Journal articles: 'Glimm approximations'

1

Hoff, David, and Joel Smoller. "Error bounds for Glimm difference approximations for scalar conservation laws." Transactions of the American Mathematical Society 289, no. 2 (February 1, 1985): 611. http://dx.doi.org/10.1090/s0002-9947-1985-0784006-5.

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2

Kutsenko, Anton A. "Classification of Integrodifferential C∗-Algebras." Symmetry 13, no. 10 (October 9, 2021): 1900. http://dx.doi.org/10.3390/sym13101900.

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The infinite product of matrices with integer entries, known as a modified Glimm–Bratteli symbol n, is a new, sufficiently simple, and very powerful tool for the characterization of approximately finite-dimensional (AF) algebras. This symbol provides a convenient algebraic representation of the Bratteli diagram for AF algebras in the same way as was previously performed by J. Glimm for more simple uniformly hyperfinite (UHF) algebras. We apply this symbol to characterize integrodifferential algebras. The integrodifferential algebra FN,M is the C∗-algebra generated by the following operators acting on L2([0,1)N→CM): (1) operators of multiplication by bounded matrix-valued functions, (2) finite-difference operators, and (3) integral operators. Most of the operators and their approximations studying in physics belong to these algebras. We give a complete characterization of FN,M. In particular, we show that FN,M does not depend on M, but depends on N. At the same time, it is known that differential algebras HN,M, generated by the operators (1) and (2) only, do not depend on both dimensions N and M; they are all ∗-isomorphic to the universal UHF algebra. We explicitly compute the Glimm–Bratteli symbols (for HN,M, it was already computed earlier) which completely characterize the corresponding AF algebras. This symbol n is an infinite product of matrices with nonnegative integer entries. Roughly speaking, all the symmetries appearing in the approximation of complex infinite-dimensional integrodifferential and differential algebras by finite-dimensional ones are coded by a product of integer matrices.

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TSIKKOU, CHARIS. "SHARPER TOTAL VARIATION BOUNDS FOR THE P-SYSTEM OF FLUID DYNAMICS." Journal of Hyperbolic Differential Equations 08, no. 02 (June 2011): 173–232. http://dx.doi.org/10.1142/s0219891611002391.

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For the p-system of fluid dynamics and under the assumption that the initial data have small oscillation but not necessarily small total variation, we reproduce the so-called Glimm–Lax estimates with greater accuracy. Our approach recovers the results known for smooth flux functions, and is based on the theory of generalized characteristics and Dafermos' technique of polygonal approximations.

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Ye, Xiao Ping, and Long Wei Lin. "Error Bounds for Glimm Difference Approximations for Scalar Conservation Laws Without Convexity." Acta Mathematica Sinica, English Series 22, no. 4 (March 23, 2006): 1271–82. http://dx.doi.org/10.1007/s10114-005-0713-3.

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Bianchini, Stefano, and Stefano Modena. "On a quadratic functional for scalar conservation laws." Journal of Hyperbolic Differential Equations 11, no. 02 (June 2014): 355–435. http://dx.doi.org/10.1142/s0219891614500118.

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We prove a quadratic interaction estimate for approximate solutions to scalar conservation laws obtained by the wavefront tracking approximation or the Glimm scheme. This quadratic estimate has been used in the literature to prove the convergence rate of the Glimm scheme. The proof is based on the introduction of a quadratic functional 𝔔(t), decreasing at every interaction, and such that its total variation in time is bounded. Differently from other interaction potentials present in the literature, the form of this functional is the natural extension of the original Glimm functional, and coincides with it in the genuinely nonlinear case.

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6

Berman, Mark, and T. Rolf Turner. "Approximating Point Process Likelihoods with GLIM." Applied Statistics 41, no. 1 (1992): 31. http://dx.doi.org/10.2307/2347614.

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Berman, M., and T. R. Turner. "Approximating point process likelihoods with GLIM." Insurance: Mathematics and Economics 13, no. 2 (November 1993): 147. http://dx.doi.org/10.1016/0167-6687(93)90845-g.

