Journal articles on the topic 'Differential equations, Partial Data processing'
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VAN GENNIP, YVES, and CAROLA-BIBIANE SCHÖNLIEB. "Introduction: Big data and partial differential equations." European Journal of Applied Mathematics 28, no. 6 (November 7, 2017): 877–85. http://dx.doi.org/10.1017/s0956792517000304.
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Partial differential equations (PDEs) are expressions involving an unknown function in many independent variables and their partial derivatives up to a certain order. Since PDEs express continuous change, they have long been used to formulate a myriad of dynamical physical and biological phenomena: heat flow, optics, electrostatics and -dynamics, elasticity, fluid flow and many more. Many of these PDEs can be derived in a variational way, i.e. via minimization of an ‘energy’ functional. In this globalised and technologically advanced age, PDEs are also extensively used for modelling social situations (e.g. models for opinion formation, mathematical finance, crowd motion) and tasks in engineering (such as models for semiconductors, networks, and signal and image processing tasks). In particular, in recent years, there has been increasing interest from applied analysts in applying the models and techniques from variational methods and PDEs to tackle problems in data science. This issue of the European Journal of Applied Mathematics highlights some recent developments in this young and growing area. It gives a taste of endeavours in this realm in two exemplary contributions on PDEs on graphs [1, 2] and one on probabilistic domain decomposition for numerically solving large-scale PDEs [3].
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Dejnožková, Eva, and Petr Dokládal. "A PARALLEL ARCHITECTURE FOR CURVE-EVOLUTION PARTIAL DIFFERENTIAL EQUATIONS." Image Analysis & Stereology 22, no. 2 (May 3, 2011): 121. http://dx.doi.org/10.5566/ias.v22.p121-132.
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The computation of the distance function is a crucial and limiting element in many applications of image processing. This is particularly true for the PDE-based methods, where the distance is used to compute various geometric properties of the travelling curve. Massive Marchinga is a parallel algorithm computing the distance function by propagating the solution from the sources and permitting simultaneous spreading of component labels in the infiuence zones. Its hardware implementation is conceivable as no sorted data structures are used. The feasibility is demonstrated here on a set of parallely-operating Processing Units arranged in a linear array. The text concludes by a study of the accuracy and the implementation cost.
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Boudellioua, M. S. "Controllable and Observable Polynomial Description for 2D Noncausal Systems." Journal of Control Science and Engineering 2007 (2007): 1–5. http://dx.doi.org/10.1155/2007/87171.
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Two-dimensional state-space systems arise in applications such as image processing, iterative circuits, seismic data processing, or more generally systems described by partial differential equations. In this paper, a new direct method is presented for the polynomial realization of a class of noncausal 2D transfer functions. It is shown that the resulting realization is both controllable and observable.
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Chevallier, Julien, María José Cáceres, Marie Doumic, and Patricia Reynaud-Bouret. "Microscopic approach of a time elapsed neural model." Mathematical Models and Methods in Applied Sciences 25, no. 14 (October 14, 2015): 2669–719. http://dx.doi.org/10.1142/s021820251550058x.
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The spike trains are the main components of the information processing in the brain. To model spike trains several point processes have been investigated in the literature. And more macroscopic approaches have also been studied, using partial differential equation models. The main aim of the present paper is to build a bridge between several point processes models (Poisson, Wold, Hawkes) that have been proved to statistically fit real spike trains data and age-structured partial differential equations as introduced by Pakdaman, Perthame and Salort.
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Jafarian, Ahmad, and Dumitru Baleanu. "Application of ANNs approach for wave-like and heat-like equations." Open Physics 15, no. 1 (December 29, 2017): 1086–94. http://dx.doi.org/10.1515/phys-2017-0135.
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Abstract Artificial neural networks are data processing systems which originate from human brain tissue studies. The remarkable abilities of these networks help us to derive desired results from complicated raw data. In this study, we intend to duplicate an efficient iterative method to the numerical solution of two famous partial differential equations, namely the wave-like and heat-like problems. It should be noted that many physical phenomena such as coupling currents in a flat multi-strand two-layer super conducting cable, non-homogeneous elastic waves in soils and earthquake stresses, are described by initial-boundary value wave and heat partial differential equations with variable coefficients. To the numerical solution of these equations, a combination of the power series method and artificial neural networks approach, is used to seek an appropriate bivariate polynomial solution of the mentioned initial-boundary value problem. Finally, several computer simulations confirmed the theoretical results and demonstrating applicability of the method.
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Miller, Andrew, Jan Petrich, and Shashi Phoha. "Advanced Image Analysis for Learning Underlying Partial Differential Equations for Anomaly Identification." Journal of Imaging Science and Technology 64, no. 2 (March 1, 2020): 20510–1. http://dx.doi.org/10.2352/j.imagingsci.technol.2020.64.2.020510.
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Abstract In this article, the authors adapt and utilize data-driven advanced image processing and machine learning techniques to identify the underlying dynamics and the model parameters for dynamic processes driven by partial differential equations (PDEs). Potential applications include non-destructive inspection for material crack detection using thermal imaging as well as real-time anomaly detection for process monitoring of three-dimensional printing applications. A neural network (NN) architecture is established that offers sufficient flexibility for spatial and temporal derivatives to capture the physical dependencies inherent in the process. Predictive capabilities are then established by propagating the process forward in time using the acquired model structure as well as individual parameter values. Moreover, deviations in the predicted values can be monitored in real time to detect potential process anomalies or perturbations. For concept development and validation, this article utilizes well-understood PDEs such as the homogeneous heat diffusion equation. Time series data governed by the heat equation representing a parabolic PDE is generated using high-fidelity simulations in order to construct the heat profile. Model structure and parameter identification are realized through a shallow residual convolutional NN. The learned model structure and associated parameters resemble a spatial convolution filter, which can be applied to the current heat profile to predict the diffusion behavior forward in time.
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Engl, H. W., O. Scherzer, and M. Yamamoto. "Uniqueness and stable determination of forcing terms in linear partial differential equations with overspecified boundary data." Inverse Problems 10, no. 6 (December 1, 1994): 1253–76. http://dx.doi.org/10.1088/0266-5611/10/6/006.
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Gillespie, Mark, Nicholas Sharp, and Keenan Crane. "Integer coordinates for intrinsic geometry processing." ACM Transactions on Graphics 40, no. 6 (December 2021): 1–13. http://dx.doi.org/10.1145/3478513.3480522.
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This paper describes a numerically robust data structure for encoding intrinsic triangulations of polyhedral surfaces. Many applications demand a correspondence between the intrinsic triangulation and the input surface, but existing data structures either rely on floating point values to encode correspondence, or do not support remeshing operations beyond basic edge flips. We instead provide an integer-based data structure that guarantees valid correspondence, even for meshes with near-degenerate elements. Our starting point is the framework of normal coordinates from geometric topology, which we extend to the broader set of operations needed for mesh processing (vertex insertion, edge splits, etc. ). The resulting data structure can be used as a drop-in replacement for earlier schemes, automatically improving reliability across a wide variety of applications. As a stress test, we successfully compute an intrinsic Delaunay refinement and associated subdivision for all manifold meshes in the Thingi10k dataset. In turn, we can compute reliable and highly accurate solutions to partial differential equations even on extremely low-quality meshes.
