Low-dimensional Model Research Articles

The basins of attraction of the global minimizers of non-convex inverse problems with low-dimensional models in infinite dimension

Abstract Non-convex methods for linear inverse problems with low-dimensional models have emerged as an alternative to convex techniques. We propose a theoretical framework where both finite dimensional and infinite dimensional linear inverse problems can be studied. We show how the size of the basins of attraction of the minimizers of such problems is linked with the number of available measurements. This framework recovers known results about low-rank matrix estimation and off-the-grid sparse spike estimation, and it provides new results for Gaussian mixture estimation from linear measurements.

Stochastic Parameterization Using Compressed Sensing: Application to the Lorenz-96 Atmospheric Model

Growing set of optimization and regression techniques, based upon sparse representations of signals, to build models from data sets has received widespread attention recently with the advent of compressed sensing. This paper deals with the parameterization of the Lorenz-96 (Lorenz, 1995) model with two time-scales that mimics mid-latitude atmospheric dynamics with microscopic convective processes. Compressed sensing is used to build models (vector fields) to emulate the behavior of the fine-scale process, so that explicit simulations become an online benchmark for parameterization. We apply compressed sensing, where the sparse recovery is achieved by constructing a sensing/dictionary matrix from ergodic samples generated by the Lorenz-96 atmospheric model, to parameterize the unresolved variables in terms of resolved variables. Stochastic parameterization is achieved by auto-regressive modelling of noise. We utilize the ensemble Kalman filter for data assimilation, where observations (direct measurements) are assimilated in the low-dimensional stochastic parameterized model to provide predictions. Finally, we compare the predictions of compressed sensing and Wilks’ polynomial regression to demonstrate the potential effectiveness of the proposed methodology.

Blind Image Inpainting Using Low-Dimensional Manifold Regularization

In this paper, we present a novel method for blind image inpainting, which can restore images with missing or corrupted pixels, or images where the location of the damaged pixels is unknown. The method applies weighted nonlocal Laplacian to address the problem of blind image inpainting using low-dimensional manifold model (LDMM) regularization, and uses semi-local blocks instead of point integrals to implement constraints in LDMM. This solves the problem of low solution efficiency caused by the asymmetry of the linear equations solved by point integration, and the problem of the high iteration count to get good restoration effect. Experiments show that our method is competitive with latest methods in terms of both repairing images with large missing pixels rate and inpainting speed.

Mean-field equationsfor neural populations with q-Gaussian heterogeneities.

Describing the collective dynamics of large neural populations using low-dimensional models for averaged variables has long been an attractive task in theoretical neuroscience. Recently developed reduction methods make it possible to derive such models directly from the microscopic dynamics of individual neurons. To simplify the reduction, the Cauchy distribution is usually assumed for heterogeneous network parameters. Here we extend the reduction method for a wider class of heterogeneities defined by the q-Gaussian distribution. The shape of this distribution depends on the Tsallis index q and gradually changes from the Cauchy distribution to the normal Gaussian distribution as this index changes. We derive the mean-field equationsfor an inhibitory network of quadratic integrate-and-fire neurons with a q-Gaussian-distributed excitability parameter. It is shown that the dynamic modes of the network significantly depend on the form of the distribution determined by the Tsallis index. The results obtained from the mean-field equationsare confirmed by numerical simulation of the microscopic model.

Experimental studies on the frequency selection in flat plate wakes: Mean-flow stability analyses and low-dimensional modeling

The global frequency selection of two-dimensional vortex shedding in the flat plate wake is investigated experimentally. By performing a local stability analysis, and low-dimensional modeling based on the time-averaged mean flow velocity profiles, we have shown that the time-averaged velocity profiles give an accurate estimate of the global shedding frequency. Furthermore, we have analyzed the interaction strengths between the mean flow and the higher harmonics thereby experimentally supporting the theoretical criterion outlined previously by Sipp and Lebedev.

Competing Models

Abstract Different agents need to make a prediction. They observe identical data, but have different models: they predict using different explanatory variables. We study which agent believes they have the best predictive ability—as measured by the smallest subjective posterior mean squared prediction error—and show how it depends on the sample size. With small samples, we present results suggesting it is an agent using a low-dimensional model. With large samples, it is generally an agent with a high-dimensional model, possibly including irrelevant variables, but never excluding relevant ones. We apply our results to characterize the winning model in an auction of productive assets, to argue that entrepreneurs and investors with simple models will be overrepresented in new sectors, and to understand the proliferation of “factors” that explain the cross-sectional variation of expected stock returns in the asset-pricing literature.

