Recent investigations have established the physical relevance of spatially localized instability mechanisms in fluid dynamics and their potential for technological innovations in flow control. In this paper, we show that the mathematical problem of identifying spatially localized optimal perturbations that maximize perturbation-energy amplification can be cast as a sparse (cardinality-constrained) optimization problem. Unfortunately, cardinality-constrained optimization problems are nonconvex and combinatorially hard to solve in general. To make the analysis viable within the context of fluid dynamics problems, we propose an efficient iterative method for computing suboptimal spatially localized perturbations. Our approach is based on a generalized Rayleigh quotient iteration algorithm followed by a variational renormalization procedure that reduces the optimality gap in the resulting solution. The approach is demonstrated on a subcritical plane Poiseuille flow at $\text{Re}=4000$, which has been a benchmark problem studied in prior investigations on identifying spatially localized flow structures. Notably, we find that a subset of the perturbations identified by our method yield a comparable degree of energy amplification as their global counterparts. We anticipate our proposed analysis tools will facilitate further investigations into spatially localized flow instabilities, including within the resolvent and input-output analysis frameworks.

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• Flow instability
• Transition to turbulence