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Kumar, Dr Amresh, and Dr Ram Kishore Singh. "A Role of Hilbert Space in Sampled Data to Reduced Error Accumulation by Over Sampling Then the Computational and Storage Cost Increase Using Signal Processing On 2-Sphere Dimension”." International Journal of Scientific Research and Management 8, no. 05 (May 15, 2020): 386–96. http://dx.doi.org/10.18535/ijsrm/v8i05.ec02.
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Hilbert Space has wide usefulness in signal processing research. It is pitched at a graduate student level, but relies only on undergraduate background material. The needs and concerns of the researchers In engineering differ from those of the pure science. It is difficult to put the finger on what distinguishes the engineering approach that we have taken. In the end, if a potential use emerges from any result, however abstract, then an engineer would tend to attach greater value to that result. This may serve to distinguish the emphasis given by a mathematician who may be interested in the proof of a fundamental concept that links deeply with other areas of mathematics or is a part of a long-standing human intellectual endeavor not that engineering, in comparison, concerns less intellectual pursuits. The theory of Hilbert spaces was initiated by David Hilbert (1862-1943), in the early of twentieth century in the context of the study of "Integral equations". Integral equations are a natural complement to differential equations and arise, for example, in the study of existence and uniqueness of function which are solution of partial differential equations such as wave equation. Convolution and Fourier transform equation also belongs to this class. Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in Hilbert space. At a deeper level, perpendicular projection onto a subspace that is the analog of "dropping the altitude" of a triangle plays a significant role in optimization problem and other aspects of the theory. An element of Hilbert space can be uniquely specified by its co-ordinates with respect to a set of coordinate axes that is an orthonormal basis, in analogy with Cartesian coordinates in the plane. When that set of axes is countably infinite, this means that the Hilbert space can also usefully be thought in terms of infinite sequences that are square summable. Linear operators on Hilbert space are ply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their spectral theory. In brief Hilbert spaces are the means by which the ordinary experience of Euclidean concepts can be extended meaningfully into idealized constructions of more complex abstract mathematics. However, in brief, the usual application demand for Hilbert spaces are integral and differential equations, generalized functions and partial differential equations, quantum mechanics, orthogonal polynomials and functions, optimization and approximation theory. In signal processing which is the main objective of the present thesis and engineering. Wavelets and optimization problem that has been dealt in the present thesis, optimal control, filtering and equalization, signal processing on 2- sphere, Shannon information theory, communication theory, linear and non-linear theory and many more is application domain of the Hilbert space.
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Rvachov, Volodimir Olexijovych, Tatiana Volodimirivna Rvachova, and Evgenia Pavlovna Tomilova. "TOMIC FUNCTIONS AND LACUNARY INTERPOLATION SERIES IN BOUNDARY VALUE PROBLEMS FOR PARTIAL DERIVATIVES EQUATIONS AND IMAGE PROCESSING." RADIOELECTRONIC AND COMPUTER SYSTEMS, no. 1 (January 28, 2020): 58–69. http://dx.doi.org/10.32620/reks.2020.1.06.
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In the paper we consider and solve the problem of construction of the so called tomic functions – the systems of infinitely differentiable functions which while retaining many important properties of the shifts of atomic function up(x) such as locality and representation of algebraic polynomials and being based on the atomic functions nevertheless have nonuniform character and therefore allow to take into account the inhomogeneous and changing character of the data encountered in real world problems in particular in boundary value problems for partial differential equations with variable coefficients and complex geometry of domains in which these boundary value problems must be solved. The same class of tomic functions can be applied to processing,denoising and sparse storage of signals and images by lacunary interpolation. The lacunary or Birkhoff interpolation of functions in which the function is being restored by the values of derivatives of orderin points in which values of function and derivatives of order k<r are unknown is of great importance in many real world problems such as remote sensing. The lacunary interpolation methods using the tomic functions possesss important advantages over currently widely applied lacunary spline interpolation in view of infinite smoothness of tomic functions.The tomic functions can also be applied to connect (to stitch) atomic expansions with different steps on different intervals preserving smoothness and optimal approximation properties. The equations for of construction oftomic functions tofuj(x) –analogues of the basic functions of the generalized atomic Taylor expansions are obtained – which are needed for lacunary (Birkhoff) interpolation. For the applications in variational and collocation methods for solving bondary value problems for partial derivative and integral equations the tomic functions ftupr,j(x) are obtained that are analogues of B-splines and atomic functions fupn(x). Using similar methods, the tomic functions based on other atomic functions such as Ξn(x) can be obtained.
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Dyvak, M., V. Manzhula, A. Melnyk, and V. Tymchyshyn. "A System for Monitoring Air Pollution by Motor Vehicles Based on an Autonomous Air-Mobile Measuring Complex." Optoelectronic Information-Power Technologies 42, no. 2 (October 26, 2022): 73–83. http://dx.doi.org/10.31649/1681-7893-2021-42-2-73-83.
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The article proposes an approach to constructing a system of complex and uninterrupted monitoring of harmful emissions of motor vehicles into the air. The architecture of the environmental monitoring system for measuring and forecasting the distribution of pollutant concentrations in motor vehicle exhaust gases, among which mainly CO, SO₂, NO₂, and СО₂, is presented. The mobile information and measurement complex Sniffer4D Hyper-local Air Quality Analyzer and a charging station based on solar batteries are used as the hardware. For modeling and forecasting the distribution of concentrations of harmful emissions, mathematical models of the dynamics of the distribution of concentrations of pollutants due to harmful emissions in the exhaust gases of motor vehicles are proposed in the form of differential equations that are analogs of differential equations in partial derivatives, as models of turbulent diffusion and interval models of the distribution of the background level of pollution concentration in the form of nonlinear algebraic equations. Implemented software for data collection, processing (model learning and prediction), and visualization.
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Dyvak, M., V. Manzhula, A. Melnyk, and V. Tymchyshyn. "A System for Monitoring Air Pollution by Motor Vehicles Based on an Autonomous Air-Mobile Measuring Complex." Optoelectronic Information-Power Technologies 42, no. 2 (October 26, 2022): 73–83. http://dx.doi.org/10.31649/1681-7893-2021-41-1-73-83.
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The article proposes an approach to constructing a system of complex and uninterrupted monitoring of harmful emissions of motor vehicles into the air. The architecture of the environmental monitoring system for measuring and forecasting the distribution of pollutant concentrations in motor vehicle exhaust gases, among which mainly CO, SO₂, NO₂, and СО₂, is presented. The mobile information and measurement complex Sniffer4D Hyper-local Air Quality Analyzer and a charging station based on solar batteries are used as the hardware. For modeling and forecasting the distribution of concentrations of harmful emissions, mathematical models of the dynamics of the distribution of concentrations of pollutants due to harmful emissions in the exhaust gases of motor vehicles are proposed in the form of differential equations that are analogs of differential equations in partial derivatives, as models of turbulent diffusion and interval models of the distribution of the background level of pollution concentration in the form of nonlinear algebraic equations. Implemented software for data collection, processing (model learning and prediction), and visualization.
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Zuo, Boxin, Xiangyun Hu, Marcelo Leão-Santos, Yi Cai, Mason Andy Kass, Lizhe Wang, and Shuang Liu. "Low-latitude reduction-to-the-pole and upward continuation between arbitrary surfaces based on the partial differential equation framework." Geophysical Journal International 226, no. 2 (March 11, 2021): 968–83. http://dx.doi.org/10.1093/gji/ggab067.
