Multidimensional Approximation Research Articles

Tensor-Based Data-Driven Identification of Partial Differential Equations

Abstract We present a tensor-based method for model selection which identifies the unknown partial differential equation that governs a dynamical system using only spatiotemporal measurements. The method circumvents a disadvantage of standard matrix-based methods which typically have large storage consumption. Using a recently developed multidimensional approximation of nonlinear dynamical systems (MANDy), we collect the nonlinear and partial derivative terms of the measured data and construct a low-rank dictionary tensor in the tensor-train (TT) format. A tensor-based linear regression problem is then built, which balances the learning accuracy, model complexity, and computational efficiency. An algebraic expression of the unknown equations can be extracted. Numerical results are demonstrated on datasets generated by the wave equation, the Burgers' equation, and a few parametric partial differential equations (PDEs).

Impact of falling‐film freeze concentration in a commercial Lager beer

AbstractInnovations and technology are the driving forces for new beer industry products, especially in exploring new flavor profiles. Freeze concentration (FC) emerges as an alternative to take advantage of this opportunity. This work studied the falling‐film freeze concentration (FFFC) application on ethanol–water solutions and commercial beer. The effect of flow rate, freezing temperature, and initial concentration of ethanol in the model solution was studied, and the best conditions were found with a surface response analysis and then applied to a commercial beer. The effect on the sensory profile was determined by multidimensional approximation for the lager beer and the concentrated liquid. A significant effect of initial concentration and freezing temperature were found in the ethanol–water solutions, and the best condition was found at −25°C, 400 L/h, and 0.03 w/w, with a concentration index of 1.35. This condition was evaluated for commercial beer, where a concentration index of 1.18 was achieved, caused by solids in beer and molecular interactions of ethanol with other compounds. The sensory profile of a commercial lager beer was modified by increasing the intensity in 38% of the flavor descriptors and the preservation of the other 41% after applying FFFC. The effect of this technique in beer could be a novel method to produce beer products with new flavor profiles.Practical applicationsInnovations in science and technology added to the search for new flavors promote the development of new products in the beer industry. In this article, a potential use of this freeze concentration (FC) technology was identified for the development of new concentrated beer products. The impact of falling‐film FC in a commercial lager beer was studied. First, obtaining a concentrated beer from this technique was possible, with a higher concentration of total solids and ethanol. In addition, the beer's sensory profile was modified by applying FC, which resulted in an increase in the intensity of 38% of the descriptors and the preservation of 41%. It also allowed the appearance of new flavors and aromas.

Methodology for multidimensional approximation of current velocity fields around offshore aquaculture installations

Consistent growth of the marine aquaculture industry over the past decades calls for potential deployments of large-scale aquaculture structures to produce finfish, shellfish and macroalgae in varying inshore and offshore environments. Numerical simulations for engineering design applications become more challenging with increase of scale since current velocity fields are no longer uniform, complicating accurate hydrodynamic load calculations. Horizontal and vertical velocity profiles in this case are spatially (depth and particular location within the deployment site) and temporary (date and time) dependent. Thus, proper representation of the current velocity field in numerical models becomes crucial for accurate predictions of structural performance of aquaculture installations. In this paper, an advanced multidimensional approximation method based on discrete current velocity data is formulated. The approach implies presenting the continuous current velocity function as a superposition of weighted radial basis functions extended by a linear polynomial. To address overfitting issues, the thin plate regularization is applied in the method. The approximation is then constrained in order to fit the velocity values on the domain boundaries. The method is implemented in finite element software Hydro-FE and its performance is compared to other approximation methods on the example of a kelp grow line deployed at the Wood Island research site, Maine, USA. It was found that the difference between regular (mean or linearly interpolated) velocity profiles and the velocity profiles approximated with the radial basis function method can reach up to 34–38 % in terms of grow line mooring tensions, and 6–18% in terms of grow line displacement.

