\mathcal{V}_M \subset H^1(\partial\Omega)$ s.t. $u_{\partial\Omega} \subset \mathcal{V}_M$. Let $P_M$ be a projection into $\mathcal{V}_M$ and $Q = 1 - P_M$. Then

$$ \|u\|_{H^1(\Omega)} \lesssim \|u\|_{L^2(\omega)} + \|Qu\|_{H^{1/2}(\partial\Omega)} + \|\Delta u\|_{H^{-1}(\Omega)} $$

<figure><img src="image_53.png" /></figure>
Solution of the linear systems
Recall $\llbracket \underline{w} \rrbracket := w_+ - w_-$

Coupling term $S^{\uparrow\downarrow}(\underline{U}_h, \underline{W}_h) = \sum_{n=1}^{N-1} S_n^{\uparrow\downarrow}$ between layers

$$S_n^{\uparrow\downarrow} = \left\{ \frac{1}{\Delta t} (\llbracket \underline{u}_1^n \rrbracket, \llbracket \underline{w}_1^n \rrbracket)_{\Omega} + \Delta t (\llbracket \nabla \underline{u}_1^n \rrbracket, \llbracket \nabla \underline{w}_1^n \rrbracket)_{\Omega} + \frac{1}{\Delta t} (\llbracket \underline{u}_2^n \rrbracket, \llbracket \underline{w}_2^n \rrbracket)_{\Omega} \right\}$$

Global space-time coupling $\rightsquigarrow$ Cannot solve time-slab per time-slab

<figure><img src="image_54.png" /></figure>
* To apply global matrix to a vector it suffices to:

* assemble the matrices local to each slab

* the coupling matrices $I_n$

* Try to solve this way using unpreconditioned GMRes.

Start for the simplest problem possible:

* $\Omega = [0, 1]$ and $\omega = [0, 1/4] \cup [3/4, 1]$

* Exact solution $u = \cos(\pi t) \sin(\pi x)$.

* Use $q = k = 1$.

* the system is terribly ill-conditioned.

* We need to find a preconditioner.

<table><thead><tr><th>N</th><th>ndof</th><th>NoPre</th><th>Block</th></tr></thead><tbody><tr><td>1</td><td>24</td><td>21</td><td>1</td></tr><tr><td>2</td><td>80</td><td>69</td><td>17</td></tr><tr><td>4</td><td>288</td><td>243</td><td>96</td></tr><tr><td>8</td><td>1,088</td><td>824</td><td>418</td></tr><tr><td>16</td><td>4,224</td><td>2,970</td><td>1,758</td></tr><tr><td>32</td><td>16,640</td><td>-</td><td>-</td></tr></tbody></table>
■ Full space-time coupling:

$$S^{\uparrow\downarrow}(\mathbf{U}_h, \mathbf{W}_h) = \sum_{n=1}^{N-1} \left\{ \frac{1}{\Delta t} (\llbracket \underline{u}_1^n \rrbracket, \llbracket \underline{w}_1^n \rrbracket)_{\Omega} \right. \\ \left. + \Delta t (\llbracket \nabla \underline{u}_1^n \rrbracket, \llbracket \nabla \underline{w}_1^n \rrbracket)_{\Omega} + \frac{1}{\Delta t} (\llbracket \underline{u}_2^n \rrbracket, \llbracket \underline{w}_2^n \rrbracket)_{\Omega} \right\}$$

<figure><img src="image_55.png" /></figure>
■ Consider only forward-coupling in time:

$$S^{\uparrow\downarrow}(\mathbf{U}_h, \mathbf{W}_h) \approx S^{\uparrow}(\mathbf{U}_h, \mathbf{W}_h) = \sum_{n=1}^{N-1} \left\{ \frac{1}{\Delta t} (\llbracket \mathbf{u}_1^n \rrbracket, \mathbf{w}_{1,+}^n)\Omega \right. \\ \left. + \Delta t (\llbracket \nabla \mathbf{u}_1^n \rrbracket, \nabla \mathbf{w}_{1,+}^n)\Omega + \frac{1}{\Delta t} (\llbracket \mathbf{u}_2^n \rrbracket, \mathbf{w}_{2,+}^n)\Omega \right\}$$

■ Allows to solve time-slab by time-slab!

