Low-dimensional Domain Research Articles

Pullback measure attractors for non-autonomous stochastic reaction-diffusion equations on thin domains

This paper is concerned with pullback measure attractors of the non-autonomous stochastic reaction-diffusion equations defined in thin domains. We first prove the existence and uniqueness of pullback measure attractors for the inhomogeneous Markov process associated with the stochastic equations. We also introduce the concept of complete orbits for this sort of systems and use these special solutions to characterize the structures of pullback measure attractors. Then we establish the upper semi-continuity of these attractors as a family of thin domains collapses into a lower-dimensional domain.

Structural Outlier Detection and Zernike-Canterakis Moments for Molecular Surface Meshes-Fast Implementation in Python.

Object retrieval systems measure the degree of similarity of the shape of 3D models. They search for the elements of the 3D model databases that resemble the query model. In structural bioinformatics, the query model is a protein tertiary/quaternary structure and the objective is to find similarly shaped molecules in the Protein Data Bank. With the ever-growing size of the PDB, a direct atomic coordinate comparison with all its members is impractical. To overcome this problem, the shape of the molecules can be encoded by fixed-length feature vectors. The distance of a protein to the entire PDB can be measured in this low-dimensional domain in linear time. The state-of-the-art approaches utilize Zernike-Canterakis moments for the shape encoding and supply the retrieval process with geometric data of the input structures. The BioZernike descriptors are a standard utility of the PDB since 2020. However, when trying to calculate the ZC moments locally, the issue of the deficiency of libraries readily available for use in custom programs (i.e., without relying on external binaries) is encountered, in particular programs written in Python. Here, a fast and well-documented Python implementation of the Pozo-Koehl algorithm is presented. In contrast to the more popular algorithm by Novotni and Klein, which is based on the voxelized volume, the PK algorithm produces ZC moments directly from the triangular surface meshes of 3D models. In particular, it can accept the molecular surfaces of proteins as its input. In the presented PK-Zernike library, owing to Numba's just-in-time compilation, a mesh with 50,000 facets is processed by a single thread in a second at the moment order 20. Since this is the first time the PK algorithm is used in structural bioinformatics, it is employed in a novel, simple, but efficient protein structure retrieval pipeline. The elimination of the outlying chain fragments via a fast PCA-based subroutine improves the discrimination ability, allowing for this pipeline to achieve an 0.961 area under the ROC curve in the BioZernike validation suite (0.997 for the assemblies). The correlation between the results of the proposed approach and of the 3D Surfer program attains values up to 0.99.

Spatiotemporal evolution of COVID-19 in Portugal’s Mainland with self-organizing maps

BackgroundSelf-Organizing Maps (SOM) are an unsupervised learning clustering and dimensionality reduction algorithm capable of mapping an initial complex high-dimensional data set into a low-dimensional domain, such as a two-dimensional grid of neurons. In the reduced space, the original complex patterns and their interactions can be better visualized, interpreted and understood.MethodsWe use SOM to simultaneously couple the spatial and temporal domains of the COVID-19 evolution in the 278 municipalities of mainland Portugal during the first year of the pandemic. Temporal 14-days cumulative incidence time series along with socio-economic and demographic indicators per municipality were analyzed with SOM to identify regions of the country with similar behavior and infer the possible common origins of the incidence evolution.ResultsThe results show how neighbor municipalities tend to share a similar behavior of the disease, revealing the strong spatiotemporal relationship of the COVID-19 spreading beyond the administrative borders of each municipality. Additionally, we demonstrate how local socio-economic and demographic characteristics evolved as determinants of COVID-19 transmission, during the 1st wave school density per municipality was more relevant, where during 2nd wave jobs in the secondary sector and the deprivation score were more relevant.ConclusionsThe results show that SOM can be an effective tool to analysing the spatiotemporal behavior of COVID-19 and synthetize the history of the disease in mainland Portugal during the period in analysis. While SOM have been applied to diverse scientific fields, the application of SOM to study the spatiotemporal evolution of COVID-19 is still limited. This work illustrates how SOM can be used to describe the spatiotemporal behavior of epidemic events. While the example shown herein uses 14-days cumulative incidence curves, the same analysis can be performed using other relevant data such as mortality data, vaccination rates or even infection rates of other disease of infectious nature.

The Hadwiger theorem on convex functions, IV: The Klain approach

New proofs of the Hadwiger theorem for smooth and for continuous valuations on convex functions are obtained, and the Klain–Schneider theorem on convex functions is established. In addition, an extension theorem for valuations defined on functions with lower dimensional domain is proved, and its connection to the Abel transform is explained.

