Identification in mixed variational problems by adjoint methods with applications (Q1695081)

Let \(V\) and \(Q\) be Hilbert spaces and let \(B\) be a Banach space. Let \(A\) be a nonempty, closed, and convex subset of \(B\). Let \(a:B\times V\times V\rightarrow \mathbb{R}\) be a trilinear map which is symmetric with respect to the second and the third arguments. Let \(b:V\times Q\rightarrow \mathbb{R} \) be a bilinear map, let \(c:Q\times Q\rightarrow \mathbb{R}\) be a symmetric bilinear map, and let \(m:V\rightarrow \mathbb{R}\) be a linear and continuous map. Under some suitable coercivity conditions and for a given \(\ell \in A\), the authors investigate the inverse problem of identifying \(\ell \in A\) for which a solution \(\left( \overline{u},p\right) \in V\times Q\), of the following mixed variational problem \[ \begin{aligned} a(\ell ,\overline{u},\overline{v})+b(\overline{v},p) & =m(\overline{v}) \\ b(\overline{u},p)-c(p,q) & =0 \end{aligned} \] is closest, in some norm, to a given measurement \((\overline{z},\widehat{z})\) of \((\overline{u},p)\). To study this inverse problem of parameter identification, the authors employ the Output Least-Squares (OLS) formulation by considering the following regularized optimization problem: \[ \underset{\ell \in A}{\min }J_{\kappa }(\ell ):=\frac{1}{2}\left\| \overline{u}(\ell )-\overline{z}\right\| _{V}^{2}+\frac{1}{2}\left\| p(\ell )-\widehat{z}\right\| _{Q}^{2}+\kappa R(\ell ) \] where \(R\) is is a regularizer from a given Hilbert space \(H\) to \(\mathbb{R}\), \(\kappa >0\) is is a regularization parameter, \((\overline{z},\widehat{z})\) is the measured data and \(\left( \overline{u}(\ell ),p(\ell )\right) \) is the unique solution of the above mixed variational problem. Then, they obtain a general existence result for the regularized optimization problem, the authors present a thorough derivation of the first-order adjoint approach for the computation of the first-order derivative of the OLS functional and also they present a second-order adjoint approach and one of its analogues for the evaluation of the second-order derivative of the OLS functional. Afterwards they prove a general existence result for the corresponding OLS formulation and present a thorough derivation of the first-order adjoint approach for the computation of the first-order derivative of the OLS functional, a second-order adjoint approach and one of its analogues for the evaluation of the second-order derivative of the OLS functional, and also a detailed discretization procedure and schemes for the gradient and the Hessian computation. The obtained results are applied to identify variables parameters in nearly incompressible elasticity, nearly incompressible Stokes equations, and fourth-order boundary value problems. For the entire collection see [Zbl 1383.49001].