Neumann--Dirichlet Nash Strategies for the Solution of Elliptic Cauchy Problems

Abstract

We consider the Cauchy problem for an elliptic operator, formulated as a Nash game. The overspecified Cauchy data are split between two players: the first player solves the elliptic equation with the Dirichlet part of the Cauchy data prescribed over the accessible boundary and a variable Neumann condition (which we call first player's strategy) prescribed over the inaccessible part of the boundary. The second player makes use correspondingly of the Neumann part of the Cauchy data, with a variable Dirichlet condition prescribed over the inaccessible part of the boundary. The first player then minimizes the gap related to the nonused Neumann part of the Cauchy data, and so does the second player with a corresponding Dirichlet gap. The two costs are coupled through a difference term. We prove that there always exists a unique Nash equilibrium, which turns out to be the reconstructed data when the Cauchy problem has a solution. We also prove that the completion Nash game has a stable solution with respect to noisy data. Some numerical two-and three-dimensional experiments are provided to illustrate the efficiency and stability of our algorithm.

A Bayesian optimization approach to find Nash equilibria

“…Let be Ω a bounded open domain in R d (d = 2, 3) with a sufficiently smooth boundary ∂Ω composed of two connected disjoint components Γ c and Γ i , with the latter being inaccessible to boundary measurements. For details, see Habbal & Kallel (2013) whence the present example is excerpt.…”

“…Theorem 1 (Habbal & Kallel, 2013) There always exists a unique Nash equilibrium (η * , ζ * ) ∈ H − 1 2 (Γ i ) × H 1 2 (Γ i ), and when the Cauchy problem has a solution u, then u 1 (η * ) = u 2 (ζ * ) = u, and (η * , ζ * ) are the missing data, namely η * = λ∇u.ν |Γi and ζ * = u |Γi .…”

“…In a realistic situation, the fields ϕ and Φ may be "polluted" by noise. Habbal & Kallel (2013) showed that the Nash equilibrium is stable with respect to small perturbations, and that the perturbed equilibrium converges strongly to the unperturbed one when the noise tends to zero. However, they did not consider directly the random Nash game:…”

A convergent data completion algorithm using surface integral equations

“…. On the other hand let (ϕ, ψ) ∈ Ker(P Γ2 − I Γ2 ) and let the function v defined by (15).Using again trace formulas for double and single layer potentials, we obtain…”

“…Consequently (ϕ, ψ) ∈ Ker(T Γ2→Γ1 ). The proof of b) uses similar arguments as the proof of a) and is based on considering v defined by (15) where Γ 2 is replaced with Γ 1 , namely…”