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We review an algorithm developed for parameter estimation within the Continuous Data Assimilation (CDA) approach. We present an alternative derivation for the algorithm presented in a paper by Carlson, Hudson, and Larios (CHL, 2021). This derivation relies on the same assumptions as the previous derivation but frames the problem as a finite dimensional root-finding problem. Within the approach we develop, the algorithm developed in (CHL, 2021) is simply a realization of Newton's method. We then consider implementing other derivative based optimization algorithms; we show that the Levenberg Maqrquardt algorithm has similar performance to the CHL algorithm in the single parameter estimation case and generalizes much better to fitting multiple parameters. We then implement these methods in three example systems: the Lorenz '63 model, the two-layer Lorenz '96 model, and the Kuramoto-Sivashinsky equation.

This article studies two particular algorithms, a Relaxation Least Squares (RLS) algorithm and a Relaxation Newton Iteration (RNI) scheme , for reconstructing unknown parameters in dissipative dynamical systems. Both algorithms are based on a continuous data assimilation (CDA) algorithm for state reconstruction of A. Azouani, E. Olson, and E.S. Titi \cite{Azouani_Olson_Titi_2014}. Due to the CDA origins of these parameter recovery algorithms, these schemes provide on-the-fly reconstruction, that is, as data is collected, of unknown state and parameters simultaneously. It is shown how both algorithms give way to a robust general framework for simultaneous state and parameter estimation. In particular, we develop a general theory, applicable to a large class of dissipative dynamical systems, which identifies structural and algorithmic conditions under which the proposed algorithms achieve reconstruction of the true parameters. The algorithms are implemented on a high-dimensional two-layer Lorenz 96 model, where the theoretical conditions of the general framework are explicitly verifiable. They are also implemented on the two-dimensional Rayleigh-B\'enard convection system to demonstrate the applicability of the algorithms beyond the finite-dimensional setting. In each case, systematic numerical experiments are carried out probing the efficacy of the proposed algorithms, in addition to the apparent benefits and drawbacks between them.

A central problem in data science is to use potentially noisy samples of an unknown function to predict values for unseen inputs. In classical statistics, predictive error is understood as a trade-off between the bias and the variance that balances model simplicity with its ability to fit complex functions. However, over-parameterized models exhibit counterintuitive behaviors, such as "double descent" in which models of increasing complexity exhibit decreasing generalization error. Others may exhibit more complicated patterns of predictive error with multiple peaks and valleys. Neither double descent nor multiple descent phenomena are well explained by the bias-variance decomposition. We introduce a novel decomposition that we call the generalized aliasing decomposition (GAD) to explain the relationship between predictive performance and model complexity. The GAD decomposes the predictive error into three parts: 1) model insufficiency, which dominates when the number of parameters is much smaller than the number of data points, 2) data insufficiency, which dominates when the number of parameters is much greater than the number of data points, and 3) generalized aliasing, which dominates between these two extremes. We demonstrate the applicability of the GAD to diverse applications, including random feature models from machine learning, Fourier transforms from signal processing, solution methods for differential equations, and predictive formation enthalpy in materials discovery. Because key components of the GAD can be explicitly calculated from the relationship between model class and samples without seeing any data labels, it can answer questions related to experimental design and model selection before collecting data or performing experiments. We further demonstrate this approach on several examples and discuss implications for predictive modeling and data science.

Recovering pressure fields from image velocimetry measurements has two general strategies: i) directly integrating the pressure gradients from the momentum equation and ii) solving or enforcing the pressure Poisson equation (divergence of the pressure gradients). In this work, we analyze the error propagation of the former strategy and provide some practical insights. For example, we establish the error scaling laws for the Pressure Gradient Integration (PGI) and the Pressure Poisson Equation (PPE). We explain why applying the Helmholtz-Hodge Decomposition (HHD) could significantly reduce the error propagation for the PGI. We also propose to use a novel HHD-based pressure field reconstruction strategy that offers the following advantages: i) effective processing of noisy scattered or structured image velocimetry data on a complex domain and ii) using Radial Basis Functions (RBFs) with curl/divergence-free kernels to provide divergence-free correction to the velocity fields for incompressible flows and curl-free correction for pressure gradients. Complete elimination of divergence-free bias in measured pressure gradient and curl-free bias in the measured velocity field results in superior accuracy. Synthetic velocimetry data based on exact solutions and high-fidelity simulations are used to validate the analysis as well as demonstrate the flexibility and effectiveness of the RBF-HHD solver.

