Speaker
Description
This talk presents an application of a linear transient PDE-constrained inverse problem to advection-diffusion problems in a real-world environment Ω ⊂ ℝⁿ for n ∈ {2,3}. We consider a forward operator F(m) = ℬu, where u ∈ V is the solution of the linear partial differential equation r(m, u) = f, and ℬ : V → ℝ^q is the observation operator. The objective is to utilize discrete measurements, represented by the vector d ∈ ℝ^q, to infer the parameter set mₒₚ , defined by
mₒₚ = arg min_ {m∈ℳ} (1/2 ‖ ℱ(m) − d ‖²_ {Γ⁻¹ₙₒᵢₛₑ} + ℛ(m)).
In this application, the transport of pollutants is simulated using the advection-diffusion equation. The inverse problem aims to identify the source of the pollutant and predict its dispersion. This leads to a highly underdetermined inverse problem, necessitating the use of a regularization term ℛ(m). We compare classical L²-regularization [1] with a sparsity-enforcing regularization [2], defined as
ℛ(m) := α ‖m‖_ℳ(Ω),
where
‖m‖ℳ(Ω) = sup {〈m,φ〉: φ ∈ C(Ω̅), ‖φ‖ C(Ω̅) = 1}.
For both methods, we present numerical implementations and evaluate their applicability to critical infrastructure protection.
Moreover, a Bayesian formulation of the inverse problem allows quantification of the uncertainty associated with the prediction of the parameter m. In particular, goal-oriented uncertainty estimation represents a promising approach for evacuation scenarios. This approach quantifies uncertainty for a specific quantity of interest [3]. In a crisis management context, this could pertain to a particular evacuation point or route. Additionally, this talk discusses the application of goal-oriented uncertainty estimation in the context of pollutant transport in complex environments.
[1] Villa, U., Petra, N., Ghattas, O. (2021). h-Adaptivity and Goal-Oriented Optimal Experimental Design for Infinite-Dimensional Bayesian Linear Inverse Problems. arXiv:1308.4084.
[2] Pieper, K., Sprungk, B., Stadler, G. (2021). Sparse Deterministic Approximation of Bayesian Inverse Problems. arXiv:1103.4522.
[3] Spantini, A., Bigoni, D., Marzouk, Y. M. (2017). Goal-Oriented Optimal Experimental Design for Infinite-Dimensional Bayesian Linear Inverse Problems. arXiv:1308.4084.
