An Efficient Polynomial Chaos Expansion Method for Uncertainty Quantification in Dynamic Systems

Uncertainty is a common feature in first-principles models that are widely used in various engineering problems. Uncertainty quantification (UQ) has become an essential procedure to improve the accuracy and reliability of model predictions. Polynomial chaos expansion (PCE) has been used as an efficient approach for UQ by approximating uncertainty with orthogonal polynomial basis functions of standard distributions (e.g., normal) chosen from the Askey scheme. However, uncertainty in practice may not be represented well by standard distributions. In this case, the convergence rate and accuracy of the PCE-based UQ cannot be guaranteed. Further, when models involve non-polynomial forms, the PCE-based UQ can be computationally impractical in the presence of many parametric uncertainties. To address these issues, the Gram–Schmidt (GS) orthogonalization and generalized dimension reduction method (gDRM) are integrated with the PCE in this work to deal with many parametric uncertainties that follow arbitrary distributions. The performance of the proposed method is demonstrated with three benchmark cases including two chemical engineering problems in terms of UQ accuracy and computational efficiency by comparison with available algorithms (e.g., non-intrusive PCE).

Four Spacetime Dimensions from Multifractal Geometry

As paradigm of complex behavior, multifractals describe the underlying geometry of self-similar objects or processes. Building on the connection between entropy and multifractals, we show here that the generalized dimension of geodesic trajectories in General Relativity coincides with the four-dimensionality of classical spacetime.

Security analysis of KXB10 QKD protocol with higher-dimensional quantum states

Quantum key distribution (QKD) is one of the exciting applications of quantum mechanics. It allows the sharing of secret keys between two communicating parties with unconditional security. A variety of QKD protocols have been proposed since the inception of the BB84 protocol. Among different implementation techniques of QKD protocols, there is a category which exploits higher dimensions qubit states to encode classical bits. In this paper, we focus on such a QKD protocol called KXB10, which uses three bases with higher dimensions. Analysis of the generalized dimension quantum states is performed by evaluating it based on the index transmission error rate ITER. We find that there is a direct relationship between qubit dimensions and ITER for the KXB10 protocol.

Multifractal and joint multifractal analysis of the spatial variability of CO2 emission and other soil properties

<p>Soil is a major source and also a sink of CO<sub>2</sub>. Agricultural management practices influence soil  carbon sequestration. Identification of CO<sub>2</sub> emission hotspots may be instrumental in implemented strategies for managing carbon cycling in agricultural soils. We used multifractal analysis to assess the spatial variability of both, soil CO<sub>2</sub> emissions and associated soil physico-chemical attributes. The objectives of this study were: i) to characterize patterns of spatial variability of CO<sub>2</sub> emissions and related soil properties using single multifractal spectra, and ii) to compare the scale‐dependent relationship between soil CO<sub>2</sub> emissions and selected soil attributes by joint multifractal analysis. The study site was an experimental field managed as a sylvopastoral system, located in Selviria, South Mato Grosso state, Brazil. The soil was an Oxisol developed over basalt. Soil CO<sub>2 </sub>emission, soil water content and soil temperature were measured at 128 points every meter. In addition<strong>, </strong>soil was sampled at the marked points to analyze clay content, macro and microporosity, air free porosity, magnetic susceptibility, bulk density, and humification index of soil organic matter in absolute values and relative to organic carbon content. The generalized dimension, D<sub>q</sub> versus q, and singularity spectra, f(α) versus α, of the spatial distributions of the 11 variables studied showed various degrees of multifractality. In general, the amplitude of the generalized dimension and singularity spectra was much higher for negative than for positive q order statistical moments. Joint multifractal spectra show a positive relationship between the scaling indices of the spatial distributions of CO<sub>2</sub> and all of the other soil variables studied. However, contour plots were diagonally oriented for higher values of scaling indices and showed no distinct trend for the lower ones. Joint multifractal analysis corroborates different degrees of association between the scaling indices of CO<sub>2</sub> and all of the remaining variables studied. It also showed that CO<sub>2</sub> was stronger correlated at multiple scales than at the observation scale. Therefore, single scale analysis may not be sufficient to fully describe relationships between soil testing methods.Our study suggests that soil factors and processes driven the spatial variability of CO<sub>2</sub> and the associated variables studied may be not very different.</p><p> </p>