"Iterates of multidimensional approximation operators via Perov theorem"

"The starting point is an approximation process consisting of linear and positive operators. The purpose of this note is to establish the limit of the iterates of some multidimensional approximation operators. The main tool is a Perov’s result which represents a generalization of Banach fixed point theorem. In order to support the theoretical aspects, we present three applications targeting respectively the operators Bernstein, Cheney-Sharma and those of binomial type. The last class involves an incursion into umbral calculus"

ЧИСЛЕННОЕ МОДЕЛИРОВАНИЕ НАПРЯЖЁННО-ДЕФОРМИРОВАННОГО СОСТОЯНИЯ МЕТАЛЛОКОНСТРУКЦИЙ С ПОМОЩЬЮ ГЕОМЕТРИЧЕСКИХ ИНТЕРПОЛЯНТОВ

The work is devoted to carrying out multidimensional interpolation and approximation methods for the numerical solution of differential equations and computer model development of the stress-strain state of metal structures. The core of the work is a fundamental computational algorithm for the numerical solution of differential equations using geometric interpolants on regular and irregular networks. On its basis, computational experiments are carried out on numerical simulation of the stress-strain state of operated reservoirs for storing petroleum products, which form a software package implemented in the Maple interpreter. At the same time, the differential equation for modelling the stress-strain state of an elastic cylindrical shell under axisymmetric loading is improved for the numerical analysis of the stress-strain state of a cylindrical reservoir with geometric imperfections. Also a new approach is proposed to take into consideration the initial conditions of the differential equation, which consists of parallel transfer of the numerical solution to the point, its coordinates correspond to the initial conditions. The advantage of the proposed approach for the numerical solution of differential equations using geometric interpolants is that it eliminates the need to coordinate geometric information in the process of interaction between CAD and FEA systems, by analogy with the isogeometric method.

Vectorial Decoding Algorithm for Fast Correlation Attack and Its Applications to Stream Cipher Grain-128a

Fast correlation attack, pioneered by Meier and Staffelbach, is an important cryptanalysis tool for LFSR-based stream cipher, which exploits the correlation between the LFSR state and key stream and targets at recovering the initial state of LFSR via a decoding algorithm. In this paper, we develop a vectorial decoding algorithm for fast correlation attack, which is a natural generalization of the original binary approach. Our approach benefits from the contributions of all correlations in a subspace. We propose two novel criteria to improve the iterative decoding algorithm. We also give some cryptographic properties of the new FCA which allows us to estimate the efficiency and complexity bounds. Furthermore, we apply this technique to the well-analyzed stream cipher Grain-128a. Based on a hypothesis, an interesting result for its security bound is deduced from the perspective of iterative decoding. Our analysis reveals the potential vulnerability for LFSRs over matrix ring and also for nonlinear functions with biased multidimensional linear approximations such as Grain-128a.

Statistical Model of Correlation Difference and Related-Key Linear Cryptanalysis

The goal of this work is to propose a related-key model for linear cryptanalysis. We start by giving the mean and variance of the difference of sampled correlations of two Boolean functions when using the same sample of inputs to compute both correlations. This result is further extended to determine the mean and variance of the difference of correlations of a pair of Boolean functions taken over a random data sample of fixed size and over a random pair of Boolean functions. We use the properties of the multinomial distribution to achieve these results without independence assumptions. Using multivariate normal approximation of the multinomial distribution we obtain that the distribution of the difference of related-key correlations is approximately normal. This result is then applied to existing related-key cryptanalyses. We obtain more accurate right-key and wrong-key distributions and remove artificial assumptions about independence of sampled correlations. We extend this study to using multiple linear approximations and propose a Χ2-type statistic, which is proven to be Χ2 distributed if the linear approximations are independent. We further examine this statistic for multidimensional linear approximation and discuss why removing the assumption about independence of linear approximations does not work in the related-key setting the same way as in the single-key setting.

Strong bounded variation estimates for the multi-dimensional finite volume approximation of scalar conservation laws and application to a tumour growth model