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8

Chou, Shih-Wei, John M. Hong, Bo-Chih Huang, and Reyna Quita. "Global bounded variation solutions describing Fanno–Rayleigh fluid flows in nozzles." Mathematical Models and Methods in Applied Sciences 28, no. 06 (May 21, 2018): 1135–69. http://dx.doi.org/10.1142/s0218202518500306.

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In this paper, we investigate the initial-boundary value problem of compressible Euler equations including friction and heating that model the transonic Fanno–Rayleigh flows through symmetric variable area nozzles. In particular, the case of contracting nozzles is considered. A new version of a generalized Glimm scheme (GGS) is presented for establishing the global existence of entropy solutions with bounded variation. Modified Riemann and boundary Riemann solutions are applied to design this GGS, which is constructed using the contraction matrices acting on the homogeneous Riemann (or boundary-Riemann) solutions. The extended Glimm–Goodman’s type of wave interaction estimates are investigated to determine the stability of the scheme and the positivity of gas velocity that results in the existence of the weak solution. The limit of approximation solutions serves as an entropy solution. Moreover, a quantitative relation between the shape of the nozzle, friction, and heat is proposed for the global existence result in the contracting nozzle. Numerical simulations of the contraction-expansion and expansion-contraction nozzles are presented to validate the scheme.

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Abgrall, Remi, and Harish Kumar. "Numerical Approximation of a Compressible Multiphase System." Communications in Computational Physics 15, no. 5 (May 2014): 1237–65. http://dx.doi.org/10.4208/cicp.110313.230913a.

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AbstractThe numerical simulation of non conservative system is a difficult challenge for two reasons at least. The first one is that it is not possible to derive jump relations directly from conservation principles, so that in general, if the model description is non ambiguous for smooth solutions, this is no longer the case for discontinuous solutions. From the numerical view point, this leads to the following situation: if a scheme is stable, its limit for mesh convergence will depend on its dissipative structure. This is well known since at least [1]. In this paper we are interested in the “dual” problem: given a system in non conservative form and consistent jump relations, how can we construct a numerical scheme that will, for mesh convergence, provide limit solutions that are the exact solution of the problem. In order to investigate this problem, we consider a multiphase flow model for which jump relations are known. Our scheme is an hybridation of Glimm scheme and Roe scheme.

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Hong, John Meng-Kai, and Reyna Marsya Quita. "Approximation of generalized Riemann solutions to compressible Euler-Poisson equations of isothermal flows in spherically symmetric space-times." Tamkang Journal of Mathematics 48, no. 1 (March 30, 2017): 73–94. http://dx.doi.org/10.5556/j.tkjm.48.2017.2274.

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In this paper, we consider the compressible Euler-Poisson system in spherically symmetric space-times. This system, which describes the conservation of mass and momentum of physical quantity with attracting gravitational potential, can be written as a $3\times 3$ mixed-system of partial differential systems or a $2\times 2$ hyperbolic system of balance laws with $global$ source. We show that, by the equation for the conservation of mass, Euler-Poisson equations can be transformed into a standard $3\times 3$ hyperbolic system of balance laws with $local$ source. The generalized approximate solutions to the Riemann problem of Euler-Poisson equations, which is the building block of generalized Glimm scheme for solving initial-boundary value problems, are provided as the superposition of Lax's type weak solutions of the associated homogeneous conservation laws and the perturbation terms solved by the linearized hyperbolic system with coefficients depending on such Lax solutions.

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Luan, Zemin, Zhaoxia Yu, Ting Zeng, Rui Wang, Maozai Tian, and Kai Wang. "A study on the factors influencing the transfer of COVID-19 severe illness patients out of the ICU based on generalized linear mixed effect model." Mathematical Biosciences and Engineering 19, no. 10 (2022): 10602–17. http://dx.doi.org/10.3934/mbe.2022495.

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<abstract><p>The clinical data of 76 severe illness patients with novel coronavirus SARS-CoV-2 from July to August, 2020 admitted to the ICU Intensive Care Unit ward in a hospital in Urumqi were collected in the paper. By using the Laplace approximation parameter estimation method based on maximum likelihood estimation, the generalized linear mixed effect model (GLMM) was established to analyze the characteristics of clinical indicators in critical patients, and to screen the main influencing factors of COVID-19 critical patients' inability to be transferred out of the ICU in a short time: age, C-reactive protein, serum creatinine and lactate dehydrogenase.</p></abstract>

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12

Abd Naeeim, Nurul Syafiah, and Nuzlinda Abdul Rahman. "Estimating relative risk for dengue disease in Peninsular Malaysia using INLA." Malaysian Journal of Fundamental and Applied Sciences 13, no. 4 (December 26, 2017): 721–27. http://dx.doi.org/10.11113/mjfas.v0n0.575.