■ Two variants:

► M-f: Using $q_* = q$, $k_* = k$.

► M-l: $q_* = 0, k_* = 1$.

<figure><img src="image_56.png" /></figure>
■ Recall that $|||(\mathbf{U}_h - \mathbf{U}, \mathbf{Z}_h)||| \lesssim h^s$, i.e. $\mathbf{Z}_h \to \mathbf{0}$,

■ Recall the first order optimality condition:

Find $(\underline{\mathbf{U}}_h, \underline{\mathbf{Z}}_h) \in \mathcal{W}_{h,\Delta t}^{k,q} \times \mathcal{W}_{h,\Delta t}^{k_*,q_*}$ such that

$$ (\underline{u}_1, \underline{w}_1)_{\omega_T} + \tilde{A}[\underline{W}_h, \underline{Z}_h] + S_h(\underline{\mathbf{U}}_h, \underline{\mathbf{W}}_h) = (u_\omega, \underline{w}_1)_{\omega_T}, \quad \forall \underline{\mathbf{W}}_h \in \mathcal{W}_{h,\Delta t}^{k,q}, \quad (1) $$

$$ \tilde{A}[\underline{\mathbf{U}}_h, \underline{\mathbf{Y}}_h] - \tilde{S}_h^*(\underline{\mathbf{Y}}_h, \underline{\mathbf{Z}}_h) = (u_\omega, \underline{y}_1)_{\omega_T}, \quad \forall \underline{\mathbf{Y}}_h \in \mathcal{W}_{h,\Delta t}^{k_*,q_*}, \quad (2) $$
■ Recall that $|||(\underline{\mathbf{U}}_h - \mathbf{U}, \underline{\mathbf{Z}}_h)||| \lesssim h^s$, i.e. $\underline{\mathbf{Z}}_h \to \mathbf{0}$,

■ Recall the first order optimality condition:

Find $(\underline{\mathbf{U}}_h, \underline{\mathbf{Z}}_h) \in \mathcal{W}_{h, \Delta t}^{k,q} \times \mathcal{W}_{h, \Delta t}^{k_*, q_*}$ such that

$$ (\underline{u}_1, \underline{w}_1)_{\omega_T} + \tilde{A}[\underline{\mathbf{W}}_h, \underline{\mathbf{Z}}_h] + S_h(\underline{\mathbf{U}}_h, \underline{\mathbf{W}}_h) = (u_\omega, \underline{w}_1)_{\omega_T}, \quad \forall \underline{\mathbf{W}}_h \in \mathcal{W}_{h, \Delta t}^{k,q}, \quad (1) $$

$$ \tilde{A}[\underline{\mathbf{U}}_h, \underline{\mathbf{Y}}_h] - \tilde{S}_h^*(\underline{\mathbf{Y}}_h, \underline{\mathbf{Z}}_h) = (u_\omega, \underline{y}_1)_{\omega_T}, \quad \forall \underline{\mathbf{Y}}_h \in \mathcal{W}_{h, \Delta t}^{k_*, q_*}, \quad (2) $$

1. Set $\underline{\mathbf{Z}}_h = \mathbf{0}$ in (2) and solve forward in time for $\underline{\mathbf{U}}_h$.
Decoupled forward-backward solve (DFB)

■ Recall that $||(\underline{\mathbf{U}}_h - \mathbf{U}, \underline{\mathbf{Z}}_h)||| \lesssim h^s$, i.e. $\underline{\mathbf{Z}}_h \to \mathbf{0}$,

■ Recall the first order optimality condition:

