GEV-Quantile Score Function Renders GKTL Memory-Stationary for Compressible SBLI

Two fields are meeting here in an unexpected way. The first is 'extreme value theory' — a branch of statistics that specializes in rare, catastrophic events. Think of it as the science of 100-year floods or once-in-a-century stock market crashes. It gives us mathematical tools to describe the tail end of distributions: the extreme outliers that are rare but matter enormously. The second field is computational fluid dynamics (CFD) — the computer simulations engineers use to model airflow over aircraft wings, turbine blades, and rocket bodies. Simulating the truly dangerous pressure events (like shock waves slamming into a wing boundary layer at near-supersonic speeds) is brutally expensive because you have to run the simulation for an extraordinarily long time just waiting for rare events to show up. One clever shortcut is called 'Adaptive Multilevel Splitting' (AMS) — essentially a way of cloning your simulation when it starts approaching a dangerous state, so you see more rare events without running forever. But AMS needs a 'score function': a way to measure how close you are to danger. This hypothesis proposes replacing the naive score (raw pressure at the shock location) with one that's been mathematically transformed using extreme value statistics. Specifically, you fit a Generalized Extreme Value distribution to pilot simulation data, then remap the score through that distribution. This means the algorithm naturally concentrates its effort right where the dangerous tail events live, rather than wasting effort on mundane fluctuations. The elegant part is that this transformation preserves all the mathematical guarantees that make AMS work correctly — it's like changing units without changing the physics. And because the score now 'sees' the tail of the distribution more clearly, the simulation should need far fewer computational steps to gather good statistics on rare, extreme aerodynamic loads. That's the core bet: better-shaped score functions mean faster, cheaper, more reliable safety calculations for aircraft and spacecraft.

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