On the choice of the low-dimensional domain for global optimization via random embeddings

The challenge of taking many variables into account in optimization problems may be overcome under the hypothesis of low effective dimensionality. Then, the search of solutions can be reduced to the random embedding of a low dimensional space into the original one, resulting in a more manageable optimization problem. Specifically, in the case of time consuming black-box functions and when the budget of evaluations is severely limited, global optimization with random embeddings appears as a sound alternative to random search. Yet, in the case of box constraints on the native variables, defining suitable bounds on a low dimensional domain appears to be complex. Indeed, a small search domain does not guarantee to find a solution even under restrictive hypotheses about the function, while a larger one may slow down convergence dramatically. Here we tackle the issue of low-dimensional domain selection based on a detailed study of the properties of the random embedding, giving insight on the aforementioned difficulties. In particular, we describe a minimal low-dimensional set in correspondence with the embedded search space. We additionally show that an alternative equivalent embedding procedure yields simultaneously a simpler definition of the low-dimensional minimal set and better properties in practice. Finally, the performance and robustness gains of the proposed enhancements for Bayesian optimization are illustrated on numerical examples.

Limiting dynamics for stochastic reaction–diffusion equations on the Sobolev space with thin domains

This paper is devoted to bi-spatial random attractors of the stochastic reaction–diffusion equation when the terminate space is the Sobolev space on a thin domain, where the nonlinearity can be decomposed into two functions with (p,q)-growth exponents. By means of a computation method of induction, it is shown that the difference of solutions near the initial time is integrable of kp−2k+2 order. This higher-order integrability shows continuity of the solution operator from the square Lebesgue space to (kp−2k+2)-order Lebesgue spaces. In particular, the (2p−2)-order integrability shows continuity of the solution operator from the square Lebesgue space to the Sobolev space, which further shows existence of a random attractor in the Sobolev space when the initial space is the square Lebesgue space. Moreover, the higher-order integrability can be uniform with respect to all thin domains, which provides uniformly asymptotic compactness of the random dynamical systems. As a conclusion, the upper semi-continuity of attractors under the Sobolev norm is established when the narrow domain degenerates onto a lower dimensional domain.

Strong convergence of bi-spatial random attractors for parabolic equations on thin domains with rough noise

This article concerns bi-spatial random dynamics for the stochastic reaction-diffusion equation on a thin domain, where the noise is described by a general stochastic process instead of the usual Wiener process. A bi-spatial attractor is obtained when the non-initial state space is the p-times Lebesgue space, meanwhile, measurability of the attractor in the Banach space is proved by using measurability of both cocycle and absorbing set. Finally, the $p$-norm convergence of attractors is obtained when the thin domain collapses onto a lower dimensional domain. The method of symbolical truncation is applied to provide some uniformly asymptotic estimates.

A probabilistic compressive sensing framework with applications to ultrasound signal processing

Abstract The field of Compressive Sensing (CS) has provided algorithms to reconstruct signals from a much lower number of measurements than specified by the Nyquist-Shannon theorem. There are two fundamental concepts underpinning the field of CS. The first is the use of random transformations to project high-dimensional measurements onto a much lower-dimensional domain. The second is the use of sparse regression to reconstruct the original signal. This assumes that a sparse representation exists for this signal in some known domain, manifested by a dictionary. The original formulation for CS specifies the use of an l 1 penalised regression method, the Lasso. Whilst this has worked well in literature, it suffers from two main drawbacks. First, the level of sparsity must be specified by the user, or tuned using sub-optimal approaches. Secondly, and most importantly, the Lasso is not probabilistic; it cannot quantify uncertainty in the signal reconstruction. This paper aims to address these two issues; it presents a framework for performing compressive sensing based on sparse Bayesian learning. Specifically, the proposed framework introduces the use of the Relevance Vector Machine (RVM), an established sparse kernel regression method, as the signal reconstruction step within the standard CS methodology. This framework is developed within the context of ultrasound signal processing in mind, and so examples and results of compression and reconstruction of ultrasound pulses are presented. The dictionary learning strategy is key to the successful application of any CS framework and even more so in the probabilistic setting used here. Therefore, a detailed discussion of this step is also included in the paper. The key contributions of this paper are a framework for a Bayesian approach to compressive sensing which is computationally efficient, alongside a discussion of uncertainty quantification in CS and different strategies for dictionary learning. The methods are demonstrated on an example dataset from collected from an aerospace composite panel. Being able to quantify uncertainty on signal reconstruction reveals that this grows as the level of compression increases. This is key when deciding appropriate compression levels, or whether to trust a reconstructed signal in applications of engineering and scientific interest.

Efficient Laplace Approximation for Bayesian Registration Uncertainty Quantification.

