In this paper, we study a mixed variational problem subject to perturbations, where the noise term is modelled by means of a bilinear form that has to be understood to be “small” in some sense. Indeed, we consider a family of such problems and provide a result that guarantees existence and uniqueness of the solution. Moreover, a stability condition for the solutions yields a Generalized Collage Theorem, which extends previous results by the same authors. We introduce the corresponding Galerkin method and study its convergence. We also analyze the associated inverse problem and we show how to solve it by means of the mentioned Generalized Collage Theorem and the use of adequate Schauder bases. Numerical examples show how the method works in a practical context.

在本文中，我们研究了一个受扰动的混合变分问题，其中噪声项是通过双线性形式建模的，该双线性形式在某种意义上必须理解为“小”。实际上，我们考虑了一系列此类问题，并提供了保证解决方案存在和唯一的结果。此外，解决方案的稳定性条件产生了广义拼贴定理，该定理扩展了同一作者的先前结果。我们介绍了相应的Galerkin方法并研究其收敛性。我们还分析了相关的逆问题，并展示了如何通过上述广义拼贴定理和适当的Schauder基的使用来解决问题。数值示例表明了该方法在实际环境中的工作方式。