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SUMMARY Magnetic surveys conducted in complex conditions, such as low magnetic latitudes, uneven observation surfaces, or above high-susceptibility sources, pose significant challenges for obtaining stable solutions for reduction-to-the-pole (RTP) and upward-continuation processing on arbitrary surfaces. To tackle these challenges, in this study, we propose constructing an equivalent-susceptibility model based on the partial differential equation (PDE) framework in the space domain. A multilayer equivalent-susceptibility method was used for RTP and upward-continuation operations, thus allowing for application on undulating observation surfaces and strong self-demagnetization effect in a non-uniform mesh. A novel positivity constraint is introduced to improve the accuracy and efficiency of the inversion. We analysed the effect of the depth-weighting function in the inversion of equivalent susceptibility for RTP and upward-continuation reproduction. Iterative and direct solvers were utilized and compared in solving the large, sparse, non-symmetric and ill-conditioned system of linear equations produced by PDE-based equivalent-source construction. Two synthetic models were used to illustrate the efficiency and accuracy of the proposed method in processing both ground and airborne magnetic data. Aeromagnetic and ground data collected in Brazil at a low magnetic latitude region were used to validate the proposed method for processing RTP and upward-continuation operations on magnetic data sets with strong self-demagnetization.
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Bhowmick, Sutanu, Satish Nagarajaiah, and Anastasios Kyrillidis. "Data- and theory-guided learning of partial differential equations using SimultaNeous basis function Approximation and Parameter Estimation (SNAPE)." Mechanical Systems and Signal Processing 189 (April 2023): 110059. http://dx.doi.org/10.1016/j.ymssp.2022.110059.
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Li, Chen. "A Partial Differential Equation-Based Image Restoration Method in Environmental Art Design." Advances in Mathematical Physics 2021 (October 28, 2021): 1–11. http://dx.doi.org/10.1155/2021/4040497.
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With the rapid development of networks and the emergence of various devices, images have become the main form of information transmission in real life. Image restoration, as an important branch of image processing, can be applied to real-life situations such as pixel loss in image transmission or network prone to packet loss. However, existing image restoration algorithms have disadvantages such as fuzzy restoration effect and slow speed; to solve such problems, this paper adopts a dual discriminator model based on generative adversarial networks, which effectively improves the restoration accuracy by adding local discriminators to track the information of local missing regions of images. However, the model is not optimistic in generating reasonable semantic information, and for this reason, a partial differential equation-based image restoration model is proposed. A classifier and a feature extraction network are added to the dual discriminator model to provide category, style, and content loss constraints to the generative network, respectively. To address the training instability problem of discriminator design, spectral normalization is introduced to the discriminator design. Extensive experiments are conducted on a data dataset of partial differential equations, and the results show that the partial differential equation-based image restoration model provides significant improvements in image restoration over previous methods and that image restoration techniques are exceptionally important in the application of environmental art design.
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Gavrilyuk, Ivan, and Boris N. Khoromskij. "Tensor Numerical Methods: Actual Theory and Recent Applications." Computational Methods in Applied Mathematics 19, no. 1 (January 1, 2019): 1–4. http://dx.doi.org/10.1515/cmam-2018-0014.
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AbstractMost important computational problems nowadays are those related to processing of the large data sets and to numerical solution of the high-dimensional integral-differential equations. These problems arise in numerical modeling in quantum chemistry, material science, and multiparticle dynamics, as well as in machine learning, computer simulation of stochastic processes and many other applications related to big data analysis. Modern tensor numerical methods enable solution of the multidimensional partial differential equations (PDE) in {\mathbb{R}^{d}} by reducing them to one-dimensional calculations. Thus, they allow to avoid the so-called “curse of dimensionality”, i.e. exponential growth of the computational complexity in the dimension size d, in the course of numerical solution of high-dimensional problems. At present, both tensor numerical methods and multilinear algebra of big data continue to expand actively to further theoretical and applied research topics. This issue of CMAM is devoted to the recent developments in the theory of tensor numerical methods and their applications in scientific computing and data analysis. Current activities in this emerging field on the effective numerical modeling of temporal and stationary multidimensional PDEs and beyond are presented in the following ten articles, and some future trends are highlighted therein.
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Hu, Jiangtao, Jianliang Qian, Jian Song, Min Ouyang, Junxing Cao, and Shingyu Leung. "Eulerian partial-differential-equation methods for complex-valued eikonals in attenuating media." GEOPHYSICS 86, no. 4 (June 1, 2021): T179—T192. http://dx.doi.org/10.1190/geo2020-0659.1.
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Seismic waves in earth media usually undergo attenuation, causing energy losses and phase distortions. In the regime of high-frequency asymptotics, a complex-valued eikonal is an essential ingredient for describing wave propagation in attenuating media, where the real and imaginary parts of the eikonal function capture dispersion effects and amplitude attenuation of seismic waves, respectively. Conventionally, such a complex-valued eikonal is mainly computed either by tracing rays exactly in complex space or by tracing rays approximately in real space so that the resulting eikonal is distributed irregularly in real space. However, seismic data processing methods, such as prestack depth migration and tomography, usually require uniformly distributed complex-valued eikonals. Therefore, we have developed a unified framework to Eulerianize several popular approximate real-space ray-tracing methods for complex-valued eikonals so that the real and imaginary parts of the eikonal function satisfy the classic real-space eikonal equation and a novel real-space advection equation, respectively, and we dub the resulting method the Eulerian partial-differential-equation method. We further develop highly efficient high-order methods to solve these two equations by using the factorization idea and the Lax-Friedrichs weighted essentially nonoscillatory schemes. Numerical examples demonstrate that our method yields highly accurate complex-valued eikonals, analogous to those from ray-tracing methods. Our methods can be useful for migration and tomography in attenuating media.
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Chitturi, Venkatratnam, and Nagi Farrukh. "Narrowband array processing beamforming technique for electrical impedance tomography." Journal of Electrical Bioimpedance 10, no. 1 (December 31, 2019): 96–102. http://dx.doi.org/10.2478/joeb-2019-0014.
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Abstract Electrical impedance tomography (EIT) has a large potential as a two dimensional imaging technique and is gaining attention among researchers across various fields of engineering. Beamforming techniques stem from the array signal processing field and is used for spatial filtering of array data to evaluate the location of objects. In this work the circular electrodes are treated as an array of sensors and beamforming technique is used to localize the object(s) in an electrical field. The conductivity distributions within a test tank is obtained by an EIT system in terms of electrode voltages. These voltages are then interpolated using elliptic partial differential equations. Finally, a narrowband beamformer detects the peak in the output response signal to localize the test object(s). Test results show that the beamforming technique can be used as a secondary method that may provide complementary information about accurate position of the test object(s) using an eight electrode EIT system. This method could possibly open new avenues for spatial EIT data filtering techniques with an understanding that the inverse problem is more likely considered here as a source localization algorithm instead as an image reconstruction algorithm.
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Gandham, Rajesh, David Medina, and Timothy Warburton. "GPU Accelerated Discontinuous Galerkin Methods for Shallow Water Equations." Communications in Computational Physics 18, no. 1 (July 2015): 37–64. http://dx.doi.org/10.4208/cicp.070114.271114a.