A uniform bounded variation estimate for finite volume approximations of the nonlinear scalar conservation law ∂tα + div(uf(α)) = 0 in two and three spatial dimensions with an initial data of bounded variation is established. We assume that the divergence of the velocity div(u) is of bounded variation instead of the classical assumption that div(u) is zero. The finite volume schemes analysed in this article are set on nonuniform Cartesian grids. A uniform bounded variation estimate for finite volume solutions of the conservation law ∂tα + div(F(t,x,α)) = 0, where divxF ≠ 0 on nonuniform Cartesian grids is also proved. Such an estimate provides compactness for finite volume approximations in Lp spaces, which is essential to prove the existence of a solution for a partial differential equation with nonlinear terms in α, when the uniqueness of the solution is not available. This application is demonstrated by establishing the existence of a weak solution for a model that describes the evolution of initial stages of breast cancer proposed by Franks et al. [J. Math. Biol. 47 (2003) 424–452]. The model consists of four coupled variables: tumour cell concentration, tumour cell velocity–pressure, and nutrient concentration, which are governed by a hyperbolic conservation law, viscous Stokes system, and Poisson equation, respectively. Results from numerical tests are provided and they complement theoretical findings.

Approximation on Manifold

The purpose of this work is to obtain an effective evaluation of the speed of convergence for multidimensional approximations of the functions define on the differential manifold. Two approaches to approximation of functions, which are given on the manifold, are considered. The firs approach is the direct use of the approximation relations for the discussed manifold. The second approach is related to using the atlas of the manifold to utilise a well-designed approximation apparatus on the plane (finit element approximation, etc.). The firs approach is characterized by the independent construction and direct solution of the approximation relations. In this case the approximation relations are considered as a system of linear algebraic equations (with respect to the unknowns basic functions ωj (ζ)). This approach is called direct approximation construction. In the second approach, an approximation on a manifold is induced by the approximations in tangent spaces, for example, the Courant or the Zlamal or the Argyris fla approximations. Here we discuss the Courant fla approximations. In complex cases (in the multidimensional case or for increased requirements of smoothness) the second approach is more convenient. Both approaches require no processes cutting the manifold into a finit number of parts and then gluing the approximations obtained on each of the mentioned parts. This paper contains two examples of Courant type approximations. These approximations illustrate the both approaches mentioned above.

Development of a Trainable Classifier of State of Rail Lines with Multiple Patterns of Image Recognition

Introduction. The necessity to classify the state of rail lines affected by significant damaging factors on the sensitive element of the information sensor providing the assurance of classification quality with the required length of the rail lines of the control section forms the task of creating a classifier with extended functionality. Extending the functionality is possible using multidimensional state images with a set of informative features and training procedures for classification models. Using the classical classification principle with a single model leads to an excessive complication of the classification algorithm with low accuracy due to inaccurate solution of the system of conditional equations with multidimensional approximation by Hermite polynomials. Materials and Methods. The principles of reducing the dimension of the features space, various procedures for trainable classifier of state of rail lines with multidimensional patterns, the selection of decisive classification rules with a hierarchical grouping of classes, and the formation of a set of models of varying degrees of complexity trained to solve an incompatible system of equations are considered to solve the problem. There were obtained various degrees of complexity used in the adaptive algorithm for classifying the rail lines states using Hermite polynomials as models. Results. The article presents the results of developing 57 classifier models using Hermite polynomials with features of 2, 3, 4, 5, 6 arguments. As an example, the procedure of developing models with 2–6 features is shown. The research results showed that with an increase in the number of features, the quality of classification improves, as when dividing the state space into several classes. Discussion and Conclusion. The results of the studies confirm the feasibility of the principle of classification of rail line states by a set of classification models, and an algorithm of recursively increasing the classification complexity using a model of increased complexity. The criterion for presenting a new, more complex model is the mismatch between the results of the class calculation by the i-th model and the real class in which the rail line is located at the moment in time.

Using Parameter Elimination to Solve Discrete Linear Chebyshev Approximation Problems

We consider discrete linear Chebyshev approximation problems in which the unknown parameters of linear function are fitted by minimizing the least maximum absolute deviation of errors. Such problems find application in the solution of overdetermined systems of linear equations that appear in many practical contexts. The least maximum absolute deviation estimator is used in regression analysis in statistics when the distribution of errors has bounded support. To derive a direct solution of the problem, we propose an algebraic approach based on a parameter elimination technique. As a key component of the approach, an elimination lemma is proved to handle the problem by reducing it to a problem with one parameter eliminated, together with a box constraint imposed on this parameter. We demonstrate the application of the lemma to the direct solution of linear regression problems with one and two parameters. We develop a procedure to solve multidimensional approximation (multiple linear regression) problems in a finite number of steps. The procedure follows a method that comprises two phases: backward elimination and forward substitution of parameters. We describe the main components of the procedure and estimate its computational complexity. We implement symbolic computations in MATLAB to obtain exact solutions for two numerical examples.