Find $(\underline{\mathbf{U}}_h, \underline{\mathbf{Z}}_h) \in W_{h, \Delta t}^{k,q} \times W_{h, \Delta t}^{k_*, q_*}$ such that

$$ \tilde{\mathcal{A}}[\underline{\mathbf{W}}_h, \underline{\mathbf{Z}}_h] = (u_\omega - \underline{u}_1, \underline{w}_1)_{\omega_T} - S_h(\underline{\mathbf{U}}_h, \underline{\mathbf{W}}_h), \quad \forall \underline{\mathbf{W}}_h \in W_{h, \Delta t}^{k,q}, \quad (1) $$

$$ \tilde{\mathcal{A}}[\underline{\mathbf{U}}_h, \underline{\mathbf{Y}}_h] - \tilde{\mathcal{S}}_h^*(\underline{\mathbf{Y}}_h, \underline{\mathbf{Z}}_h) = (u_\omega, \underline{y}_1)_{\omega_T}, \quad \forall \underline{\mathbf{Y}}_h \in W_{h, \Delta t}^{k_*, q_*}, \quad (2) $$

1. Set $\underline{\mathbf{Z}}_h = \mathbf{0}$ in (2) and solve forward in time for $\underline{\mathbf{U}}_h$.

2. Insert $\underline{\mathbf{U}}_h$ in (1) and solve backward in time for $\underline{\mathbf{Z}}_h$.

Require: $\tilde{\mathcal{A}}[\cdot, \cdot]$ provides a consistent and stable discretization of wave equation on its own.
$$ \tilde{A}[\underline{\mathbf{U}}_h, \underline{\mathbf{Y}}_h] = A[\underline{\mathbf{U}}_h, \underline{\mathbf{Y}}_h] + (\underline{u}_1, \underline{y}_1)_{\omega_T} + \frac{\lambda}{h} \sum_{n=0}^{N-1} (\underline{u}_1, \underline{y}_1)^{\Sigma^n} + \sum_{n=1}^{N-1} \left\{ (\llbracket \underline{u}_1^n \rrbracket, \underline{y}_{2,+}^n)_{\Omega} + (\llbracket \underline{u}_2^n \rrbracket, \underline{y}_{1,+}^n)_{\Omega} \right\} $$

with Nitsche parameter $\lambda > 0$ sufficiently large as usual.

* Enriched bilinear form is similar to method first proposed in [Johnson, '93].

* Luenberger observer type term on $\omega_T$ motivated by [Chapelle, Cîndea, De Buhan, & Moireau, '12].

* Notice that $\underline{\mathbf{U}}_h \in W_{h,\Delta t}^{k,q}$ and $\underline{\mathbf{Y}}_h \in W_{h,\Delta t}^{k*,q*}$. We require $q = q_*$ and $k = k_*$ to apply this preconditioner!
<figure><img src="image_57.png" /></figure>

## Monolithic:

* Need to factorize $(u, z)$-matrix with stabilization.

* But can choose $q_* = 0, k_* = 1$.

<figure><img src="image_58.png" /></figure>

## Splitting:

* Only need to factorize individual matrices without stabilization.

* Have to choose $q = q_*$ and $k = k_*$.
Preconditioning results for $\Omega = [0, 1]$

$q = k = 1$

<table><thead><tr><th>$N$</th><th>ndof</th><th>NoPre</th><th>Block</th><th>M-I □</th><th>DFB ○</th><th>M-f △</th></tr></thead><tbody><tr><td>1</td><td>24</td><td>21</td><td>1</td><td>2</td><td>5</td><td>1</td></tr><tr><td>2</td><td>80</td><td>69</td><td>17</td><td>7</td><td>12</td><td>8</td></tr><tr><td>4</td><td>288</td><td>243</td><td>96</td><td>22</td><td>26</td><td>19</td></tr><tr><td>8</td><td>1,088</td><td>824</td><td>418</td><td>66</td><td>67</td><td>36</td></tr><tr><td>16</td><td>4,224</td><td>2,970</td><td>1,758</td><td>189</td><td>155</td><td>74</td></tr><tr><td>32</td><td>16,640</td><td>-</td><td>-</td><td>523</td><td>337</td><td>176</td></tr></tbody></table>

