Failure-aware fluid UQ callback and certified sampling budget

Fixed-size callback mode. Inputs are a finite integer $N\ge 2$, the specified input law $\mu$ and exact/numerical observable interface, an IID sampler for $\mu$, and a solver callback with a supplied finite computational budget for each invocation. The callback is total, with a measurable success set and measurable successful outputs: for every input $\theta$ it returns either solver-failed or a numerical state $s_h(\theta )$ with finite real observable $q_h(\theta )=J_h(s_h(\theta ))$, together with a finite $b(\theta )\ge 0$ and its supplied pointwise certificate $\mathrm{\mid }Y(\theta )-q_h(\theta )\mathrm{\mid }\le b(\theta )$. A stated failure result is allowed when the finite budget is exhausted; the procedure asserts no universal numerical solver termination theorem.

Draw ${\theta }_1,\dots ,{\theta }_N$ independently with law $\mu$ and call the callback exactly once on each draw, including the draws after any earlier failure. If any call fails, return incomplete with the failed indices and the collected outputs, and no IID mean/error certificate for $\mu$. Never delete, replace or resample failed cases. If all calls succeed with finite arithmetic values, set ${\bar{q}}_h=N^{-1}{\sum }_{i=1}^N{q}_i,s_N^2=(N-1{)}^{-1}{\sum }_{i=1}^N(q_i-{\bar{q}}_h{)}^2,{\widehat{se}}_N=s_N\mathrm{/}\sqrt{N},\bar{b}=N^{-1}{\sum }_{i=1}^N{b}_i\mathrm{.}$ Return $N,{\bar{q}}_h,s_N^2,{\widehat{se}}_N,\bar{b}$ and complete; if $s_N^2=0$, also return zero-sample-variance. This diagnostic asserts neither a deterministic population nor zero estimation error. The sample mean of certificates $\bar{b}$ is a diagnostic, not a rigorous upper bound on $\mathbb{E}b$. The plug-in standard error has only the moment/asymptotic interpretation supplied by the sampling theorem, not an exact confidence guarantee. If callback failure has positive $\mu$-probability, conditioning on complete changes the input law and does not certify IID sampling of the original $\mu$.

Optional certified-budget mode. This retains a separate finite-sample guarantee when externally proved global bounds are available. Inputs are the numerical output law $Y_h$, a total oracle returning independent finite real samples of that law, finite ${ϵ}_b,V_h\ge 0$, ${ϵ}_s>0$, $\delta \in (0,1)$, and a certificate that $Y\in L^1$, $Y_h\in L^2$, $\mathrm{\mid }\mathbb{E}{Y}_h-\mathbb{E}Y\mathrm{\mid }\le {ϵ}_b$, $\mathrm{Var}(Y_h)\le V_h$, and the oracle has that IID law. The algorithm does not validate a supplied mathematical certificate. Apply the input guards in this order: if the certificate or oracle is absent, return not-certified; if both are supplied but any of ${ϵ}_b,V_h,{ϵ}_s,\delta$ is nonfinite or outside its stated admissible range, return invalid-input. Otherwise set $N:=\max \{1,\lceil V_h\mathrm{/}(\delta {ϵ}_s^2)\rceil \}\mathrm{.}$ Request exactly $N$ samples and return conditional-budget, $N$, their mean ${\widehat{\theta }}_{h,N}$, tolerance ${ϵ}_b+{ϵ}_s$, failure bound $\delta$ and RMS upper bound $\sqrt{V_h\mathrm{/}N+{ϵ}_b^2}$, conditional on the certificate. If an oracle invocation instead fails its total-return contract, return incomplete and no guarantee. This mode permits $N=1$ but then supplies no sample variance or plug-in standard error. When implemented through the callback mode, its certificate additionally requires callback success $\mu$-almost surely; the probability guarantee is not conditional on deleting failed inputs.

Neither mode infers a bias/variance certificate from a sample or mesh trend. Numerical bias, physical model discrepancy and measurement uncertainty are different quantities; the latter two are absent unless separately quantified. Both tolerance allocations use the declared observable unit, variance bounds use its square and $\delta$ is dimensionless. The operation counts concern the specified finite callback/oracle contract, not uncontrolled floating-point error.

Execution and conditional error guarantees of the fluid UQ budget algorithm proves the execution, completion-law and conditional error properties of this procedure.

Random fluid observable and numerical approximation

IID Monte Carlo sample mean, variance, and standard error

Exact sampling variance and a numerical-bias error bound

Art B. Owen, Monte Carlo Theory, Methods and Examples, Chapter 2, §§2.1–2.2

NASA Glenn Research Center, Uncertainty and Error in CFD Simulations