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Abstract

Discrete Empirical Interpolation Method (DEIM) is a simple and effective method for reconstructing a function from its incomplete pointwise observations. However, applying DEIM to functions with physically constrained ranges can produce reconstructions with values outside the prescribed physical bounds. Such physically constrained quantities occur routinely in applications, e.g., mass density whose range is nonnegative. The DEIM reconstructions which violate these physical constraints are not usable in downstream tasks such as forecasting and control. To address this issue, we develop Constrained DEIM (C-DEIM) whose reconstructions are guaranteed to respect the physical bounds of the quantity of interest. C-DEIM enforces the bounds as soft constraints, in the form of a carefully designed penalty term, added to the underlying least squares problem. We prove that the C-DEIM reconstructions satisfy the physical constraints asymptotically, i.e., as the penalty parameter increases towards infinity. We also derive a quantitative upper bound for the observation residual of C-DEIM. Based on these theoretical results, we devise an efficient algorithm for practical implementation of C-DEIM. The efficacy of the method and the accompanying algorithm are demonstrated on several examples, including a heat transfer problem from fluid dynamics and a cellular automaton model of wildfire spread.