Indonesia is one of the world's most densely populated regions and lies among the epicenters of Earth's greatest natural hazards. Effectively reducing the disaster potential of these hazards through resource allocation and preparedness first requires an analysis of the risk factors of the region. Since destructive tsunamis present one of the most eminent dangers to coastal communities, understanding their sources and geological history is necessary to determine the potential future risk. Inspired by results from Cummins et al. 2020, and previous efforts that identified source parameters for earthquake-generated tsunamis, we consider landslide-generated tsunamis. This is done by constructing a probability distribution of potential landslide sources based on anecdotal observations of the 1852 Banda Sea tsunami, using Bayesian inference and scientific computing. After collecting over 100,000 samples (simulating 100,000 landslide induced tsunamis), we conclude that a landslide event provides a reasonable match to the tsunami reported in the anecdotal accounts. However, the most viable landslides may push the boundaries of geological plausibility. Future work creating a joint landslide-earthquake model may compensate for the weaknesses associated with an individual landslide or earthquake source event.

We propose a simple boundary condition regularization strategy to reduce error propagation in pressure field reconstruction from corrupted image velocimetry data. The core idea is to replace the canonical Neumann boundary conditions with Dirichlet ones obtained by integrating the tangential part of the pressure gradient along the boundaries. Rigorous analysis and numerical experiments justify the effectiveness of this regularization.

Using limited observations of the velocity field of the two-dimensional Navier-Stokes equations, we successfully reconstruct the steady body force that drives the flow. The number of observed data points is less than 10\% of the number of modes that describes the full flow field, indicating that the method introduced here is capable of identifying complicated forcing mechanisms from a relatively small collection of observations. In addition to demonstrating the efficacy of this method on turbulent flow data generated by simulations of the two-dimensional Navier-Stokes equations, we also rigorously justify convergence of the derived algorithm. Beyond the practical applicability of such an algorithm, the reliance of this method on the dynamical evolution of the system yields physical insight into the turbulent cascade., Comment: 18 pages, 4 figures

Motivated by recent progress in data assimilation, we develop an algorithm to dynamically learn the parameters of a chaotic system from partial observations. Under reasonable assumptions, we rigorously establish the convergence of this algorithm to the correct parameters when the system in question is the classic three-dimensional Lorenz system. Computationally, we demonstrate the efficacy of this algorithm on the Lorenz system by recovering any proper subset of the three non-dimensional parameters of the system, so long as a corresponding subset of the state is observable. We also provide computational evidence that this algorithm works well beyond the hypotheses required in the rigorous analysis, including in the presence of noisy observations, stochastic forcing, and the case where the observations are discrete and sparse in time.

Improving understanding of current seismic risk is often dependent on developing a more complete characterization of earthquakes that have occurred in the past, and in particular before the development of modern sensing equipment in the middle of the twentieth century. However, accounts of such events are typically anecdotal in nature, limiting efforts to model them to more heuristic approaches. To address this shortfall, we develop a framework based in Bayesian inference to provide a more rigorous methodology for estimating pre-instrumental earthquakes. By directly modeling accounts of resultant tsunamis via probability distributions, the framework allows practitioners to make principled estimates of key characteristics (e.g., magnitude and location) of historical earthquakes. To illustrate this idea, we apply the methodology to the estimation of an earthquake in Eastern Indonesia in the mid 19th century, the source of which is currently the subject of considerable debate in the geological community. The approach taken here gives evidence that even "small data" that is limited in scope and extremely uncertain can still be used to yield information on past seismic events. Moreover, sensitivity bounds indicate that the results obtained here are robust despite the inherent uncertainty in the observations.

We develop an algorithm based on the nudging data assimilation scheme for the concurrent (on-the-fly) estimation of scalar parameters for a system of evolutionary dissipative partial differential equations in which the state is partially observed. The algorithm takes advantage of the error that results from nudging a system with incorrect parameters with data from the true system. The intuitive nature of the algorithm makes its extension to several different systems immediate, and it allows for recovery of multiple parameters simultaneously. We test the method on the Kuramoto-Sivashinsky equation in one dimension and demonstrate its efficacy in this context.

We demonstrate the efficacy of a Bayesian statistical inversion framework for reconstructing the likely characteristics of large pre-instrumentation earthquakes from historical records of tsunami observations. Our framework is designed and implemented for the estimation of the location and magnitude of seismic events from anecdotal accounts of tsunamis including shoreline wave arrival times, heights, and inundation lengths over a variety of spatially separated observation locations. As an initial test case we use our framework to reconstruct the great 1852 earthquake and tsunami of eastern Indonesia. Relying on the assumption that these observations were produced by a subducting thrust event, the posterior distribution indicates that the observables were the result of a massive mega-thrust event with magnitude near 8.8 Mw and a likely rupture zone in the north-eastern Banda arc. The distribution of predicted epicentral locations overlaps with the largest major seismic gap in the region as indicated by instrumentally recorded seismic events. These results provide a geologic and seismic context for hazard risk assessment in coastal communities experiencing growing population and urbanization in Indonesia. In addition, the methodology demonstrated here highlights the potential for applying a Bayesian approach to enhance understanding of the seismic history of other subduction zones around the world.

