Random fluid observable and numerical approximation

Let $(\mathrm{\Theta },\mathcal{A},\mu )$ be a specified probability space of inputs, let $I\subset \mathbb{R}$ be a nonempty common time interval, and let $(X,\mathcal{X})$ be a specified measurable state or path space. An exact selected solution map is a supplied measurable map $S:\mathrm{\Theta }\to X$ whose values solve the specified mathematical model on $I$. The equation, solution class and admissible input set are supplied as data; a smooth uniqueness theorem may identify a single-valued map, but measurability is a separate required hypothesis. In a nonunique weak theory an actual measurable selection must be given before taking a pushforward law; this definition asserts no selection theorem. More generally a supplied measurable random state is allowed without any PDE interpretation. A real observable is a map $J:X\to \mathbb{R}$ such that $Y:=J\circ S$ is measurable. It is integrable or square-integrable when ${\mathbb{E}}_{\mu }\mathrm{\mid }Y\mathrm{\mid }<\mathrm{\infty }$ or ${\mathbb{E}}_{\mu }{Y}^2<\mathrm{\infty }$, respectively.

For a nonempty set $H$ of numerical settings, supplied measurable numerical composites $Y_h=J_h\circ S_h$ define approximate outputs on the same input law; the maps $S_h$ may take values in specified approximation spaces $X_h$, with $J_h:X_h\to \mathbb{R}$. When $Y,Y_h$ are integrable, define $\theta ={\mathbb{E}}_{\mu }Y$, ${\theta }_h={\mathbb{E}}_{\mu }{Y}_h$ and signed mean bias $b_h={\theta }_h-\theta$. Output variances require finite second moments. This definition implies neither IID sampling nor numerical convergence, a bias bound, or physical applicability. All exact and approximate outputs must use the same declared scalar unit before forming their bias.

The finite-second-moment spatial velocity law and sensor are included as follows. Let $H_D=L^2(D,\mathcal{D},\lambda ;{\mathbb{R}}^d)$ and let ${\mu }_H$ be the supplied finite-second-moment probability law on $(H_D,\mathcal{B}(H_D))$. Take $\mathrm{\Theta }=X=H_D$, $\mathcal{A}=\mathcal{X}=\mathcal{B}(H_D)$, $\mu ={\mu }_H$, $S={\mathrm{id}}_{H_D}$, and $J=S_{\psi }$ for the supplied weight $\psi \in H_D$. Thus the input itself is the random velocity class and $Y(v)=S_{\psi }(v)$. The identity map is measurable, and the pairing lemma gives measurability and a finite second moment under exactly ${\mu }_H$. This use is on quotient classes and requires no measurable choice of pointwise representatives. In the general separable Banach-state branch, bounded linear sensors have the same measurability and moment property by their operator bounds. Time-series sampling, a Markov chain, and mixing remain additional assumptions; dependent consecutive readings are not IID merely because this definition applies.

The earlier pointwise-field formulation is also included: let $d_s\in \{2,3\}$, let $D\subseteq {\mathbb{R}}^{d_s}$ be nonempty and Lebesgue-measurable, and let $(\mathcal{V},\mathcal{V})$ be any specified measurable space of product-measurable fields $u:D\times I\to {\mathbb{R}}^{d_s}$. Use $X=\mathcal{V}$, a supplied measurable random-field map, and measurable scalar composites. Passing from an $L^2$ class to such a pointwise random field requires a supplied jointly measurable representative; it is not part of the quotient-sensor specialization. Velocity components use one fixed velocity unit, and sensor weights carry the corresponding integral units.

Finite-second-moment random velocity sensor law

The L2 velocity sensor is well-defined

Art B. Owen, Monte Carlo Theory, Methods and Examples, Chapter 2

Olivier Le Maître and Omar Knio, Introduction to Uncertainty Quantification, DTU lecture notes

NASA Glenn Research Center, Uncertainty and Error in CFD Simulations