Seismic data reconstruction based on low dimensional manifold model

Seismic data reconstruction is an essential and yet fundamental step in seismic data processing workflow, which is of profound significance to improve migration imaging quality, multiple suppression effect, and seismic inversion accuracy. Regularization methods play a central role in solving the under-determined inverse problem of seismic data reconstruction. In this paper, a novel regularization approach is proposed, the low dimensional manifold model (LDMM), for reconstructing the missing seismic data. Our work relies on the fact that seismic patches always occupy a low dimensional manifold. Specifically, we exploit the dimension of the seismic patches manifold as a regularization term in the reconstruction problem, and reconstruct the missing seismic data by enforcing low dimensionality on this manifold. The crucial procedure of the proposed method is to solve the dimension of the patches manifold. Toward this, we adopt an efficient dimensionality calculation method based on low-rank approximation, which provides a reliable safeguard to enforce the constraints in the reconstruction process. Numerical experiments performed on synthetic and field seismic data demonstrate that, compared with the curvelet-based sparsity-promoting L1-norm minimization method and the multichannel singular spectrum analysis method, the proposed method obtains state-of-the-art reconstruction results.

Denoising for 3D Point Cloud Based on Regularization of a Statistical Low-Dimensional Manifold.

A point cloud obtained by stereo matching algorithm or three-dimensional (3D) scanner generally contains much complex noise, which will affect the accuracy of subsequent surface reconstruction or visualization processing. To eliminate the complex noise, a new regularization algorithm for denoising was proposed. In view of the fact that 3D point clouds have low-dimensional structures, a statistical low-dimensional manifold (SLDM) model was established. By regularizing its dimensions, the denoising problem of the point cloud was expressed as an optimization problem based on the geometric constraints of the regularization term of the manifold. A low-dimensional smooth manifold model was constructed by discrete sampling, and solved by means of a statistical method and an alternating iterative method. The performance of the denoising algorithm was quantitatively evaluated from three aspects, i.e., the signal-to-noise ratio (SNR), mean square error (MSE) and structural similarity (SSIM). Analysis and comparison of performance showed that compared with the algebraic point-set surface (APSS), non-local denoising (NLD) and feature graph learning (FGL) algorithms, the mean SNR of the point cloud denoised using the proposed method increased by 1.22 DB, 1.81 DB and 1.20 DB, respectively, its mean MSE decreased by 0.096, 0.086 and 0.076, respectively, and its mean SSIM decreased by 0.023, 0.022 and 0.020, respectively, which shows that the proposed method is more effective in eliminating Gaussian noise and Laplace noise in common point clouds. The application cases showed that the proposed algorithm can retain the geometric feature information of point clouds while eliminating complex noise.

Implications of Multivariate Non-Gaussian Data Assimilation for Multi-scale Weather Prediction

Abstract Weather prediction models currently operate within a probabilistic framework for generating forecasts conditioned on recent measurements of Earth’s atmosphere. This framework can be conceptualized as one that approximates parts of a Bayesian posterior density estimated under assumptions of Gaussian errors. Gaussian error approximations are appropriate for synoptic-scale atmospheric flow, which experiences quasi-linear error evolution over time scales depicted by measurements, but are often hypothesized to be inappropriate for highly nonlinear, sparsely-observed mesoscale processes. The current study adopts an experimental regional modeling system to examine the impact of Gaussian prior error approximations, which are adopted by ensemble Kalman filters (EnKFs) to generate probabilistic predictions. The analysis is aided by results obtained using recently-introduced particle filter (PF) methodology that relies on an implicit non-parametric representation of prior probability densities—but with added computational expense. The investigation focuses on EnKF and PF comparisons over month-long experiments performed using an extensive domain, which features the development and passage of numerous extratropical and tropical cyclones. The experiments reveal spurious small-scale corrections in EnKF members, which come about from inappropriate Gaussian approximations for priors dominated by alignment uncertainty in mesoscale weather systems. Similar behavior is found in PF members, owing to the use of a localization operator, but to a much lesser extent. This result is reproduced and studied using a low-dimensional model, which permits the use of large sample estimates of the Bayesian posterior distribution. Findings from this study motivate the use of data assimilation techniques that provide a more appropriate specification of multivariate non-Gaussian prior densities or a multi-scale treatment of alignment errors during data assimilation.

