Conditional stability for an inverse problem of a fully-discrete stochastic hyperbolic equation

Technical Summary: Conditional Stability for an Inverse Problem of a Fully Discrete Stochastic Hyperbolic Equation

Problem Statement
This paper investigates a discrete inverse problem associated with a fully discrete approximation of a one-dimensional stochastic hyperbolic equation. The continuous model, defined on a space-time domain $Q = (0,1) \times (0,T)$, describes a system subject to random forces. The primary objective is to determine three unknowns simultaneously: the initial displacement ($y_0$), the initial velocity ($y_1$), and the random source term ($g$).

The inverse problem is formulated using two types of observation data:

1. Discrete spatial derivative data at the left boundary ($x=0$).
2. Measurements of the solution at the final time ($t=T$).

The authors focus on the fully discrete setting, employing an explicit finite difference scheme for the spatial and temporal discretization of the stochastic partial differential equation (SPDE). This contrasts with previous works that primarily addressed continuous settings or semi-discrete approximations.

Methodology
The core methodology relies on the derivation and application of a Carleman estimate tailored for fully discrete stochastic hyperbolic equations. The approach involves several technical steps:

1. Discrete Calculus Framework: The authors establish a rigorous discrete calculus framework, defining discrete function spaces, translation operators, average and difference operators, and discrete integration by parts formulas. Special attention is paid to the dual meshes required for handling discrete derivatives.
2. Weight Function Construction: A specific weight function, $r = e^{s\phi(x,t)}$, is constructed using large parameters $s$ and $\lambda$. This function is designed to handle the hyperbolic nature of the equation and the stochastic terms.
3. Derivation of the Carleman Estimate:
   • A change of variable $y = \rho Y$ (where $\rho = r^{-1}$) is introduced to transform the operator.
   • The proof involves decomposing the transformed operator into terms $A(Y)$, $B(Y,Z)$, and a remainder $R(Y,Z)$.
   • Crucially, the authors utilize the independence between the solution and the future increment of the Brownian motion ($B^{n+1} - B^n$) to show that the expectation of the stochastic integral term vanishes ($E[\dots] = 0$). This allows the elimination of the discrete Itô integral term, a necessary step for establishing the estimate.
   • The proof carefully handles error terms arising from discretization, particularly those involving the mesh sizes $\Delta x$ and $\Delta t$.
4. Stability Analysis: The derived Carleman estimate is applied to the difference of two solutions corresponding to different unknowns. By combining this with discrete energy estimates, the authors derive a conditional stability result for the inverse problem.

Key Contributions and Results

• New Discrete Carleman Estimate: The paper establishes a novel Carleman estimate for fully discrete stochastic hyperbolic equations. Unlike continuous counterparts, this estimate includes additional terms on the right-hand side that depend on the mesh sizes ($\Delta x$ and $\Delta t$). Specifically, it contains a term involving the discrete time derivative of the spatial derivative ($|D_t D_x y|^2$) and a boundary term localized at $t=T$ involving the source $g$.
• Conditional Stability: Based on the Carleman estimate, the authors prove a conditional stability result. They show that the difference in the unknowns ($g, y_0, y_1$) can be bounded by the difference in the observation data (boundary flux and final time solution), provided the unknowns belong to an admissible set with a priori bounds.
• Mesh Dependence: The stability estimate explicitly accounts for the discretization parameters. The error term associated with the source $g$ at the final time scales with $(\Delta t)^{1/2}$. The authors note that as $\Delta x, \Delta t \to 0$, these discrete terms vanish, recovering the continuous stability result.
• Relaxed Final Time Assumption: A significant technical improvement over previous continuous studies (e.g., [28]) is the treatment of the final time condition. While continuous uniqueness often assumes $y^{(1)}(T) = y^{(2)}(T)$, this paper derives a stability estimate involving the term $\|y^{(1)}(T) - y^{(2)}(T)\|^{1/2}$. This Hölder-type dependence is more suitable for numerical analysis where exact equality at the final time is rarely achieved, only small errors.

Significance and Claims
The paper claims to fill a gap in the literature regarding fully discrete stochastic inverse problems. While extensive work exists for deterministic discrete inverse problems and continuous stochastic inverse problems, this work is the first to address the fully discrete stochastic hyperbolic case.

The authors highlight that the discrete setting introduces specific challenges, such as the need to manage additional error terms arising from the discretization of random terms and the specific structure of the discrete Itô formula. The derived stability result is presented as particularly valuable for error analysis in numerical implementations, as it quantifies how discretization errors affect the reconstruction of unknown parameters.

The paper concludes by noting that while Lipschitz stability is expected for the deterministic case, the stochastic nature and discrete setting currently yield a conditional stability result with a power of $1/2$ for the final time error. Future work, as suggested by the authors, would involve investigating stronger stability properties and the numerical implementation of these inverse problems.