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AbstractWe discuss the development, verification, and performance of a GPU accelerated discontinuous Galerkin method for the solutions of two dimensional nonlinear shallow water equations. The shallow water equations are hyperbolic partial differential equations and are widely used in the simulation of tsunami wave propagations. Our algorithms are tailored to take advantage of the single instruction multiple data (SIMD) architecture of graphic processing units. The time integration is accelerated by local time stepping based on a multi-rate Adams-Bashforthscheme. A total variational bounded limiter is adopted for nonlinear stability of the numerical scheme. This limiter is coupled with a mass and momentum conserving positivity preserving limiter for the special treatment of a dry or partially wet element in the triangulation. Accuracy, robustness and performance are demonstrated with the aid of test cases. Furthermore, we developed a unified multi-threading model OCCA. The kernels expressed in OCCA model can be cross-compiled with multi-threading models OpenCL, CUDA, and OpenMP. We compare the performance of the OCCA kernels when cross-compiled with these models.
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Wang, X. Joey, John R. J. Thompson, W. John Braun, and Douglas G. Woolford. "Fitting a stochastic fire spread model to data." Advances in Statistical Climatology, Meteorology and Oceanography 5, no. 1 (April 16, 2019): 57–66. http://dx.doi.org/10.5194/ascmo-5-57-2019.
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Abstract. As the climate changes, it is important to understand the effects on the environment. Changes in wildland fire risk are an important example. A stochastic lattice-based wildland fire spread model was proposed by Boychuk et al. (2007), followed by a more realistic variant (Braun and Woolford, 2013). Fitting such a model to data from remotely sensed images could be used to provide accurate fire spread risk maps, but an intermediate step on the path to that goal is to verify the model on data collected under experimentally controlled conditions. This paper presents the analysis of data from small-scale experimental fires that were digitally video-recorded. Data extraction and processing methods and issues are discussed, along with an estimation methodology that uses differential equations for the moments of certain statistics that can be derived from a sequential set of photographs from a fire. The interaction between model variability and raster resolution is discussed and an argument for partial validation of the model is provided. Visual diagnostics show that the model is doing well at capturing the distribution of key statistics recorded during observed fires.
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Xu, Qingzhen. "A Novel Machine Learning Strategy Based on Two-Dimensional Numerical Models in Financial Engineering." Mathematical Problems in Engineering 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/659809.
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Machine learning is the most commonly used technique to address larger and more complex tasks by analyzing the most relevant information already present in databases. In order to better predict the future trend of the index, this paper proposes a two-dimensional numerical model for machine learning to simulate major U.S. stock market index and uses a nonlinear implicit finite-difference method to find numerical solutions of the two-dimensional simulation model. The proposed machine learning method uses partial differential equations to predict the stock market and can be extensively used to accelerate large-scale data processing on the history database. The experimental results show that the proposed algorithm reduces the prediction error and improves forecasting precision.
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Sun, Yuhui, Peiru Wu, G. W. Wei, and Ge Wang. "Evolution-Operator-Based Single-Step Method for Image Processing." International Journal of Biomedical Imaging 2006 (2006): 1–27. http://dx.doi.org/10.1155/ijbi/2006/83847.
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This work proposes an evolution-operator-based single-time-step method for image and signal processing. The key component of the proposed method is a local spectral evolution kernel (LSEK) that analytically integrates a class of evolution partial differential equations (PDEs). From the point of view PDEs, the LSEK provides the analytical solution in a single time step, and is of spectral accuracy, free of instability constraint. From the point of image/signal processing, the LSEK gives rise to a family of lowpass filters. These filters contain controllable time delay and amplitude scaling. The new evolution operator-based method is constructed by pointwise adaptation of anisotropy to the coefficients of the LSEK. The Perona-Malik-type of anisotropic diffusion schemes is incorporated in the LSEK for image denoising. A forward-backward diffusion process is adopted to the LSEK for image deblurring or sharpening. A coupled PDE system is modified for image edge detection. The resulting image edge is utilized for image enhancement. Extensive computer experiments are carried out to demonstrate the performance of the proposed method. The major advantages of the proposed method are its single-step solution and readiness for multidimensional data analysis.
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Zaręba, Mateusz, and Tomasz Danek. "Nonlinear anisotropic diffusion techniques for seismic signal enhancing - Carpathian Foredeep study." E3S Web of Conferences 66 (2018): 01016. http://dx.doi.org/10.1051/e3sconf/20186601016.
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The use of nonlinear anisotropic diffusion algorithm for advanced seismic signal processing in the complicated geological region of Carpathian Foredeep was examined. This technique allows for an improvement of seismic data quality and for more accurate interpretation by the recovery of a significant amount of structural information in the form of a correlating seismic reflections and by preserving true DHI indicators. It also allows searching for more subtle geological structures. Anisotropic diffusion is an iterative image processing algorithm that removes noise by modifying the data by solving partial differential equations. Moreover, it can reduce image noise without blurring the edges between regions of different chrominance or brightness. This filter preserves edges, lines, or other features relevant to the seismic structural and stratigraphic interpretation. The algorithm also enables noise reduction without removing significant information from a seismic section even for high dips values. For a better estimation of anisotropic diffusion structure tensor, the parameterization is done using the depth field and the calculations in the two-way travel time field. The presented research shows the results of using an anisotropic diffusion algorithm for post-stack and migration processing of seismic 3D data collected in Carpathian reservoir rocks of southern Poland.
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Alexandre dit Sandretto, Julien, Olivier Mullier, and Alexandre Chapoutot. "Preface." Acta Cybernetica 25, no. 1 (May 31, 2021): 3. http://dx.doi.org/10.14232/actacyb.293250.
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The Summer Workshop on Interval Methods (SWIM) is an annual meeting initiated in 2008 by the French MEA working group on Set Computation and Interval Techniques of the French research group on Automatic Control. A special focus of the MEA group is on promoting interval analysis techniques and applications to a broader community of researchers, facilitated by such multidisciplinary workshops. Since 2008, SWIM has become a keystone event for researchers dealing with various aspects of interval and set-based methods.
In 2019, the 12th edition in this workshop series was held at ENSTA Paris, France, with a total of 25 talks.
Traditionally, workshops in the series of SWIM provide a platform for both theoretical and applied researchers who work on the development, implementation, and application of interval methods, verified numerics, and other related (set-membership) techniques.For this edition, given talks were in the fields of
the verified solution of initial value problems for ordinary differential equations, differential-algebraic system models, and partial differential equations,
scientific computing with guaranteed error bounds,
the design of robust and fault-tolerant control systems,
the implementation of corresponding software libraries, and
the usage of the mentioned approaches for a large variety of system models in areas such as control engineering, data analysis, signal and image processing.
Seven papers were selected for submission to this Acta Cybernetica special issue. After a two turn peer-review process, six high-quality articles were selected for publication in this special issue. Three papers propose a contribution regarding differential equations, two papers focus on robust control, and one paper considers fault detection.
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Auer, Ekaterina, Julia Kersten, and Andreas Rauh. "Preface." Acta Cybernetica 24, no. 3 (March 16, 2020): 265–66. http://dx.doi.org/10.14232/actacyb.24.3.2020.1.
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The Summer Workshop on Interval Methods (SWIM) is an annual meeting initiated in 2008 by the French MEA working group on Set Computation and Interval Techniques of the French research group on Automatic Control. A special focus of the MEA group is on promoting interval analysis techniques and applications to a broader community of researchers, facilitated by such multidisciplinary workshops. Since 2008, SWIM has become a keystone event for researchers dealing with various aspects of interval and set-based methods.