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Study in spatio-temporal disease mapping models give a great worth in epidemiology, in describing the pattern of disease incidence across geographical space and time. This paper studies generalized linear mixed models (GLMM) for the analysis of spatial and temporal variability of dengue disease rates. For spatio-temporal study, the models accommodate spatially correlated random effects as well as temporal effects together with the space time interaction. The space time interaction is used to capture any additional effects that are not explained by the main factors of space and time. However, as study including time dimension is quite complex for disease mapping, the temporal effects that only relate to structured and unstructured time pattern are considered in these models as initial screening in studying disease pattern and time trend. The models are fitted within a hierarchical Bayesian framework using Integrated Nested Laplace Approximation (INLA) methodology. For this study, there are three main objectives. First, to choose the best model that represent the disease phenomenon. Second, to estimate the relative risk of disease based on the model selected and lastly, to visualize the risk spatial pattern and temporal trend using graphical representation. The models are applied to monthly dengue fever data in Peninsular Malaysia reported to Ministry of Health Malaysia for year 2015 by district level.

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13

Li, Wentao, Jiayi Tong, Md Monowar Anjum, Noman Mohammed, Yong Chen, and Xiaoqian Jiang. "Federated learning algorithms for generalized mixed-effects model (GLMM) on horizontally partitioned data from distributed sources." BMC Medical Informatics and Decision Making 22, no. 1 (October 16, 2022). http://dx.doi.org/10.1186/s12911-022-02014-1.

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Abstract Objectives This paper developed federated solutions based on two approximation algorithms to achieve federated generalized linear mixed effect models (GLMM). The paper also proposed a solution for numerical errors and singularity issues. And showed the two proposed methods can perform well in revealing the significance of parameter in distributed datasets, comparing to a centralized GLMM algorithm from R package (‘lme4’) as the baseline model. Methods The log-likelihood function of GLMM is approximated by two numerical methods (Laplace approximation and Gaussian Hermite approximation, abbreviated as LA and GH), which supports federated decomposition of GLMM to bring computation to data. To solve the numerical errors and singularity issues, the loss-less estimation of log-sum-exponential trick and the adaptive regularization strategy was used to tackle the problems caused by federated settings. Results Our proposed method can handle GLMM to accommodate hierarchical data with multiple non-independent levels of observations in a federated setting. The experiment results demonstrate comparable (LA) and superior (GH) performances with simulated and real-world data. Conclusion We modified and compared federated GLMMs with different approximations, which can support researchers in analyzing versatile biomedical data to accommodate mixed effects and address non-independence due to hierarchical structures (i.e., institutes, region, country, etc.).

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Abdelrahman, Mahmoud A. E., and Hanan A. Alkhidhr. "A new variation for the relativistic Euler equations." Advances in Difference Equations 2020, no. 1 (September 29, 2020). http://dx.doi.org/10.1186/s13662-020-02990-6.

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Abstract The Glimm scheme is one of the so famous techniques for getting solutions of the general initial value problem by building a convergent sequence of approximate solutions. The approximation scheme is based on the solution of the Riemann problem. In this paper, we use a new strength function in order to present a new kind of total variation of a solution. Based on this new variation, we use the Glimm scheme to prove the global existence of weak solutions for the nonlinear ultra-relativistic Euler equations for a class of large initial data that involve the interaction of nonlinear waves.

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15

Gurinovich, Anastasia, Mengze Li, Anastasia Leshchyk, Harold Bae, Zeyuan Song, Konstantin G. Arbeev, Marianne Nygaard, Mary F. Feitosa, Thomas T. Perls, and Paola Sebastiani. "Evaluation of GENESIS, SAIGE, REGENIE and fastGWA-GLMM for genome-wide association studies of binary traits in correlated data." Frontiers in Genetics 13 (September 23, 2022). http://dx.doi.org/10.3389/fgene.2022.897210.

