Many hairbraned schemes for calculating the expectation

Dissertation/ann_product.tex

@@ -1386,7 +1386,7 @@ Let $\mathfrak{p}_i$ for $i \in \{1,2,...\}$ be the set of functions defined for
 		This completes the proof of the Lemma.
 \end{proof}
 \subsection{$\xpn_n^{q,\ve}$, $\csn_n^{q,\ve}$, $\sne_n^{q,\ve}$, and Neural Network Approximations of $e^x$, $\cos(x)$, and $\sin(x)$.}
-Once we have neural network polynomials, we may take the next leap to transcendental functions. Here, we will explore neural network approximations for three common transcendental functions: $e^x$, $\cos(x)$, and $\sin(x)$. 
+Once we have neural network polynomials, we may take the next leap to transcendental functions. For approximating them we will use Taylor expansions which will swiftly give us our approximations for our desired functions. Here, we will explore neural network approximations for three common transcendental functions: $e^x$, $\cos(x)$, and $\sin(x)$. 
 
 \begin{lemma}
 	Let $\nu_1,\nu_2 \in \neu$, $f,g \in C \lp \R, \R \rp$, and $\ve_1,\ve_2 \in \lp 0 ,\infty \rp$ such that for all $x\in \R$ it holds that $\left| f(x) - \real_{\rect} \lp \nu_1 \rp \right| \les \ve_1 $ and $\left| g(x) - \real_{\rect} \lp \nu_2 \rp \right| \les \ve_2$. It is then the case for all $x \in \R$ that:

Dissertation/ann_rep_brownian_motion_monte_carlo.tex

@@ -864,9 +864,9 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
 %		&\les \left| \exp \lp \int^T_tf\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u^T_d\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp - \real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \right| \nonumber
 %	\end{align}
 	\begin{corollary}
-		Let $N,n,\fn \in \N$, $h,\ve \in \lp 0,\infty\rp$, $q\in\lp 2,\infty\rp$, given $\mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d, \Omega, \fn}$, the Monte Carlo standard error for approximating $\exp \lp \int_t^T f\lp \mathcal{X}^{d,t,x}_{r,\Omega}\rp ds \cdot u_d^T\lp \mathcal{X}^{d,t,x}_{r,\Omega}\rp\rp$ is:
+		Let $N,n,\fn \in \N$, $h,\ve \in \lp 0,\infty\rp$, $q\in\lp 2,\infty\rp$, given $\mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d, \Omega, \fn} \subsetneq \neu $, it is the case that:
 		\begin{align}
-			s
+			\left\| \exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot u\lp T,\cX^{d,t,x}_{r,\Omega}\rp-\frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T \alpha_d \circ \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rb \rb \right\|\nonumber
 		\end{align}
 	\end{corollary}
 	\begin{proof}

Dissertation/main.bib

@@ -624,6 +624,22 @@ version = {0.10},
 year = {2024}
 }
 
+@article{https://doi.org/10.1002/cnm.3535,
+author = {Rego, Bruno V. and Weiss, Dar and Bersi, Matthew R. and Humphrey, Jay D.},
+title = {Uncertainty quantification in subject-specific estimation of local vessel mechanical properties},
+journal = {International Journal for Numerical Methods in Biomedical Engineering},
+volume = {37},
+number = {12},
+pages = {e3535},
+keywords = {digital image correlation, image-based modeling, subject-specific model, uncertainty quantification},
+doi = {https://doi.org/10.1002/cnm.3535},
+url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/cnm.3535},
+eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/cnm.3535},
+abstract = {Abstract Quantitative estimation of local mechanical properties remains critically important in the ongoing effort to elucidate how blood vessels establish, maintain, or lose mechanical homeostasis. Recent advances based on panoramic digital image correlation (pDIC) have made high-fidelity 3D reconstructions of small-animal (e.g., murine) vessels possible when imaged in a variety of quasi-statically loaded configurations. While we have previously developed and validated inverse modeling approaches to translate pDIC-measured surface deformations into biomechanical metrics of interest, our workflow did not heretofore include a methodology to quantify uncertainties associated with local point estimates of mechanical properties. This limitation has compromised our ability to infer biomechanical properties on a subject-specific basis, such as whether stiffness differs significantly between multiple material locations on the same vessel or whether stiffness differs significantly between multiple vessels at a corresponding material location. In the present study, we have integrated a novel uncertainty quantification and propagation pipeline within our inverse modeling approach, relying on empirical and analytic Bayesian techniques. To demonstrate the approach, we present illustrative results for the ascending thoracic aorta from three mouse models, quantifying uncertainties in constitutive model parameters as well as circumferential and axial tangent stiffness. Our extended workflow not only allows parameter uncertainties to be systematically reported, but also facilitates both subject-specific and group-level statistical analyses of the mechanics of the vessel wall.},
+year = {2021}
+}
+
+
 
 
 

Dissertation/main.tex

@@ -24,7 +24,7 @@
 
 \include{ann_product}
 
-\include{modified_mlp_associated_nn}
+%\include{modified_mlp_associated_nn}
 
 \include{ann_first_approximations}