Characterizing stages of COVID-19 epidemics: a nonlinear physics perspective based on amplitude equations.

The relevant dynamics underlying COVID-19 waves is described from an amplitude space perspective. To this end, the amplitude dynamics of infected populations is considered in different stages of epidemic waves. Eigenvectors and their corresponding amplitudes are derived analytically for low-dimensional models and by means of computational methods for high-dimensional models. It is shown that the amplitudes of all eigenvectors as functions of time can be tracked through the diverse stages of COVID-19 waves featuring jumps at the stage boundaries. In particular, it is shown that under certain circumstances the initial, outbreak stage and the final, subsiding stage of an epidemic wave are primarily determined by the unstable eigenvector of the initial stage and its corresponding remnant vector of the final stage. The corresponding amplitude captures most of the dynamics of the emerging and subsiding epidemics such that the problem at hand effectively becomes one dimensional leading to a dramatic reduction of the complexity of the problem at hand. Explicitly demonstrated for the first-wave COVID-19 epidemics of the year 2020 in the state of New York and Pakistan are given.

SNP-based heritability and selection analyses: Improved models and new results.

Complex-trait genetics has advanced dramatically through methods to estimate the heritability tagged by SNPs, both genome-wide and in genomic regions of interest such as those defined by functional annotations. The models underlying many of these analyses are inadequate, and consequently many SNP-heritability results published to date are inaccurate. Here, we review the modelling issues, both for analyses based on individual genotype data and association test statistics, highlighting the role of a low-dimensional model for the heritability of each SNP. We use state-of-art models to present updated results about how heritability is distributed with respect to functional annotations in the human genome, and how it varies with allele frequency, which can reflect purifying selection. Our results give finer detail to the picture that has emerged in recent years of complex trait heritability widely dispersed across the genome. Confounding due to population structure remains a problem that summary statistic analyses cannot reliably overcome. Also see the video abstract here: https://youtu.be/WC2u03V65MQ.

Implementation of subsurface transport processes in the low-dimensional integrated hydrological model NIHM

Solute transport models based on the resolution of the 3-D Advection-Dispersion (AD) equation are frequently plagued by several numerical problems, which add to the high computational cost. The hydrological model NIHM (Normally Integrated Hydrological Model) was recently proposed as a tool simulating the hydrological responses of watersheds with shallow saturated aquifers by coupling surface flow and a low-dimensional subsurface system, including the vadose zone. In this paper, we couple the low-dimensional flow model NIHM with a transport module solving the AD to propose an approach that enables to reduce the dimensionality of both the flow and transport problems. In NIHM, the low-dimensionality in the subsurface compartment results from an integration along the local direction normal to the bedrock of the aquifer. NIHM was previously evaluated and applied to actual hydrosystems—without addressing mass transfers—and it showed its ability to capture various hydrological responses even from complex systems. However, the relevance of a low-dimensional approach to transport is not proven yet as the model reduction could also render approximated velocity fields inappropriate to mass transfer problems. The accuracy and computational efficiency of the proposed model have been thoroughly examined through various synthetic test cases under different hydrodynamic conditions to assess the influence of the reduction of dimensionality on solute transport simulations. The findings of this study demonstrate that the reduction of dimension remains suited to predicting solute transport behaviors in shallow subsurface systems while providing an important gain in computation time. This might be promising for various applications dealing with groundwater quality.

A Bayesian Framework for Addressing the Uncertainty in Production Forecasts of Tight-Oil Reservoirs Using a Physics-Based Two-Phase Flow Model