In 2018, the 11th edition in this workshop series was held at the University of Rostock, Germany, with a focus on research topics in the fields of engineering, computer science, and mathematics. A total of 31 talks were given during this workshop, covering the following areas:
verified solution of initial value problems for ordinary differential equations, differential-algebraic system models, and partial differential equations,
scientific computing with guaranteed error bounds,
design of robust and fault-tolerant control systems,
modeling and quantification of errors in engineering tasks,
implementation of software libraries, and
usage of the aforementioned approaches for system models in control engineering, data analysis, signal and image processing.
After a peer-review process, 15 high-quality articles were selected for publication in this special issue. They are roughly divided into two thematic groups: Uncertainty Modeling, Software, Verified Computing and Optimization as well as Interval Methods in Control and Robotics.
The first part, Uncertainty Modeling, Software, Verified Computing and Optimization, contains methodological aspects concerning reliable modeling of dynamic systems as well as visualization and quantification of uncertainty in the fields of measurement and simulation. Moreover, existence proofs for solutions of partial differential equations and their reliable optimal control synthesis are considered. A paper making use of quantifier elimination for robust linear output feedback control by means of eigenvalue placement concludes this section.
The second part of this special issue, Interval Methods in Control and Robotics, is focused on the design as well as numerical and experimental validation of robust state observation and control procedures along with reliable parameter and state estimation approaches in the fields of control for thermal systems, robotics, localization of drones and global positioning systems.
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Ismailov, Bakhtiyar, Zhanat Umarova, Khairulla Ismailov, Aibarsha Dosmakanbetova, and Saule Meldebekova. "Mathematical modeling and algorithm for calculation of thermocatalytic process of producing nanomaterial." Indonesian Journal of Electrical Engineering and Computer Science 23, no. 3 (September 1, 2021): 1590. http://dx.doi.org/10.11591/ijeecs.v23.i3.pp1590-1601.
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<p>At present, when constructing a mathematical description of the pyrolysis reactor, partial differential equations for the components of the gas phase and the catalyst phase are used. In the well-known works on modeling pyrolysis, the obtained models are applicable only for a narrow range of changes in the process parameters, the geometric dimensions are considered constant. The article poses the task of creating a complex mathematical model with additional terms, taking into account nonlinear effects, where the geometric dimensions of the apparatus and operating characteristics vary over a wide range. An analytical method has been developed for the implementation of a mathematical model of catalytic pyrolysis of methane for the production of nanomaterials in a continuous mode. The differential equation for gaseous components with initial and boundary conditions of the third type is reduced to a dimensionless form with a small value of the peclet criterion with a form factor. It is shown that the laplace transform method is mainly suitable for this case, which is applicable both for differential equations for solid-phase components and calculation in a periodic mode. The adequacy of the model results with the known experimental data is checked.</p>
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Peng, Jianping, Peiwen Guo, Meiwen Guo, and Guoying Zhang. "IT Application Maturity in China." Journal of Global Information Management 28, no. 3 (July 2020): 99–122. http://dx.doi.org/10.4018/jgim.2020070106.
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In order to investigate the relationship between IT application maturity and management capabilities, the authors conducted a survey study to collect related company information for analysis. Data processing was conducted to obtain valid and reliable variables representing IT application maturity, management institutional capability, and process management capability. Then, they adopted a partial differential equation approach to capture the time dynamics of these variables. The equations were solved analytically, and further empirically estimated through our processed survey data. The validated model demonstrates that both management capabilities have direct enhancement effects on IT application maturity. In addition, process management capability has a greater influence on IT application maturity in comparison with management institutional capability. Furthermore, it is found that there exist local maximums for both enhancement effects, provided that the two management capabilities are well balanced. The findings not only offer practical implications, but also supplement the literature of factors for IS success in light of the dynamic relationship between IT application maturity and management capabilities.
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Yaparov, D. D., and A. L. Shestakov. "Numerical Method for Processing the Results of Dynamic Measurements." Bulletin of the South Ural State University. Ser. Computer Technologies, Automatic Control & Radioelectronics 21, no. 4 (November 2021): 115–25. http://dx.doi.org/10.14529/ctcr210410.
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The problem of processing data obtained during dynamic measurements is one of the central problems in measuring technology. Purpose of the study. The article is devoted to the study of the stability of the method for solving the problem of processing the results of dynamic measurements with respect to the error in the initial data. Therefore, an urgent task is the development of algorithms for processing the results of dynamic measurements. Materials and methods. This article proposes an algorithm for processing the data obtained during dynamic measurements based on the finite-difference approach. The main prerequisites of the mathematical model of the problem of dynamic measurements associated with the processes of restoration of the input signal in conditions of incomplete and noisy initial data are as follows. Initially, the function of the noisy output signal is known. The restoration of the input signal is carried out using the transfer function of the sensor. The transfer function of the sensor is presented in the form of a differential equation. This equation describes the state of a dynamic system in real time. The proposed computational scheme of the method is based on finite-difference analogs of partial derivatives and the Tikhonov regularization method was used to construct a numerical model of the sensor. The problem of stability of the method for solving high-order differential equations is also one of the central problems of data processing in automatic control systems. Based on the approach of the generalized quasi-optimal choice of the regularization parameter in the Lavrent'ev method, the dependence of the regularization parameter, the parameters of the dynamic measuring system, the noise index and the required level of accuracy was found. Results. The main goal of the computational experiment was to construct a numerical solution to the problem under consideration. Standard test functions were considered as input signals. Test signals simulating various physical processes were used as an input signal. The function of the output signal was found using the proposed numerical method, the found function was noisy with an additive noise of 5 %. Conclusion. The input signal was restored from the noisy signal. The deviation of the reconstructed signal from the initial one in all experiments was no more than 0.05, which indicates the stability of this method with respect to noisy data.
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Leung, Shingyu, Jiangtao Hu, and Jianliang Qian. "Liouville partial-differential-equation methods for computing 2D complex multivalued eikonals in attenuating media." GEOPHYSICS 87, no. 2 (December 30, 2021): T71—T84. http://dx.doi.org/10.1190/geo2021-0113.1.
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We have developed a Liouville partial-differential-equation (PDE)-based method for computing complex-valued eikonals in real phase space in the multivalued sense in attenuating media with frequency-independent qualify factors, where the new method computes the real and imaginary parts of the complex-valued eikonal in two steps by solving Liouville equations in real phase space. Because the earth is composed of attenuating materials, seismic waves usually attenuate so that seismic data processing calls for properly treating the resulting energy losses and phase distortions of wave propagation. In the regime of high-frequency asymptotics, the complex-valued eikonal is one essential ingredient for describing wave propagation in attenuating media because this unique quantity summarizes two wave properties into one function: Its real part describes the wave kinematics and its imaginary part captures the effects of phase dispersion and amplitude attenuation. Because some popular ordinary-differential-equation (ODE)-based ray-tracing methods for computing complex-valued eikonals in real space distribute the eikonal function irregularly in real space, we are motivated to develop PDE-based Eulerian methods for computing such complex-valued eikonals in real space on regular meshes. Therefore, we solved novel paraxial Liouville PDEs in real phase space so that we can compute the real and imaginary parts of the complex-valued eikonal in the multivalued sense on regular meshes. We call the resulting method the Liouville PDE method for complex-valued multivalued eikonals in attenuating media; moreover, this new method provides a unified framework for Eulerianizing several popular approximate real-space ray-tracing methods for complex-valued eikonals, such as viscoacoustic ray tracing, real viscoelastic ray tracing, and real elastic ray tracing. In addition, we also provide Liouville PDE formulations for computing multivalued ray amplitudes in a weakly viscoacoustic medium. Numerical examples, including a synthetic gas-cloud model, demonstrate that our methods yield highly accurate complex-valued eikonals in the multivalued sense.