Rayleigh-B\'enard convection (RBC) and Taylor-Couette Flow (TCF) are two paradigmatic fluid dynamical systems frequently discussed together because of their many similarities despite their different geometries and forcing. Often these analogies require approximations, but in the limit of large radii where TCF becomes rotating plane Couette flow (RPC) exact relations can be established. When the flows are restricted to two spatial degrees of freedom there is an exact specification that maps the three velocity components in RPC to the two velocity components and one temperature field in RBC. Using this, we deduce several relations between both flows: (i) The Rayleigh number $Ra$ in convection and the Reynolds $Re$ and rotation $R_\Omega$ number in RPC flow are related by $Ra= Re^2 R_\Omega (1-R_\Omega)$. (ii) Heat and angular momentum transport differ by $(1-R_\Omega)$, explaining why angular momentum transport is not symmetric around $R_\Omega=1/2$ even though the relation between $Ra$ and $R_\Omega$ has this symmetry. This relationship leads to a predicted value of $R_\Omega$ that maximizes the angular momentum transport that agrees remarkably well with existing numerical simulations of the full 3D system. (iii) One variable in both flows satisfy a maximum principle i.e., the fields' extrema occur at the walls. Accordingly, backflow events in shear flow \emph{cannot} occur in this two-dimensional setting. (iv) For free slip boundary conditions on the axial and radial velocity components, previous rigorous analysis for RBC implies that the azimuthal momentum transport in RPC is bounded from above by $Re^{5/6}$ with a scaling exponent smaller than the anticipated $Re^1$., Comment: 10 pages, 2 figures

The Rayleigh-B\'enard system with stress-free boundary conditions is shown to have a global attractor in each affine space where velocity has fixed spatial average. The physical problem is shown to be equivalent to one with periodic boundary conditions and certain symmetries. This enables a Gronwall estimate on enstrophy. That estimate is then used to bound the $L^2$ norm of the temperature gradient on the global attractor, which, in turn, is used to find a bounding region for the attractor in the enstrophy, palinstrophy-plane. All final bounds are algebraic in the viscosity and thermal diffusivity, a significant improvement over previously established estimates. The sharpness of the bounds are tested with numerical simulations.

An analytical framework for the propagation of velocity errors into PIV-based pressure calculation is extended. Based on this framework, the optimal spatial resolution and the corresponding minimum field-wide error level in the calculated pressure field are determined. This minimum error can be viewed as the smallest resolvable pressure. We find that the optimal spatial resolution is a function of the flow features (patterns and length scales), fundamental properties of the flow domain (e.g., geometry of the flow domain and the type of the boundary conditions), in addition to the error in the PIV experiments, and the choice of numerical methods. Making a general statement about pressure sensitivity is difficult. The minimum resolvable pressure depends on competing effects from the experimental error due to PIV and the truncation error from the numerical solver, which is affected by the formulation of the solver. This means that PIV experiments motivated by pressure measurements should be carefully designed so that the optimal resolution (or close to the optimal resolution) is used. Flows ($Re = 1.27 \times 10^4$ and $5\times 10^4$) with exact solutions are used as examples to validate the theoretical predictions of the optimal spatial resolutions and pressure sensitivity. The numerical experimental results agree well with the rigorous analytical predictions. We also propose an \textit{a posterior} method to estimate the contribution of truncation error using Richardson extrapolation and that of PIV error by adding artificially overwhelming noise. We also provide an introductory analysis of the effects of interrogation window overlap in PIV in the context of the pressure calculation., Comment: This is a significant extension of the previous version. Correspondence and requests should be addressed to Zhao Pan (email: zhao.pan@uwaterloo.ca)

We investigate the stability of statistically stationary conductive states for Rayleigh-B\'enard convection that arise due to a bulk stochastic internal heating. Our results indicate that stochastic forcing at small magnitude has little to no effect, while strong stochastic forcing has a destabilizing effect. The methodology put forth in this article, which combines rigorous analysis with careful computation, provides an approach to hydrodynamic stability which is applicable to a variety of systems subject to a large scale stochastic forcing.

