Browsing by Subject "Multiscale analysis"

This work deals with the problem of estimating the intrinsic dimension of noisy, high-dimensional point clouds. A general class of sets which are locally well-approximated by k dimensional planes but which are embedded in a D>>k dimensional Euclidean space are considered. Assuming one has samples from such a set, possibly corrupted by high-dimensional noise, if the data is linear the dimension can be recovered using PCA. However, when the data is non-linear, PCA fails, overestimating the intrinsic dimension. A multiscale version of PCA is thus introduced which is robust to small sample size, noise, and non-linearities in the data.

One of the key mechanisms that mediate renal autoregulation is the tubuloglomerular feedback (TGF) system, which is a negative feedback loop in the kidney that balances glomerular filtration with tubular reabsorptive capacity. In this dissertation, we develop several mathematical models of the TGF system to study TGF-mediated model dynamics.

First, we develop a mathematical model of compliant thick ascending limb (TAL) of a short loop of Henle in the rat kidney, called TAL model, to investigate the effects of spatial inhomogeneous properties in TAL on TGF-mediated dynamics. We derive a characteristic equation that corresponds to a linearized TAL model, and conduct a bifurcation analysis by finding roots of that equation. Results of the bifurcation analysis are also validated via numerical simulations of the full model equations.

We then extend the TAL model to explicitly represent an entire short-looped nephron including the descending segments and having compliant tubular walls, developing a short-looped nephron model. A bifurcation analysis for the TGF loop-model equations is similarly performed by computing parameter boundaries, as functions of TGF gain and delay, that separate differing model behaviors. We also use the loop model to better understand the effects of transient as well as sustained flow perturbations on the TGF system and on distal NaCl delivery.

To understand the impacts of internephron coupling on TGF dynamics, we further develop a mathematical model of a coupled-TGF system that includes any finite number of nephrons coupled through their TGF systems, coupled-nephron model. Each model nephron represents a short loop of Henle having compliant tubular walls, based on the short-looped nephron model, and is assumed to interact with nearby nephrons through electrotonic signaling along the pre-glomerular vasculature. The characteristic equation is obtained via linearization of the loop-model equations as in TAL model. To better understand the impacts of parameter variability on TGF-mediated dynamics, we consider special cases where the relation between TGF delays and gains among two coupled nephrons is specifically chosen. By solving the characteristic equation, we determine parameter regions that correspond to qualitatively differing model behaviors.

TGF delays play an essential role in determining qualitatively and quantitatively different TGF-mediated dynamic behaviors. In particular, when noise arising from external sources of system is introduced, the dynamics may become significantly rich and complex, revealing a variety of model behaviors owing to the interaction with delays. In our next study, we consider the effect of the interactions between time delays and noise, by developing a stochastic model. We begin with a simple time-delayed transport equation to represent the dynamics of chloride concentration in the rigid-TAL fluid. Guided by a proof for the existence and uniqueness of the steady-state solution to the deterministic Dirichlet problem, obtained via bifurcation analysis and the contraction mapping theorem, an analogous proof for stochastic system with random boundary conditions is presented. Finally we conduct multiscale analysis to study the effect of the noise, specifically when the system is in subcritical region, but close enough to the critical delay. To analyze the solution behaviors in long time scales, reduced equations for the amplitude of solutions are derived using multiscale method.

Uncertainty quantification (UQ) plays a critical role in computational science and engineering. The representation of uncertainties stands at the core of UQ frameworks and encompasses the modeling of parametric uncertainties --- which are uncertainties affecting parameters in a well-known model --- and model-form uncertainties --- which are uncertainties defined at the operator level. Past contributions in the field have primarily focused on parametric uncertainties in idealized environments involving simple state spaces and index sets. On the other hand, the consideration of model-form uncertainties (beyond model error correction) is still in its infancy. In this context, this dissertation aims to develop stochastic modeling approaches to represent these two forms of uncertainties in multiscale and multimodel settings.

The case of spatially-varying geometrical perturbations on nonregular index sets is first addressed. We propose an information-theoretic formulation where a push-forward map is used to induce bounded variations and the latent Gaussian random field is implicitly defined through a stochastic partial differential equation on the manifold defining the surface of interest. Applications to a gyroid structure and patient-specific brain interfaces are presented. We then address operator learning in a multiscale setting where we propose a data-free training method, applied to Fourier neural operators. We investigate the homogenization of random media defined at microscopic and mesoscopic scales. Next, we develop a Riemannian probabilistic framework to capture operator-form uncertainties in the multimodel setting (i.e., when a family of model candidates is available). The proposed methodology combines a proper-orthogonal-decomposition reduced-order model with Riemannian operators ensuring admissibility in the almost sure sense. The framework exhibits several key advantages, including the ability to generate a model ensemble within the convex hull defined by model proposals and to constrain the mean in the Fréchet sense, as well as ease of implementation. The method is deployed to investigate model-form uncertainties in various molecular dynamics simulations on graphene sheets. We finally propose an extension of this framework to systems described by coupled partial differential equations, with emphasis on the phase-field approach to brittle fracture.