<figure><img src="image_59.png" /></figure>

$q = k = 2$

<table><thead><tr><th>$N$</th><th>ndof</th><th>M-I</th><th>DFB</th><th>M-f</th></tr></thead><tbody><tr><td>1</td><td>60</td><td>1</td><td>6</td><td>1</td></tr><tr><td>2</td><td>216</td><td>10</td><td>18</td><td>10</td></tr><tr><td>4</td><td>816</td><td>22</td><td>42</td><td>23</td></tr><tr><td>8</td><td>3,168</td><td>53</td><td>106</td><td>52</td></tr><tr><td>16</td><td>12,480</td><td>135</td><td>624</td><td>133</td></tr></tbody></table>

<figure><img src="image_60.png" /></figure>
- 'ndof-tot': total number of dofs for entire space-time system

- 'ndof-inv': size of linear system to be inverted to apply preconditioner.

$q = k = 2$

<table><thead><tr><th rowspan="2">N</th><th colspan="3">DFB</th><th colspan="3">M-I</th></tr><tr><th>ndof-tot</th><th>ndof-inv</th><th>iter</th><th>ndof-tot</th><th>ndof-inv</th><th>iter</th></tr></thead><tbody><tr><td>1</td><td>1,500</td><td>750</td><td>14</td><td>804</td><td>804</td><td>2</td></tr><tr><td>2</td><td>17,496</td><td>4,374</td><td>38</td><td>9,248</td><td>4,624</td><td>11</td></tr><tr><td>4</td><td>2.36 · 10<sup>5</sup></td><td>29,478</td><td>105</td><td>1.24 · 10<sup>5</sup></td><td>30,936</td><td>25</td></tr><tr><td>8</td><td>3.45 · 10<sup>6</sup></td><td>2.16 · 10<sup>5</sup></td><td>239</td><td>1.8 · 10<sup>6</sup></td><td>2.25 · 10<sup>5</sup></td><td>81</td></tr></tbody></table>

<figure><img src="image_61.png" /></figure>

For higher polynomial order the M-I preconditioner is preferred!
Find $(\underline{\mathbf{U}}_h, \underline{\mu}_h, \underline{\mathbf{Z}}_h) \in \mathcal{W}_{h,\Delta t}^{k,q} \times V_M \times \mathcal{W}_{h,\Delta t}^{k_*,q_*}$ such that

$$B^{\#}[(\underline{\mathbf{U}}_h, \underline{\mu}_h, \underline{\mathbf{Z}}_h), (\underline{\mathbf{W}}_h, \underline{\eta}_h, \underline{\mathbf{Y}}_h)] = (\tilde{u}_{\omega}, \underline{w}_1)_{\omega_T}$$

for all $(\underline{\mathbf{W}}_h, \underline{\eta}_h, \underline{\mathbf{Y}}_h) \in \mathcal{W}_{h,\Delta t}^{k,q} \times V_M \times \mathcal{W}_{h,\Delta t}^{k_*,q_*}$, where

$$B^{\#}[(\underline{\mathbf{U}}_h, \underline{\mu}_h, \underline{\mathbf{Z}}_h), (\underline{\mathbf{W}}_h, \underline{\eta}_h, \underline{\mathbf{Y}}_h)] := B^0[(\underline{\mathbf{U}}_h, \underline{\mathbf{Z}}_h), (\underline{\mathbf{W}}_h, \underline{\mathbf{Y}}_h)] + (\underline{u}_1 - \underline{\mu}_h, \underline{w}_1 - \underline{\eta}_h)_{\Sigma}.$$

* Replace global space $V_M$ by its 'broken-in-time' version $V_M^{\text{disc}} := \bigotimes_{n=0}^{N-1} V_M|_{Q^n}$

* To recover continuity in time add additional term

$$\sum_{n=1}^{N-1} \frac{1}{h} ( [\underline{\mu}_h^n], [\underline{\eta}_h^n] ) \partial \Omega$$