diff --git a/.DS_Store b/.DS_Store index 4455a1e..be8fb8c 100644 Binary files a/.DS_Store and b/.DS_Store differ diff --git a/Dissertation/ann_product.tex b/Dissertation/ann_product.tex index f17b1e7..db758af 100644 --- a/Dissertation/ann_product.tex +++ b/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: diff --git a/Dissertation/ann_rep_brownian_motion_monte_carlo.tex b/Dissertation/ann_rep_brownian_motion_monte_carlo.tex index baf7cd2..88b0f41 100644 --- a/Dissertation/ann_rep_brownian_motion_monte_carlo.tex +++ b/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} diff --git a/Dissertation/main.bib b/Dissertation/main.bib index d719af5..c5fcd84 100644 --- a/Dissertation/main.bib +++ b/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} +} + + diff --git a/Dissertation/main.pdf b/Dissertation/main.pdf index 30fd6bc..f37219b 100644 Binary files a/Dissertation/main.pdf and b/Dissertation/main.pdf differ diff --git a/Dissertation/main.tex b/Dissertation/main.tex index 347cd69..54739c3 100644 --- a/Dissertation/main.tex +++ b/Dissertation/main.tex @@ -1,7 +1,7 @@ \include{preamble} \include{commands} -\title{Artificial Neural Networks Applied to Stochastic Monte Carlo as a Way to Approximate Modified Heat Equations, and Their Associated Parameters, Depths, and Accuracies.} +\title{Artificial Neural Networks Applied to Stochastic Monte Carlo as a Way to Approximate Modified Heat Equations, and Their Associated Parameters, Depths, and Accuracies.} \author{Shakil Rafi} \begin{document} \maketitle @@ -24,7 +24,7 @@ \include{ann_product} -\include{modified_mlp_associated_nn} +%\include{modified_mlp_associated_nn} \include{ann_first_approximations} diff --git a/MLP and DNN Material/.DS_Store b/MLP and DNN Material/.DS_Store index 8964485..06b708d 100644 Binary files a/MLP and DNN Material/.DS_Store and b/MLP and DNN Material/.DS_Store differ diff --git a/Templates/.DS_Store b/Templates/.DS_Store index c9df383..713dfaa 100644 Binary files a/Templates/.DS_Store and b/Templates/.DS_Store differ