We investigate the dynamics of turbulent flows and scalar fields based on multiscale analysis, numerical simulations, and modeling. Specifically, we study the fundamental mechanisms of multiscale energy transfers in stratified turbulence where both the turbulent fluid flow and scalar field are present and exchanging energies (i.e., kinetic and potential energies). We also have developed a Lagrangian model which shows great capabilities for predicting the important dynamics of passive scalars in isotropic turbulence. Further evaluations and analysis of the scalar gradient diffusion term (which is approximated by the Lagrangian closure model) are also performed based on direct numerical simulation (DNS) data at higher Reynolds numbers $Re$, to potentially improve the model capability for higher $Re$.

In the first part of the work, we analyze the budgets of turbulent kinetic energy (TKE) and turbulent potential energy (TPE) at different scales $\ell$ in sheared, stably stratified turbulence using a filtering approach. We consider the competing effects in the flow along with the physical mechanisms governing the energy fluxes between scales. The theoretical work of our energy budget analysis is used to analyze data from direct numerical simulation (DNS) at buoyancy Reynolds number $Re_b=O(100)$. Various quantities in the energy budget equations are evaluated based on DNS data of SSST, with detailed discussions on both the mean-field behavior of the flow, as well as fluctuations about this mean-field state. Importantly, it is shown that the TKE and TPE fluxes between scales are both downscale on average and their instantaneous values are positively correlated, but not strongly. The relative weak correlation occurs mainly due to the different physical mechanisms that govern the TKE and TPE fluxes. Moreover, the contribution to these fluxes arising from the sub-grid fields (i.e., small scales) are shown to be significant, in addition to the filtered scale contributions associated with the processes of strain-self amplification, vortex stretching, and density gradient amplification.

Motivated by our findings that the average downscale flux of TKE and TPE are due to different mechanisms and that the contributions to the energy fluxes from small scale (i.e., sub-grid) dynamics are significant, in the second part we develop a Lagrangian model for studying the small-scale scalar dynamics in isotropic turbulence. It is known that the equation for the fluid velocity gradient along a Lagrangian trajectory immediately follows from the Navier-Stokes equation, and such an equation involves two terms that cannot be determined from the velocity gradient along the chosen Lagrangian path: the pressure Hessian and the viscous Laplacian; similarly, the equation for passive scalar gradients also involves an unclosed term in the Lagrangian frame, namely the scalar gradient diffusion term which needs to be closed. For the fluid velocity gradient, a recent model handles the unclosed terms using a multi-level version of the recent deformation of Gaussian fields (RDGF) closure (Johnson \& Meneveau, Phys.~Rev.~Fluids, 2017). The model is in remarkable agreement with DNS data and works for arbitrary Taylor Reynolds numbers $Re_\lambda$. Inspired by this, our Lagrangian model for the passive scalar gradients is developed using the RDGF approach. However, comparisons of the statistics obtained from this model with direct numerical simulation (DNS) data reveal substantial errors due to erroneously large fluctuations generated by the model. We address this defect by incorporating into the closure approximation information regarding the scalar gradient production along the local trajectory history of the particle. This modified model makes predictions for the scalar gradients, their production rates, and alignments with the strain-rate eigenvectors that are in very good agreement with DNS data. However, while the model yields valid predictions up to around $Re_\lambda\approx 500$, beyond this, the model breaks down.

In consideration of the model failure beyond $Re_\lambda\approx 500$, the final part of work conducts further investigations via theoretical analysis and computations of more DNS data at various Reynolds numbers $Re_\lambda$. We theoretically analyzed the governing equations and identified two key mechanisms preventing the divergence of the scalar gradient magnitude. The conditional average of the scalar gradient diffusion term is also analyzed via its reduced forms which are used to test the model closure against DNS results. The model closure shows considerable errors in terms of its linear predictions of the conditional averages, in contrast to the strongly nonlinear dependencies on the condition quantities shown in DNS data. Such revealed errors potentially could be the reason why the model collapses beyond $Re_\lambda\approx 500$. Also discussed are the local relations of the scalar gradient diffusion term and various relevant quantities. It has been found that the diffusion term acts strictly to dissipate fluctuations of the scalar gradients in all regions where the scalar gradients are being either amplified or suppressed. Scalar gradients are dissipated most strongly in the regions where the straining motions are strong and the TKE are most strongly dissipated. Overall, the presented work here gains novel insights into the dynamics of scalar turbulence, reveals important implications/defects of the existing model closures, and in the meantime provides useful guidance for further improvements of existing model closures or for developing new models so that the complex scalar dynamics can be better captured in a more accurate manner.