A recent study investigated the propagation of error in a Velocimetry-based Pressure (V-Pressure) field reconstruction problem by directly analyzing the properties of the pressure Poisson equation (Pan et al., 2016). In the present work, we extend these results by quantifying the effect of the error profile in the data field (shape/structure of the error in space) on the resultant error in the reconstructed pressure field. We first calculate the mode of the error in the data that maximizes error in the pressure field, which is the most dangerous error (called the worst error in the present work). This calculation of the worst error is equivalent to finding the principle mode of, for example, an Euler-Bernoulli beam problem in one-dimension and the Kirchhoff-Love plate in two-dimensions, thus connecting the V-Pressure problem from experimental fluid mechanics to buckling elastic bodies from elastic mechanics. Taking advantage of this analogy, we then analyze how the error profile (e.g., spatial frequency of the error and the location of the most concentrated error) in the data field coupled with fundamental features of the flow domain (i.e., size, shape, and dimension of the domain, and the configuration of boundary conditions) significantly affects the error propagation from data to the reconstructed pressure. Our analytical results lend to practical applications in two ways. First, minimization of error propagation can be achieved by avoiding low-frequency error profiles in data similar to the worst case scenarios and error concentrated at sensitive locations. Second, small amounts of the error in the data, if the error profile is similar to the worst error case, can cause significant error in the reconstructed pressure field; such a synthetic error can be used to benchmark V-Pressure algorithms., Comment: this article is significantly extended from the previous version

Obtaining pressure field data from particle image velocimetry (PIV) is an attractive technique in fluid dynamics due to its noninvasive nature. The application of this technique generally involves integrating the pressure gradient or solving the pressure Poisson equation using a velocity field measured with PIV. However, very little research has been done to investigate the dynamics of error propagation from PIV-based velocity measurements to the pressure field calculation. Rather than measure the error through experiment, we investigate the dynamics of the error propagation by examining the Poisson equation directly. We analytically quantify the error bound in the pressure field, and are able to illustrate the mathematical roots of why and how the Poisson equation based pressure calculation propagates error from the PIV data. The results show that the error depends on the shape and type of boundary conditions, the dimensions of the flow domain, and the flow type.

We study the long-time behavior an extended Navier-Stokes system in $\R^2$ where the incompressibility constraint is relaxed. This is one of several "reduced models" of Grubb and Solonnikov '89 and was revisited recently (Liu, Liu, Pego '07) in bounded domains in order to explain the fast convergence of certain numerical schemes (Johnston, Liu '04). Our first result shows that if the initial divergence of the fluid velocity is mean zero, then the Oseen vortex is globally asymptotically stable. This is the same as the Gallay Wayne '05 result for the standard Navier-Stokes equations. When the initial divergence is not mean zero, we show that the analogue of the Oseen vortex exists and is stable under small perturbations. For completeness, we also prove global well-posedness of the system we study., Comment: 24 pages, 1 figure, updated to add authors' contact information and to address referee's comments

The heat transport in rotating Rayleigh-B\'enard convection is considered in the limit of rapid rotation (small Ekman number $E$) and strong thermal forcing (large Rayleigh number $Ra$). The analysis proceeds from a set of asymptotically reduced equations appropriate for rotationally constrained dynamics; the conjectured range of validity for these equations is $Ra \lesssim E^{-8/5}$. A rigorous bound on heat transport of $Nu \le 20.56Ra^3E^4$ is derived in the limit of infinite Prandtl number using the background method. We demonstrate that the exponent in this bound cannot be improved on using a piece-wise monotonic background temperature profile like the one used here. This is true for finite Prandtl numbers as well, i.e. $Nu \lesssim Ra^3$ is the best upper bound for this particular setup of the background method. The feature that obstructs the availability of a better bound in this case is the appearance of small-scale thermal plumes emanating from (or entering) the thermal boundary layer.

A fluid dynamics video of the rotating, weakly stratified Boussinesq equations is presented that illustrates the spontaneous formation of columnar vortices in the presence of stochastic, white noise forcing., Comment: includes a video

We derive an improved rigorous bound on the space and time averaged temperature $$ of an infinite Prandtl number Boussinesq fluid contained between isothermal no-slip boundaries thermally driven by uniform internal heating. A novel approach is used wherein a singular stable stratification is introduced as a perturbation to a non-singular background profile, yielding the estimate $\geq 0.419[R\log(R)]^{-1/4}$ where $R$ is the heat Rayleigh number. The analysis relies on a generalized Hardy-Rellich inequality that is proved in the appendix.

Rigorous upper limits on the vertical heat transport in two dimensional Rayleigh-Benard convection between stress-free isothermal boundaries are derived from the Boussinesq approximation of the Navier-Stokes equations. The Nusselt number Nu is bounded in terms of the Rayleigh number Ra according to $Nu \leq 0.2295 Ra^{5/12}$ uniformly in the Prandtl number Pr. This Nusselt number scaling challenges some theoretical arguments regarding the asymptotic high Rayleigh number heat transport by turbulent convection., Comment: 4 pages