This paper presents a novel approach to modeling the pos terior distribution in image registration that is computationally efficient for large deformation diffeomorphic metric mapping (LDDMM). We develop a Laplace approximation of Bayesian registration models entirely in a bandlimited space that fully describes the properties of diffeomorphic transformations. In contrast to current methods, we compute the inverse Hessian at the mode of the posterior distribution of diffeomorphisms directly in the low dimensional frequency domain. This dramatically reduces the computational complexity of approximating posterior marginals in the high dimensional imaging space. Experimental results show that our method is significantly faster than the state-of-the-art diffeomorphic image registration uncertainty quantification algorithms, while producing comparable results. The efficiency of our method strengthens the feasibility in prospective clinical applications, e.g., real- time image-guided navigation for brain surgery.

Out-of-Sample Generalizations for Supervised Manifold Learning for Classification.

Supervised manifold learning methods for data classification map high-dimensional data samples to a lower dimensional domain in a structure-preserving way while increasing the separation between different classes. Most manifold learning methods compute the embedding only of the initially available data; however, the generalization of the embedding to novel points, i.e., the out-of-sample extension problem, becomes especially important in classification applications. In this paper, we propose a semi-supervised method for building an interpolation function that provides an out-of-sample extension for general supervised manifold learning algorithms studied in the context of classification. The proposed algorithm computes a radial basis function interpolator that minimizes an objective function consisting of the total embedding error of unlabeled test samples, defined as their distance to the embeddings of the manifolds of their own class, as well as a regularization term that controls the smoothness of the interpolation function in a direction-dependent way. The class labels of test data and the interpolation function parameters are estimated jointly with an iterative process. Experimental results on face and object images demonstrate the potential of the proposed out-of-sample extension algorithm for the classification of manifold-modeled data sets.

Composition of Local Normal Coordinates and Polyhedral Geometry in Riemannian Manifold Learning

The Local Riemannian Manifold Learning (LRML) recovers the manifold topology and geometry behind database samples through normal coordinate neighborhoods computed by the exponential map. Besides, LRML uses barycentric coordinates to go from the parameter space to the Riemannian manifold in order to perform the manifold synthesis. Despite of the advantages of LRML, the obtained parameterization cannot be used as a representational space without ambiguities. Besides, the synthesis process needs a simplicial decomposition of the lower dimensional domain to be efficiently performed, which is not considered in the LRML proposal. In this paper, the authors address these drawbacks of LRML by using a composition procedure to combine the normal coordinate neighborhoods for building a suitable representational space. Moreover, they incorporate a polyhedral geometry framework to the LRML method to give an efficient background for the synthesis process and data analysis. In the computational experiments, the authors verify the efficiency of the LRML combined with the composition and discrete geometry frameworks for dimensionality reduction, synthesis and data exploration.

Large time and space size behaviour of the heat equation in non-cylindrical domains

In this note we consider a linear parabolic problem defined on a non-cylindrical unbounded domain Q. If Ω t denotes the section of Q above t, the Ω t size goes to +∞, when t → +∞, i.e. the sections Ω t become unbounded in some directions when the time t becomes large. Here a model problem is studied, but the technique used can be applied for a wide class of problems, as nonlinear ones, defined on more general domains Q as those introduced by Lions [11]. An asymptotically exponential convergence of the solutions of such problems towards the solution of an elliptic problem defined on a lower dimensional domain is established.

A multi-zone chemistry mapping approach for direct numerical simulation of auto-ignition and flame propagation in a constant volume enclosure

A direct numerical simulation (DNS) coupling with multi-zone chemistry mapping (MZCM) is presented to simulate flame propagation and auto-ignition in premixed fuel/air mixtures. In the MZCM approach, the physical domain is mapped into a low-dimensional phase domain with a few thermodynamic variables as the independent variables. The approach is based on the fractional step method, in which the flow and transport are solved in the flow time steps whereas the integration of the chemical reaction rates and heat release rate is performed in much finer time steps to accommodate the small time scales in the chemical reactions. It is shown that for premixed mixtures, two independent variables can be sufficient to construct the phase space to achieve a satisfactory mapping. The two variables can be the temperature of the mixture and the specific element mass ratio of H atom for fuels containing hydrogen atoms. An aliasing error in the MZCM is investigated. It is shown that if the element mass ratio is based on the element involved in the most diffusive molecules, the aliasing error of the model can approach zero when the grid in the phase space is refined. The results of DNS coupled with MZCM (DNS-MZCM) are compared with full DNS that integrates the chemical reaction rates and heat release rate directly in physical space. Application of the MZCM to different mixtures of fuel and air is presented to demonstrate the performance of the method for combustion processes with different complexity in the chemical kinetics, transport and flame–turbulence interaction. Good agreement between the results from DNS and DNS-MZCM is obtained for different fuel/air mixtures, including H2/air, CO/H2/air and methane/air, while the computational time is reduced by nearly 70%. It is shown that the MZCM model can properly address important phenomena such as differential diffusion, local extinction and re-ignition in premixed combustion.