Summary Extrapolation of history matched single-phase flow solutions is a common practice to forecast production from tight-oil reservoirs. Nonetheless, this approach (a) omits multiphase flow effects that exist below bubblepoint conditions and (b) has not included the quantification of the uncertainty of the estimated ultimate recovery (EUR). This work combines a new two-phase (oil and gas) flow solution within a Bayesian framework to address the uncertainty in the EUR. We illustrate the application of the procedure to tight-oil wells of west Texas. First, we combine the oil and the gas flow equations into a single dimensionless two-phase flow equation. The solution is a dimensionless flow rate model that can be easily scaled using two parameters: hydrocarbon pore volume and characteristic time. Second, this study generates the probabilistic production forecasts using a Bayesian approach in which the parameters of the model are treated as random variables. We construct parallel Markov chains of the parameters using an adaptative Metropolis-Hastings (M-H) Markov chain Monte Carlo (MCMC) for this purpose. Third, we evaluate the robustness of our inferences using posterior predictive checks (PPCs). Finally, we quantify the uncertainty in the EUR percentiles using the bootstrap method. The results of this research are as follows. First, this work shows that EUR estimates based on single-phase flow solutions will consistently underestimate the ultimate oil recovery factors in solution-gas drives where the reservoir pressure is less than the bubblepoint. The degree of underestimation will depend on the reservoir and flowing conditions as well as the fluid properties. Second, the application of parallel Markov chains using an adaptative M-H MCMC scheme that addresses the correlation between the model’s parameters solves the issues of mixing and autocorrelation of Markov chains and, thus, it speeds up speeding up the convergence of the Markov chains. Third, we generate replicated data from our posterior distributions to assess the robustness of our inferences (PPCs). Finally, we use hindcasting to calibrate and strengthen our inferences. To our knowledge, all these approaches are novel in EUR forecasting. Using a Bayesian framework with a low-dimensional (two-parameter) physics-based model provides a fast and reliable technique to quantify the uncertainty in production forecasts. In addition, the use of parallel chains with an adaptative M-H MCMC accelerates the rate of convergence and increases the robustness of the method.

Device performances analysis of p-type doped silicene-based field effect transistor using SPICE-compatible model.

Moore’s Law is approaching its end as transistors are scaled down to tens or few atoms per device, researchers are actively seeking for alternative approaches to leverage more-than-Moore nanoelectronics. Substituting the channel material of a field-effect transistors (FET) with silicene is foreseen as a viable approach for future transistor applications. In this study, we proposed a SPICE-compatible model for p-type (Aluminium) uniformly doped silicene FET for digital switching applications. The performance of the proposed device is benchmarked with various low-dimensional FETs in terms of their on-to-off current ratio, subthreshold swing and drain-induced barrier lowering. The results show that the proposed p-type silicene FET is comparable to most of the selected low-dimensional FET models. With its decent performance, the proposed SPICE-compatible model should be extended to the circuit-level simulation and beyond in future work.

Asynchronous and synchronous quenching of a globally unstable jet via axisymmetry breaking

We explore experimentally whether axisymmetry breaking can be exploited for the open-loop control of a prototypical hydrodynamic oscillator, namely a low-density inertial jet exhibiting global self-excited axisymmetric oscillations. We find that when forced transversely or axially at a low amplitude, the jet always transitions first from a period-1 limit cycle to $\mathbb {T}^2$ quasiperiodicity via a Neimark–Sacker bifurcation. However, we find that the subsequent transition, from $\mathbb {T}^2$ quasiperiodicity to $1:1$ lock-in, depends on the spatial symmetry of the applied perturbations: axial forcing induces a saddle-node bifurcation at small detuning but an inverse Neimark–Sacker bifurcation at large detuning, whereas transverse forcing always induces an inverse Neimark–Sacker bifurcation irrespective of the detuning. Crucially, we find that only transverse forcing can enable both asynchronous and synchronous quenching of the natural mode to occur without resonant or non-resonant amplification of the forced mode, resulting in substantially lower values of the overall response amplitude across all detuning values. From this, we conclude that breaking the jet axisymmetry via transverse forcing is a more effective control strategy than preserving the jet axisymmetry via axial forcing. Finally, we show that the observed synchronization phenomena can be modelled qualitatively with just two forced coupled Van der Pol oscillators. The success of such a simple low-dimensional model in capturing the complex synchronization dynamics of a multi-modal hydrodynamic system opens up new opportunities for axisymmetry breaking to be exploited for the open-loop control of other globally unstable flows.