On the Usage of Deterministic (Related-Key) Truncated Differentials and Multidimensional Linear Approximations for SPN Ciphers

Among the few works realising the search of truncated differentials (TD) and multidimensional linear approximations (MDLA) holding for sure, the optimality of the distinguisher should be confirmed via an exhaustive search over all possible input differences/masks, which cannot be afforded when the internal state of the primitive has a considerable number of words. The incomplete search is also a long-term problem in the search of optimal impossible differential (ID) and zerocorrelation linear approximation (ZCLA) since all available automatic tools operate under fixed input and output differences/masks, and testing all possible combinations of differences/masks is impracticable for now. In this paper, we start by introducing an automatic approach based on the constraint satisfaction problem for the exploration of deterministic TDs and MDLAs. Since we transform the exhaustive search into an inherent feature of the searching model, the issue of incomplete search is settled. This tool is applied to search for related-key (RK) TDs of AES-192, and a new related-key differential-linear (DL) distinguisher is identified with a TD with certainty. Due to the novel property of the distinguisher, the previous RK DL attack on AES-192 is improved. Also, the new distinguisher is explained from the viewpoint of differentiallinear connectivity table (DLCT) and thus can be regarded as the first application of DLCT in the related-key attack scenario. As the second application of the tool, we propose a method to construct (RK) IDs and ZCLAs automatically. Benefiting from the control of the nonzero fixed differential pattern and the inherent feature of exhaustive search, the new searching scheme can discover longer distinguishers and hence possesses some superiorities over the previous methods. This technique is implemented with several primitives, and the provable security bounds of SKINNY and Midori64 against impossible differential distinguishing attack are generalised.

Optimization of Abnormal Point Cloud Recognition in Robot Vision Grinding System Based on Multidimensional Improved Eigenvalue Method (MIEM)

To improve the accurate and sufficient recognition of abnormal points on the workpiece, a multidimensional anomaly point identification approach based on an improved eigenvalue method is proposed in this paper. Whether a point is normal or not depends on the angle between the two adjacent vectors which consisted of four adjacent points around the current focus. The comprehensive judgment is carried out by multidimensional approximation. The numerical simulation and actual experiment validate the efficiency of the proposed method to quickly and accurately identify the abnormal point cloud in the surface point cloud data.

Multidimensional spline–interpolation–extrapolation methods for approximating hydraulics of steering actuating units for rocket stages

The paper discusses development principles of multidimensional approximation methods for hydraulic properties of throttling windows in slide hydraulic valves of electrohydraulic amplifiers of steering actuator, that are based on cubic spline-interpolation-extrapolation (SIE) of functions. A distinctive feature of the methods that were developed consists in that math simulations do not require complex analytical dependences of the properties, but rather only tabular representation of experimental data is used for their approximation in the course of calculation. The high efficiency of cubic SIE-based multidimensional approximation methods presented in the paper has been proven and demonstrated in math models for static analysis of autonomous one-stage electrohydraulic steering actuators with two- and four-throttle electrohydraulic amplifiers. Thanks to these methods the calculations of static properties of steering actuators within wide ranges of working fluid temperatures and power supply voltages are at present accurate to within ±2%. Key words: multidimensional approximation method, cubic spline-interpolation-extrapolation, electrohydraulic steering actuator.

Geometric modeling of multifactor processes and phenomena by the multidimensional parabolic interpolation method

The paper presents the results of multidimensional parabolic interpolation studies, (one of the special cases of the multidimensional interpolation method), applied to solve problems of modelling multifactor processes and phenomena using geometric objects of multidimensional affine space. The authors describe the technique of geometric model tree forming of the process under study and its analytical description based on computational point algorithms with subsequent implementation on a computer. Such an approach makes it possible to effectively use multidimensional interpolation instead of multidimensional approximation (based on the least squares method) for solving problems of mathematical and computer modelling of multifactor 3-level processes and phenomena of animate and inanimate nature, technology, economy, construction, and architecture The study gives an example of multidimensional parabolic interpolation application to simulate the dependence of the fine-grained tar-polymer concrete compressive strength on 4 factors: tar viscosity, polyvinyl chloride dropout concentration in coal tar, activator concentration on the mineral powder surface and temperature, followed by optimization of the composition and operating conditions road pavement.