MULTIFRACTAL ANALYSIS OF SOIL RESISTANCE TO PENETRATION IN DIFFERENT PEDOFORMS

ABSTRACT Soils are highly variable across landscapes, which can be assessed and characterized according to scale, as well as fractal and multifractal concepts of scale. Thus, the objective of this study was to analyze the multifractality of the penetration resistance (PR) of vertical profiles from different slope forms (concave and convex). The experimental plot incorporated 44.75 ha, and the PR was measured at 70 sampling points in the 0-0.6 m layer, distributed in concave (Type A: 38 sampling points) and convex pedoforms (Type B: 32 sampling points). Data analysis was performed using the PR value (every 0.01 m depth) for each of the sampling points (PRmean), and their respective maximum (Prmaximun) and minimum (PRminimum) values. Multifractal analysis was performed to assess the changes in the structure, heterogeneity, and uniformity of the vertical profiles according to the scale, characterizing the partition function, generalized dimension, and singularity spectrum. The multifractal parameters of the generalized dimension and singularity spectrum demonstrated greater homogeneity and uniformity in the vertical PR profiles of pedoform B (convex) compared to those of pedoform A (concave). The minimum PR values in pedoform A (PRminimum) showed the greatest scale heterogeneity, indicating that in terms of soil management, it is more relevant to monitor the minimum values than the maximum values. The fractal analysis allowed us to describe the heterogeneity of the data on scales not evaluated by conventional analysis methods, with high potential for use in precision agriculture and the delimitation of specific management zones.

Modified Polynomial Chaos Expansion for Efficient Uncertainty Quantification in Biological Systems

Uncertainty quantification (UQ) is an important part of mathematical modeling and simulations, which quantifies the impact of parametric uncertainty on model predictions. This paper presents an efficient approach for polynomial chaos expansion (PCE) based UQ method in biological systems. For PCE, the key step is the stochastic Galerkin (SG) projection, which yields a family of deterministic models of PCE coefficients to describe the original stochastic system. When dealing with systems that involve nonpolynomial terms and many uncertainties, the SG-based PCE is computationally prohibitive because it often involves high-dimensional integrals. To address this, a generalized dimension reduction method (gDRM) is coupled with quadrature rules to convert a high-dimensional integral in the SG into a few lower dimensional ones that can be rapidly solved. The performance of the algorithm is validated with two examples describing the dynamic behavior of cells. Compared to other UQ techniques (e.g., nonintrusive PCE), the results show the potential of the algorithm to tackle UQ in more complicated biological systems.

The classification of certain ASH C*-algebras of real rank zero

In this paper, using ordered total K-theory, we give a K-theoretic classification for the real rank zero inductive limits of direct sums of generalized dimension drop interval algebras.

Multifractal analysis of soil penetration resistance under sugarcane cultivation

ABSTRACT Soil resistance to penetration (PR) is an indirect measure of the state of soil compaction. Thus, the objective of this study was to characterize PR in vertical profiles in an area cultivated with sugarcane using multifractal models for different relief units. The experiment was carried out in an Oxisol with a clay texture, with 6.85 ha in the municipality of Coelho Neto (Maranhão state, Brazil), where 60 sampling points were demarcated. The area was divided into four relief units (Type A > 74 m, Type B from 71 to 74 m, Type C from 68 to 71 m and Type D from 65 to 68 m). The PR was measured at the 60 sampling points using an impact penetrometer, and the PR determined in the 0-0.60 m depth layer every 0.01 m. The multifractal analysis was performed considering the scale property of each profile and typified the singularity and Rènyi spectra estimated using the current method. Multifractal analysis allowed the identification of patterns at different scales and with high heterogeneity. The multifractal behavior was represented by the singularity spectrum (α), versus f(α), and the generalized dimension (Dq). The multifractal analysis allowed the differentiation between the profiles of the relief units (Types A, B, C and D), resulting in an important tool for studies of soil resistance to penetration.