Nonlinear dynamic modeling and responses of a cable dragged flexible spacecraft

The space debris removal system (SDRS) of tethered space tug is modelled as a cable dragged flexible spacecraft. The main goal of this paper is to develop a dynamic modeling approach for mode characteristics analysis and forced vibration analysis of the planar motion of a cable dragged flexible spacecraft. Solar arrays of the spacecraft are modelled as multi-beams connected by joints with additional rotating spring where the nonlinear stiffness, damping and friction are considered. Using the Global mode method (GMM), a novel analytical and low-dimensional nonlinear dynamic model is developed for vibration analysis of SDRS to enhance the design capacity for better fulfillment of space tasks. The linear and nonlinear partial differential equations that governing transverse vibration of solar arrays, transverse and longitudinal vibrations of cable are derived, along with the matching and boundary conditions. The natural frequencies and analytical global mode shapes of SDRS are determined, and orthogonality relations of the global mode shapes are established. Dynamical equations of the system are truncated to a set of ordinary differential equations with multiple-DOF. The validity of the method is verified by comparing the natural frequencies obtained from the characteristic equation with those obtained from FEM. Interesting mode localization and mode shift phenomena are observed in mode analysis. Dynamic responses of the system excitated by fluctuation of attitude control torque and short-time attitude control torque are worked out, respectively. Nonlinear behaviors are observed such as hardening, jump and super-harmonic resonances. Residual vibration of the overall system with considering the varous values of nonlinear stiffness, damping coefficient and friction coefficient has shown that the nonlinearity of joints has a great influence on the vibration of the overall system.

Cross-Regional Asymmetric Volatility Connectedness in the Korean Housing Markets

This paper investigates asymmetries in the volatility connectedness of the Korean housing market. For this purpose, we employ the asymmetric connectedness approach proposed by Baruink et al.(2017). While the analysis of Baruink et al.(2017) is based on low dimensional VAR model, we consider 16 regional markets, by using high dimensional LASSO-VAR model. The main results are as follows. First, the total CAM(connectedness asymmetric measure) is estimated at 27.08%, indicating a high asymmetry in the housing market. In particular, the impact of good volatility strictly dominates that of bad volatility. Second, the total CAM shows time-varying patterns, with a surge in December 2020. Third, the contribution of Seoul to the aggregate housing market had gradually decreased since 2010 but started to rise again in 2017. The findings of this study suggest that macroprudential policy instruments are necessary to stabilize the housing market in the face of the recent accumulation of financial imbalance risks.

Data-driven modeling and prediction of non-linearizable dynamics via spectral submanifolds

We develop a methodology to construct low-dimensional predictive models from data sets representing essentially nonlinear (or non-linearizable) dynamical systems with a hyperbolic linear part that are subject to external forcing with finitely many frequencies. Our data-driven, sparse, nonlinear models are obtained as extended normal forms of the reduced dynamics on low-dimensional, attracting spectral submanifolds (SSMs) of the dynamical system. We illustrate the power of data-driven SSM reduction on high-dimensional numerical data sets and experimental measurements involving beam oscillations, vortex shedding and sloshing in a water tank. We find that SSM reduction trained on unforced data also predicts nonlinear response accurately under additional external forcing.

Simulating heterogeneous corporate dynamics via the Rulkov map

This work deals with corporate dynamics as they emerge from mutual interaction between firms competing on the market. As it is commonly observed, corporate dynamics may oscillate between periods of fierce competition and calmer periods when companies settle in their niche. This is much like mutual synchronization and chaos regularization of bursts in a group of chaotically bursting cells modelled by the Rulkov map. We find that our framework can accommodate dynamic changes between multiple equilibria, model skewness, and idiosyncrasy even in the case of homogeneous firms. These features allow the link with empirical findings in labour and industrial economics. This paper supports the idea that nature is governed by simple laws from physics to engineering, from biology to economics, etc. For this reason, a low-dimensional deterministic model like the Rulkov map may be applied to economics and game theory for describing corporate dynamics.

Blade Envelopes Part I: Concept and Methodology

Abstract Blades manufactured through flank and point milling will likely exhibit geometric variability. Gauging the aerodynamic repercussions of such variability, prior to manufacturing a component, is challenging enough, let alone trying to predict what the amplified impact of any in-service degradation will be. While rules of thumb that govern the tolerance band can be devised based on expected boundary layer characteristics at known regions and levels of degradation, it remains a challenge to translate these insights into quantitative bounds for manufacturing. In this work, we tackle this challenge by leveraging ideas from dimension reduction to construct low-dimensional representations of aerodynamic performance metrics. These low-dimensional models can identify a subspace which contains designs that are invariant in performance—the inactive subspace. By sampling within this subspace, we design techniques for drafting manufacturing tolerances and for quantifying whether a scanned component should be used or scrapped. We introduce the blade envelope as a computational manufacturing guide for a blade that is also amenable to qualitative visualizations. In this paper, the first of two parts, we discuss its underlying concept and detail its computational methodology, assuming one is interested only in the single objective of ensuring that the loss of all manufactured blades remains constant. To demonstrate the utility of our ideas, we devise a series of computational experiments with the Von Karman Institute’s LS89 turbine blade.