Tensor-Based Algorithms for Image Classification

Interest in machine learning with tensor networks has been growing rapidly in recent years. We show that tensor-based methods developed for learning the governing equations of dynamical systems from data can, in the same way, be used for supervised learning problems and propose two novel approaches for image classification. One is a kernel-based reformulation of the previously introduced multidimensional approximation of nonlinear dynamics (MANDy), the other an alternating ridge regression in the tensor train format. We apply both methods to the MNIST and fashion MNIST data set and show that the approaches are competitive with state-of-the-art neural network-based classifiers.

A\xa0duration approach for the measurement of biometric risks in life insurance

Duration concepts are standard methods for measuring interest rate risks of portfolios, liabilities or other cash flows. Macaulay duration, effective duration and key-rate duration are widely used in practice according to different types of yield curves. In this paper, we will present a formulation for a forward rate duration measure by using multi-dimensional Taylor series approximation. It allows to measure the interest rate risk based on arbitrary forward rates. This approach can easily be adopted for biometric risk management, allowing the definition of a biometric duration. The biometric duration will be applied to actuarial present values of premiums and benefits as well as the actuarial reserve to quantify the corresponding biometric risk. It proves to be an easy-to-use tool in the actuarial practise.

GEOMETRIC MEANING OF LEAST SQUARES METHOD

The geometric interpretation of the least squares method is presents. At the same time, one of its possible generalizations to multidimensional space is proposed. This approach makes it possible to expand the capabilities one of the key methods of multidimensional approximation and effectively use it for geometric modeling of multifactor processes and phenomena. The analytical description of the proposed method is performed using point equations. The geometric interpretation of the generalized least squares method, which consists in determining the linear surface of the minimum width between two hypersurfaces in the hyperspace of the General position, extends the tools of geometric modeling objects in multidimensional space and can be effectively used for geometric modeling of multifactor processes and phenomena’s, by presenting them in the form of geometric multiparameter objects passing through predetermined points. In this case, the approximation process is reduced to determining the coordinates of the nodal points of the geometric object of multidimensional space that satisfy the condition of minimizing the sum of the lengths of the segments between the nodal points and the given ones. Also, using the proposed approach, the generalization of the coefficient of determination on the multidimensional space as a tool for assessing the accuracy of the results of multidimensional approximation is performed. An example of using the proposed generalization for the geometric modeling of a three-parameter hypersurface of the response belonging to a four-dimensional space in relation to the determination of strength characteristics over the entire volume of a concrete column is given.

Unimodular Invariance of Karyon Expansions of Algebraic Numbers in Multidimensional Continued Fractions

By the method of differentiation of induced toric tilings, periodic expansions for algebraic irrationalities in multidimensional continued fractions are found. These expansions give the best karyon approximations with respect to polyhedral norms. The above irrationalities are obtained by the composition of backward continued fraction mappings and unimodular transformations of algebraic units that are expanded in purely periodic continued fractions. Karyon expansions have several invariants: recurrence relations for the numerators and denominators of the convergents of continued fractions and the rate of multidimensional approximation of irrationalities by rational numbers.

Multidimensional Approximation of Nonlinear Dynamical Systems

A key task in the field of modeling and analyzing nonlinear dynamical systems is the recovery of unknown governing equations from measurement data only. There is a wide range of application areas for this important instance of system identification, ranging from industrial engineering and acoustic signal processing to stock market models. In order to find appropriate representations of underlying dynamical systems, various data-driven methods have been proposed by different communities. However, if the given data sets are high-dimensional, then these methods typically suffer from the curse of dimensionality. To significantly reduce the computational costs and storage consumption, we propose the method multidimensional approximation of nonlinear dynamical systems (MANDy) which combines data-driven methods with tensor network decompositions. The efficiency of the introduced approach will be illustrated with the aid of several high-dimensional nonlinear dynamical systems.