Exact operator inference decouples parametric problems

The application of operator inference to parametric problems whose operators exhibit affine parameter dependency results in a minimization problem with three issues. First, the solution to the minimization problem is generally not unique, so we might obtain a different solution than the one we are interested in. Second, time discretization inconsistencies and non-Markovian information can cause errors in the inferred operators. Third, the large dimensions of the minimization problem can cause prohibitively high computational costs. In this work, we resolve these issues by extending previous work on exact operator inference to the parametric case. Concretely, we propose a snapshot data generation method that guarantees the uniqueness of the solution and the absence of time discretization inconsistencies and non-Markovian information, and thereby the exact reconstruction of the corresponding intrusive operators of projection-based reduced order models. Furthermore, we show that the monolithic problem that comprises the snapshot data of all sampled parameters can be decoupled into separate problems for only one parameter sample each. This decoupling results in smaller subproblems that are embarrassingly parallel, numerically more stable and can be solved more efficiently. In numerical experiments, we demonstrate this decoupling for the heat equation with two different temperature-, parameter- and space-dependent thermal conductivities.

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