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Chi, Jieru, Lei Guo, Aurelien Destruel, Yaohui Wang, Chunyi Liu, Mingyan Li, Ewald Weber, et al. "Magnetic Resonance-Electrical Properties Tomography by Directly Solving Maxwell’s Curl Equations." Applied Sciences 10, no. 9 (May 10, 2020): 3318. http://dx.doi.org/10.3390/app10093318.
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Magnetic Resonance-Electrical Properties Tomography (MR-EPT) is a method to reconstruct the electrical properties (EPs) of bio-tissues from the measured radiofrequency (RF) field in Magnetic Resonance Imaging (MRI). Current MR-EPT approaches reconstruct the EP profile by solving a second-order partial differential wave equation problem, which is sensitive to noise and can induce large reconstruction artefacts near tissue boundaries and areas with inaccurate field measurements. In this paper, a novel MR-EPT approach is proposed, which is based on a direct solution to Maxwell’s curl equations. The distribution of EPs is calculated by iteratively fitting the RF field calculated by the finite-difference-time-domain (FDTD) technique to the measured values. To solve the time-consuming problem of the iterative fitting, a graphics processing unit (GPU) is used to accelerate the FDTD technique to process the field calculation kernel. The new EPT method was evaluated by investigating a numerical head phantom, and it was found that EP values can be accurately calculated and were relatively insensitive to simulated RF field errors. The feasibility of the proposed method was further validated using phantom-based experimental data collected from a 9.4 Tesla (T) Magnetic Resonance Imaging (MRI) system. The results also indicated that more accurate EP values could be reconstructed from the new method compared with conventional methods. Moreover, even in the absence of phase information of the RF field, the proposed approach is still capable of offering robust EPT solutions, thus having improved practicality for potential clinical applications.
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Castro, C. E., J. Behrens, and C. Pelties. "CUDA-C implementation of the ADER-DG method for linear hyperbolic PDEs." Geoscientific Model Development Discussions 6, no. 3 (July 13, 2013): 3743–86. http://dx.doi.org/10.5194/gmdd-6-3743-2013.
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Abstract. We implement the ADER-DG numerical method using the CUDA-C language to run the code in a Graphic Processing Unit (GPU). We focus on solving linear hyperbolic partial differential equations where the method can be expressed as a combination of precomputed matrix multiplications becoming a good candidate to be used on the GPU hardware. Moreover, the method is arbitrarily high-order involving intensive work on local data, a property that is also beneficial for the target hardware. We compare our GPU implementation against CPU versions of the same method observing similar convergence properties up to a threshold where the error remains fixed. This behaviour is in agreement with the CPU version but the threshold is larger that in the CPU case. We also observe a big difference when considering single and double precision where in the first case the threshold error is significantly larger. Finally, we did observe a speed up factor in computational time but this is relative to the specific test or benchmark problem.
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Waidyasooriya, Hasitha Muthumala, Tsukasa Endo, Masanori Hariyama, and Yasuo Ohtera. "OpenCL-Based FPGA Accelerator for 3D FDTD with Periodic and Absorbing Boundary Conditions." International Journal of Reconfigurable Computing 2017 (2017): 1–11. http://dx.doi.org/10.1155/2017/6817674.
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Finite difference time domain (FDTD) method is a very poplar way of numerically solving partial differential equations. FDTD has a low operational intensity so that the performances in CPUs and GPUs are often restricted by the memory bandwidth. Recently, deeply pipelined FPGA accelerators have shown a lot of success by exploiting streaming data flows in FDTD computation. In spite of this success, many FPGA accelerators are not suitable for real-world applications that contain complex boundary conditions. Boundary conditions break the regularity of the data flow, so that the performances are significantly reduced. This paper proposes an FPGA accelerator that computes commonly used absorbing and periodic boundary conditions in many 3D FDTD applications. Accelerator is designed using a “C-like” programming language called OpenCL (open computing language). As a result, the proposed accelerator can be customized easily by changing the software code. According to the experimental results, we achieved over 3.3 times and 1.5 times higher processing speed compared to the CPUs and GPUs, respectively. Moreover, the proposed accelerator is more than 14 times faster compared to the recently proposed FPGA accelerators that are capable of handling complex boundary conditions.
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Sripanich, Yanadet, and Sergey Fomel. "Fast time-to-depth conversion and interval velocity estimation in the case of weak lateral variations." GEOPHYSICS 83, no. 3 (May 1, 2018): S227—S235. http://dx.doi.org/10.1190/geo2017-0338.1.
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Time-domain processing has a long history in seismic imaging and has always been a powerful workhorse that is routinely used. It generally leads to an expeditious construction of the subsurface velocity model in time, which can later be expressed in the Cartesian depth coordinates via a subsequent time-to-depth conversion. The conventional practice of such a conversion is done using Dix inversion, which is exact in the case of laterally homogeneous media. For other media with lateral heterogeneity, the time-to-depth conversion involves solving a more complex system of partial differential equations (PDEs). We have developed an efficient alternative for time-to-depth conversion and interval velocity estimation based on the assumption of weak lateral velocity variations. By considering only first-order perturbative effects from lateral variations, the exact system of PDEs required to accomplish the exact conversion reduces to a simpler system that can be solved efficiently in a layer-stripping (downward-stepping) fashion. Numerical synthetic and field data examples show that our method can achieve reasonable accuracy and is significantly more efficient than previously proposed methods with a speedup by an order of magnitude.
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Van, Sin'tun, Aleksey Kolos, and Andrey Petryaev. "Mathematical Modeling of the Process of Soil Freezing of Railway Subgrade in Cold Climate Con-ditions." Proceedings of Petersburg Transport University 19, no. 4 (December 20, 2022): 820–31. http://dx.doi.org/10.20295/1815-588x-2022-4-820-831.
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Purpose: Mathematical model for studying the process of soil freezing given moisture migration is developed. Numerical modeling of temperature mode of railway subgrade at non-stationary pro-cess in cold climate conditions is performed. Methods: Numerical implementation of the model is performed by mathematical module processing in COMSOL Multiphysics program, which’s based on partial differential equations (PDE), with finite element method. The model reliability is con-firmed by the comparison with previous experimental data and the results of simulation by other authors. Results: Calculation results on the developed model basis show the best correlation with experimental data in comparison with the results for other models. Calculation example and calcu-lation results for subgrade temperature mode in freezing-thawing fifth cycle are presented. The analysis of soil freezing depth change and soil temperature fluctuation change by depth by fifth year are carried out. Practical significance: The developed mathematical model makes it possible to predict soil freezing depth, taking into account moisture migration, including freezing and thawing depth changes caused by climate warming. The developed model can be used both, to study the mechanism of subgrade temperature mode distribution at freezing and thawing, and to improve subgrade construction for to protect it from soil frost heaving.
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Hsu, Gerald C. "Application of Robotic Software on Effective Diabetes Control (GH-Method: Math-Physical Medicine)." Series of Endocrinology, Diabetes and Metabolism 2, no. 2 (July 13, 2020): 27–31. http://dx.doi.org/10.54178/jsedmv2i2002.