Voice Pathology Detection and Discrimination Based on Modulation Spectral Features

In this paper, we explore the information provided by a joint acoustic and modulation frequency representation, referred to as modulation spectrum, for detection and discrimination of voice disorders. The initial representation is first transformed to a lower dimensional domain using higher order singular value decomposition (HOSVD). From this dimension-reduced representation a feature selection process is suggested using an information-theoretic criterion based on the mutual information between voice classes (i.e., normophonic/dysphonic) and features. To evaluate the suggested approach and representation, we conducted cross-validation experiments on a database of sustained vowel recordings from healthy and pathological voices, using support vector machines (SVMs) for classification. For voice pathology detection, the suggested approach achieved a classification accuracy of 94.1±0.28% (95% confidence interval), which is comparable to the accuracy achieved using cepstral-based features. However, for voice pathology classification the suggested approach significantly outperformed the performance of cepstral-based features.

Model-based predictive control for spatially-distributed systems using dimensional reduction models

In this paper, a low-dimensional multiple-input and multiple-output (MIMO) model predictive control (MPC) configuration is presented for partial differential equation (PDE) unknown spatially-distributed systems (SDSs). First, the dimension reduction with principal component analysis (PCA) is used to transform the high-dimensional spatio-temporal data into a low-dimensional time domain. The MPC strategy is proposed based on the online correction low-dimensional models, where the state of the system at a previous time is used to correct the output of low-dimensional models. Sufficient conditions for closed-loop stability are presented and proven. Simulations demonstrate the accuracy and efficiency of the proposed methodologies.

Hodgkin-Huxley Model-Based Analysis of Electric-Field Effect on Nerve Cell Using Self-Organizing Map

This paper describes an analysis of the effects of electric field on nerve cells by using the Hodgkin-Huxley model. When evaluating our model, which combines an additional ionic current source and generated membrane potential, we derive the peak-to-peak value, the accumulated square of variation, and Kolmogorov-Sinai (KS) entropy of the cell-membrane potential excited by 10, 100, 1k, and 10kHz-sinusoidal electric fields. In addition, to obtain a comprehensive view of the time-variation patterns of our model, we used a self-organizing map, which provides a way to map high-dimensional data onto a low-dimensional domain. Simulation results confirmed that lower-frequency electric fields tended to increase fluctuations of the cell-membrane potential, and the additional ionic current source was a more dominant factor for fluctuations of the cell-membrane potential. On the basis of our model, we visually confirmed that the obtained data could be projected onto the map in accordance with responses of cell-membrane potential excited by electric fields, resulting in a combined depiction of the effects of KS entropy and other parameters.

Homogenization and reduction of dimension for nonlinear parametric variational inequalities

In this paper we study the limit behavior of the solution u ϵ of a parametric variational inequality, governed by a nonlinear differential operator, the gradient operator ∇ being replaced by another operator ∇ ϵ , with the positive parameter ϵ ( ϵ → 0 ). Generalizing earlier results of Courilleau and Mossino [P. Courilleau, J. Mossino, Compensated compactness for nonlinear homogenization and reduction of dimension, Calculus of Variations 20 (2004) 65–91], we show that, up to a subsequence, u ϵ weakly converges to a solution of a limit problem. For quasilinear operators, we show that the limit problem can be formulated on a lower dimensional domain.

Dimensional Reduction in Electromagnetic Boundary Value Problems

If an electromagnetic boundary value problem exhibits appropriate symmetry, it is possible to solve it in a lower dimensional domain. We discuss symmetries in general and in particular show how dimensional reduction is based on symmetry.

Extinction, stable pattern and their transition in a diffusive single species population with distributed maturity

We consider a single-species structured population with distributed maturity and spatial diffusion in a cylindrical domain subject to Neumann and Robin boundary conditions. We first establish the threshold property of the reaction–diffusion system with distributed delay and non-local interaction in a corresponding lower-dimensional domain, so that the system approaches either an extinction state or a stable spatially varying pattern. We then investigate the transition from the extinction state to the stable pattern of the system in the cylindrical domain.

APPROXIMATE IMAGE QUALITY MEASURE IN LOW-DIMENSIONAL DOMAIN BASED ON RANDOM PROJECTION

Image Quality Measure (IQM) is used to automatically measure the degree of image artifacts such as blocking, ringing and blurring effects. It is calculated traditionally in the image spatial domain. In this paper, we present a new method of transforming an image into a low-dimensional domain based on random projection, so we can efficiently obtain the compatible IQM. From the transformed domain, we can calculate the Peak Signal-to-Noise Ratio (PSNR) and apply fuzzy logic to generate a Low-Dimensional Quality Index (LDQI). Experimental results show that the LDQI can approximate the IQM in the image spatial domain. We observe that the LDQI is suited for measuring the compression blur due to its relatively low distortion. The relative error is about 0.15 as the compression blur increases.