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Performing a genome-wide association study (GWAS) with a binary phenotype using family data is a challenging task. Using linear mixed effects models is typically unsuitable for binary traits, and numerical approximations of the likelihood function may not work well with rare genetic variants with small counts. Additionally, imbalance in the case-control ratios poses challenges as traditional statistical methods such as the Score test or Wald test perform poorly in this setting. In the last couple of years, several methods have been proposed to better approximate the likelihood function of a mixed effects logistic regression model that uses Saddle Point Approximation (SPA). SPA adjustment has recently been implemented in multiple software, including GENESIS, SAIGE, REGENIE and fastGWA-GLMM: four increasingly popular tools to perform GWAS of binary traits. We compare Score and SPA tests using real family data to evaluate computational efficiency and the agreement of the results. Additionally, we compare various ways to adjust for family relatedness, such as sparse and full genetic relationship matrices (GRM) and polygenic effect estimates. We use the New England Centenarian Study imputed genotype data and the Long Life Family Study whole-genome sequencing data and the binary phenotype of human extreme longevity to compare the agreement of the results and tools’ computational performance. The evaluation suggests that REGENIE might not be a good choice when analyzing correlated data of a small size. fastGWA-GLMM is the most computationally efficient compared to the other three tools, but it appears to be overly conservative when applied to family-based data. GENESIS, SAIGE and fastGWA-GLMM produced similar, although not identical, results, with SPA adjustment performing better than Score tests. Our evaluation also demonstrates the importance of adjusting by full GRM in highly correlated datasets when using GENESIS or SAIGE.

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16

Bologa, Cristian G., Vernon Shane Pankratz, Mark L. Unruh, Maria Eleni Roumelioti, Vallabh Shah, Saeed Kamran Shaffi, Soraya Arzhan, John Cook, and Christos Argyropoulos. "High performance implementation of the hierarchical likelihood for generalized linear mixed models: an application to estimate the potassium reference range in massive electronic health records datasets." BMC Medical Research Methodology 21, no. 1 (July 24, 2021). http://dx.doi.org/10.1186/s12874-021-01318-6.

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Abstract Background Converting electronic health record (EHR) entries to useful clinical inferences requires one to address the poor scalability of existing implementations of Generalized Linear Mixed Models (GLMM) for repeated measures. The major computational bottleneck concerns the numerical evaluation of multivariable integrals, which even for the simplest EHR analyses may involve millions of dimensions (one for each patient). The hierarchical likelihood (h-lik) approach to GLMMs is a methodologically rigorous framework for the estimation of GLMMs that is based on the Laplace Approximation (LA), which replaces integration with numerical optimization, and thus scales very well with dimensionality. Methods We present a high-performance, direct implementation of the h-lik for GLMMs in the R package TMB. Using this approach, we examined the relation of repeated serum potassium measurements and survival in the Cerner Real World Data (CRWD) EHR database. Analyzing this data requires the evaluation of an integral in over 3 million dimensions, putting this problem beyond the reach of conventional approaches. We also assessed the scalability and accuracy of LA in smaller samples of 1 and 10% size of the full dataset that were analyzed via the a) original, interconnected Generalized Linear Models (iGLM), approach to h-lik, b) Adaptive Gaussian Hermite (AGH) and c) the gold standard for multivariate integration Markov Chain Monte Carlo (MCMC). Results Random effects estimates generated by the LA were within 10% of the values obtained by the iGLMs, AGH and MCMC techniques. The H-lik approach was 4–30 times faster than AGH and nearly 800 times faster than MCMC. The major clinical inferences in this problem are the establishment of the non-linear relationship between the potassium level and the risk of mortality, as well as estimates of the individual and health care facility sources of variations for mortality risk in CRWD. Conclusions We found that the direct implementation of the h-lik offers a computationally efficient, numerically accurate approach for the analysis of extremely large, real world repeated measures data via the h-lik approach to GLMMs. The clinical inference from our analysis may guide choices of treatment thresholds for treating potassium disorders in the clinic.

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Journal articles on the topic 'Glimm approximations'

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