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This paper focuses on the author’s invented robotic software technology, the artificial intelligence glucometer (AIG) product, to provide a diagnosis for diabetes disease and glucose control. From 2010–2013, he self-studied internal medicine and food nutrition. In 2014, he further utilized topology concept, partial differential equation, non-linear algebra, and finite element engineering concept to develop a human metabolism’s mathematical model. It consists of 10 categories and ~500 elements with ~1.5 million collected data of his own body health, disease conditions, and lifestyle details. Starting from 2015, he focused on the root cause of diabetes, which is “glucose”. By applying wave theory, signal processing, energy theory, optical physics, structural & fluid dynamics from physics and engineering modeling; pattern and segmentation analysis, time/space/frequency domain analyses, big data analytics, machine learning and self-correction, prediction equations from mathematics and computer science, he decided to utilize his robotic software as the foundation to further build up his needed medical research and clinical tools. By using the artificial intelligence (AI) robotic software, the author’s average glucose decreased from 280 mg/dL to 118 mg/dL and his hemoglobin A1C (HbA1C or A1C) reduced from 10%+ to below 6.5%, without diabetes medications. All his diabetes complications are either under control or have subsided. This innovative technology of his robotic software for glucose prediction and diabetes control has also been proven by many other patients, who have achieved equally remarkable medical results.
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van Gennip, Yves. "An MBO Scheme for Minimizing the Graph Ohta–Kawasaki Functional." Journal of Nonlinear Science 30, no. 5 (June 1, 2018): 2325–73. http://dx.doi.org/10.1007/s00332-018-9468-8.
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Abstract We study a graph-based version of the Ohta–Kawasaki functional, which was originally introduced in a continuum setting to model pattern formation in diblock copolymer melts and has been studied extensively as a paradigmatic example of a variational model for pattern formation. Graph-based problems inspired by partial differential equations (PDEs) and variational methods have been the subject of many recent papers in the mathematical literature, because of their applications in areas such as image processing and data classification. This paper extends the area of PDE inspired graph-based problems to pattern-forming models, while continuing in the tradition of recent papers in the field. We introduce a mass conserving Merriman–Bence–Osher (MBO) scheme for minimizing the graph Ohta–Kawasaki functional with a mass constraint. We present three main results: (1) the Lyapunov functionals associated with this MBO scheme $$\Gamma $$ Γ -converge to the Ohta–Kawasaki functional (which includes the standard graph-based MBO scheme and total variation as a special case); (2) there is a class of graphs on which the Ohta–Kawasaki MBO scheme corresponds to a standard MBO scheme on a transformed graph and for which generalized comparison principles hold; (3) this MBO scheme allows for the numerical computation of (approximate) minimizers of the graph Ohta–Kawasaki functional with a mass constraint.
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Alzahrani, Abdullah K. "Effects of Hall Current and Viscous Dissipation on Bioconvection Transport of Nanofluid Over a Rotating Disk with Motile Microorganisms." Nanomaterials 12, no. 22 (November 16, 2022): 4027. http://dx.doi.org/10.3390/nano12224027.
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The study of rotating-disk heat-flow problems is relevant to computer storage devices, rotating machineries, heat-storage devices, MHD rotators, lubrication, and food-processing devices. Therefore, this study investigated the effects of a Hall current and motile microorganisms on nanofluid flow generated by the spinning of a disk under multiple slip and thermal radiation conditions. The Buongiorno model of a nonhomogeneous nanofluid under Brownian diffusion and thermophoresis was applied. Using the Taylor series, the effect of Resseland radiation was linearized and included in the energy equation. By implementing the appropriate transformations, the governing partial differential equations (PDEs) were simplified into a two-point ordinary boundary value problem. The classical Runge–Kutta dependent shooting method was used to find the numerical solutions, which were validated using the data available in the literature. The velocity, motile microorganism distribution, temperature, and concentration of nanoparticles were plotted and comprehensively analyzed. Moreover, the density number, Sherwood number, shear stresses, and Nusselt number were calculated. The radial and tangential velocity declined with varying values of magnetic numbers, while the concentration of nanoparticles, motile microorganism distribution, and temperature increased. There was a significant reduction in heat transfer, velocities, and motile microorganism distribution under the multiple slip conditions. The Hall current magnified the velocities and reduced the heat transfer. Thermal radiation improved the Nusselt number, while the thermal slip conditions reduced the Nusselt number.
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Li, Zhi, Lei Liu, Jiaqiang Wang, Li Lin, Jichang Dong, and Zhi Dong. "Design and Analysis of an Effective Multi-Barriers Model Based on Non-Stationary Gaussian Random Fields." Electronics 12, no. 2 (January 9, 2023): 345. http://dx.doi.org/10.3390/electronics12020345.
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In this paper, we propose an extension to the barrier model, i.e., the Multi-Barriers Model, which could characterize an area of interest with different types of obstacles. In the proposed model, the area of interest is divided into two or more areas, which include a general area of interest with sampling points and the rest of the area with different types of obstacles. Firstly, the correlation between the points in space is characterized by the obstruction degree of the obstacle. Secondly, multiple Gaussian random fields are constructed. Then, continuous Gaussian fields are expressed by using stochastic partial differential equations (SPDEs). Finally, the integrated nested Laplace approximation (INLA) method is employed to calculate the posterior mean of parameters and the posterior parameters to establish a spatial regression model. In this paper, the Multi-Barriers Model is also verified by using the geostatistical model and log-Gaussian Cox model. Furthermore, the stationary Gaussian model, the barrier model and the Multi-Barriers Model are investigated in the geostatistical data, respectively. Real data sets of burglaries in a certain area are used to compare the performance of the stationary Gaussian model, barrier model and Multi-Barriers Model. The comparison results suggest that the three models achieve similar performance in the posterior mean and posterior distribution of the parameters, as well as the deviance information criteria (DIC) value. However, the Multi-Barriers Model can better interpret the spatial model established based on the spatial data of the research areas with multiple types of obstacles, and it is closer to reality.
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Sarti, Alessandro, Karol Mikula, Fiorella Sgallari, and Claudio Lamberti. "Evolutionary partial differential equations for biomedical image processing." Journal of Biomedical Informatics 35, no. 2 (April 2002): 77–91. http://dx.doi.org/10.1016/s1532-0464(02)00502-6.
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Bähr, Martin, and Michael Breuß. "Efficient Long-Term Simulation of the Heat Equation with Application in Geothermal Energy Storage." Mathematics 10, no. 13 (July 1, 2022): 2309. http://dx.doi.org/10.3390/math10132309.
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Long-term evolutions of parabolic partial differential equations, such as the heat equation, are the subject of interest in many applications. There are several numerical solvers marking the state-of-the-art in diverse scientific fields that may be used with benefit for the numerical simulation of such long-term scenarios. We show how to adapt some of the currently most efficient numerical approaches for solving the fundamental problem of long-term linear heat evolution with internal and external boundary conditions as well as source terms. Such long-term simulations are required for the optimal dimensioning of geothermal energy storages and their profitability assessment, for which we provide a comprehensive analytical and numerical model. Implicit methods are usually considered the best choice for resolving long-term simulations of linear parabolic problems; however, in practice the efficiency of such schemes in terms of the combination of computational load and obtained accuracy may be a delicate issue, as it depends very much on the properties of the underlying model. For example, one of the challenges in long-term simulation may arise by the presence of time-dependent boundary conditions, as in our application. In order to provide both a computationally efficient and accurate enough simulation, we give a thorough discussion of the various numerical solvers along with many technical details and own adaptations. By our investigation, we focus on two largely competitive approaches for our application, namely the fast explicit diffusion method originating in image processing and an adaptation of the Krylov subspace model order reduction method. We validate our numerical findings via several experiments using synthetic and real-world data. We show that we can obtain fast and accurate long-term simulations of typical geothermal energy storage facilities. We conjecture that our techniques can be highly useful for tackling long-term heat evolution in many applications.
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Rudy, Samuel H., Steven L. Brunton, Joshua L. Proctor, and J. Nathan Kutz. "Data-driven discovery of partial differential equations." Science Advances 3, no. 4 (April 2017): e1602614. http://dx.doi.org/10.1126/sciadv.1602614.
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Trogdon, Thomas, and Gino Biondini. "Evolution partial differential equations with discontinuous data." Quarterly of Applied Mathematics 77, no. 4 (November 28, 2018): 689–726. http://dx.doi.org/10.1090/qam/1526.
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Zhao, Weng Cang, and Fan Wang. "Face Image Denoising Method Based on Fourth-Order Partial Differential Equations." Advanced Engineering Forum 6-7 (September 2012): 700–703. http://dx.doi.org/10.4028/www.scientific.net/aef.6-7.700.
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In order to improve the effect of face image denoising, this paper put forward several face image denoising methods based on partial differential equations, including P-M non-linear diffusion equations and fourth-order partial differential equations. We use those two methods by establishing non-linear diffusion equations and fourth-order anisotropic diffusion partial differential equation. The P-M non-linear diffusion denoising method can remove noise in intra-regions sufficiently but noise at edges can not be eliminated successfully and line-like structures can not be held very well.While the fourth-order partial differential equations denoising can retain the local detail characteristics of the original face image. Finally, through the experimental results we can see the effect of the fourth-order partial differential equations denoising is better, which makes the later face image processing more accurate and promotes the development of face image processing.
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Pesenson, Meyer, William Roby, and Bruce McCollum. "Multiscale Astronomical Image Processing Based on Nonlinear Partial Differential Equations." Astrophysical Journal 683, no. 1 (August 10, 2008): 566–76. http://dx.doi.org/10.1086/589276.
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Bar-Sinai, Yohai, Stephan Hoyer, Jason Hickey, and Michael P. Brenner. "Learning data-driven discretizations for partial differential equations." Proceedings of the National Academy of Sciences 116, no. 31 (July 16, 2019): 15344–49. http://dx.doi.org/10.1073/pnas.1814058116.
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The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length- and timescales. Often, it is computationally intractable to resolve the finest features in the solution. The only recourse is to use approximate coarse-grained representations, which aim to accurately represent long-wavelength dynamics while properly accounting for unresolved small-scale physics. Deriving such coarse-grained equations is notoriously difficult and often ad hoc. Here we introduce data-driven discretization, a method for learning optimized approximations to PDEs based on actual solutions to the known underlying equations. Our approach uses neural networks to estimate spatial derivatives, which are optimized end to end to best satisfy the equations on a low-resolution grid. The resulting numerical methods are remarkably accurate, allowing us to integrate in time a collection of nonlinear equations in 1 spatial dimension at resolutions 4× to 8× coarser than is possible with standard finite-difference methods.
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Rudy, Samuel, Alessandro Alla, Steven L. Brunton, and J. Nathan Kutz. "Data-Driven Identification of Parametric Partial Differential Equations." SIAM Journal on Applied Dynamical Systems 18, no. 2 (January 2019): 643–60. http://dx.doi.org/10.1137/18m1191944.
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Jiang, Xin, and Ren Jie Zhang. "Image Restoration Based on Partial Differential Equations (PDEs)." Advanced Materials Research 647 (January 2013): 912–17. http://dx.doi.org/10.4028/www.scientific.net/amr.647.912.
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Image restoration plays an important role in both the quantitative analysis and qualitative analysis of image. It directly affects the further works of analysis and processing. At present, a large number of image restoration methods are recorded in the literatures. And image restoration method based on partial differential equations(PDEs) is one of the main tools in this area. Although these methods often seem powerless for the images with complex features, image restoration method based on PDEs still has its advantages cannot be replaced. In this paper, we make a summary and appraisal on image restoration methods based on PDEs on basis of the analysis for image characteristics and predict the development trend of image restoration methods based on PDEs.
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Wang, Zhi, and Sinan Fang. "Three-Dimensional Inversion of Borehole-Surface Resistivity Method Based on the Unstructured Finite Element." International Journal of Antennas and Propagation 2021 (September 24, 2021): 1–13. http://dx.doi.org/10.1155/2021/5154985.
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The electromagnetic wave signal from the electromagnetic field source generates induction signals after reaching the target geological body through the underground medium. The time and spatial distribution rules of the artificial or the natural electromagnetic fields are obtained for the exploration of mineral resources of the subsurface and determining the geological structure of the subsurface to solve the geological problems. The goal of electromagnetic data processing is to suppress the noise and improve the signal-to-noise ratio and the inversion of resistivity data. Inversion has always been the focus of research in the field of electromagnetic methods. In this paper, the three-dimensional borehole-surface resistivity method is explored based on the principle of geometric sounding, and the three-dimensional inversion algorithm of the borehole-surface resistivity method in arbitrary surface topography is proposed. The forward simulation and calculation start from the partial differential equation and the boundary conditions of the total potential of the three-dimensional point current source field are satisfied. Then the unstructured tetrahedral grids are used to discretely subdivide the calculation area that can well fit the complex structure of subsurface and undulating surface topography. The accuracy of the numerical solution is low due to the rapid attenuation of the electric field at the point current source and the nearby positions and sharply varying potential gradients. Therefore, the mesh density is defined at the local area, that is, the vicinity of the source electrode and the measuring electrode. The mesh refinement can effectively reduce the influence of the source point and its vicinity and improve the accuracy of the numerical solution. The stiffness matrix is stored with Compressed Row Storage (CSR) format, and the final large linear equations are solved using the Super Symmetric Over Relaxation Preconditioned Conjugate Gradient (SSOR-PCG) method. The quasi-Newton method with limited memory (L_BFGS) is used to optimize the objective function in the inversion calculation, and a double-loop recursive method is used to solve the normal equation obtained at each iteration in order to avoid computing and storing the sensitivity matrix explicitly and reduce the amount of calculation. The comprehensive application of the above methods makes the 3D inversion algorithm efficient, accurate, and stable. The three-dimensional inversion test is performed on the synthetic data of multiple theoretical geoelectric models with topography (a single anomaly model under valley and a single anomaly model under mountain) to verify the effectiveness of the proposed algorithm.
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Buet, Blanche, Jean-Marie Mirebeau, Yves van Gennip, François Desquilbet, Johann Dreo, Frédéric Barbaresco, Gian Paolo Leonardi, Simon Masnou, and Carola-Bibiane Schönlieb. "Partial differential equations and variational methods for geometric processing of images." SMAI journal of computational mathematics S5 (2019): 109–28. http://dx.doi.org/10.5802/smai-jcm.55.
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Do Hyun Chung and G. Sapiro. "Segmenting skin lesions with partial-differential-equations-based image processing algorithms." IEEE Transactions on Medical Imaging 19, no. 7 (July 2000): 763–67. http://dx.doi.org/10.1109/42.875204.
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