Abstract Mathematical models in immunology differ enormously in the dimensionality of the state space, the number of parameters and the parameterizations used to describe the immune processes. The ongoing diversification of the models needs to be complemented by rigorous ways to evaluate their complexity and select the parsimonious ones in relation to the data available/used for their calibration. A broadly applied metrics for ranking the models in mathematical immunology with respect to their complexity/parsimony is provided by the Akaike information criterion. In the present study, a computational framework is elaborated to characterize the complexity of mathematical models in immunology using a more general approach, namely, the Minimum Description Length criterion. It balances the model goodness-of-fit with the dimensionality and geometrical complexity of the model. Four representative models of the immune response to acute viral infection formulated with either ordinary or delay differential equations are studied. Essential numerical details enabling the assessment and ranking of the viral infection models include: (1) the optimization of the likelihood function, (2) the computation of the model sensitivity functions, (3) the evaluation of the Fisher information matrix and (4) the estimation of multidimensional integrals over the model parameter space.
{"title":"Application of minimum description length criterion to assess the complexity of models in mathematical immunology","authors":"D. Grebennikov, V. V. Zheltkova, G. Bocharov","doi":"10.1515/rnam-2022-0022","DOIUrl":"https://doi.org/10.1515/rnam-2022-0022","url":null,"abstract":"Abstract Mathematical models in immunology differ enormously in the dimensionality of the state space, the number of parameters and the parameterizations used to describe the immune processes. The ongoing diversification of the models needs to be complemented by rigorous ways to evaluate their complexity and select the parsimonious ones in relation to the data available/used for their calibration. A broadly applied metrics for ranking the models in mathematical immunology with respect to their complexity/parsimony is provided by the Akaike information criterion. In the present study, a computational framework is elaborated to characterize the complexity of mathematical models in immunology using a more general approach, namely, the Minimum Description Length criterion. It balances the model goodness-of-fit with the dimensionality and geometrical complexity of the model. Four representative models of the immune response to acute viral infection formulated with either ordinary or delay differential equations are studied. Essential numerical details enabling the assessment and ranking of the viral infection models include: (1) the optimization of the likelihood function, (2) the computation of the model sensitivity functions, (3) the evaluation of the Fisher information matrix and (4) the estimation of multidimensional integrals over the model parameter space.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":"37 1","pages":"253 - 261"},"PeriodicalIF":0.6,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45560982","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
S. Simakov, T. Gamilov, A. Danilov, F. Liang, P. Chomakhidze, Mariam K. Gappoeva, Alina A. Rebrova, P. Kopylov
Abstract Coronary artery disease is the leading cause of mortality worldwide, accounting for 12.8% of all deaths. Although the clinical benefits of treating stenosis with percutaneous coronary intervention (PCI) have been extensively demonstrated, residual myocardial ischemia remains in about 30–50% of patients even after a formally successful PCI. We apply previously developed and validated 1D model of haemodynamics, which distributes terminal hydraulic resistance based on the diameters of the parent vessels and Murray’s law by a recursive algorithm. In our new model the terminal resistance is decreased according to a transmural perfusion ratio increase. In contrast to our previous work we calculate the transmural perfusion ratio for personally defined zones. Thus, peripheral hydraulic resistance of myocardial perfusion is personalized based on patient data, whichwere extracted from computed tomography perfusion images. The model serves as a computational tool for simulating pre- to post-PCI changes in coronary haemodynamics of four patients. We simulate fractional flow reserve (FFR), coronary flow reserve (CFR), instantaneous wave-free ratio (iFR), average flow in selected arteries in hyperemic and rest conditions before PCI and after PCI immediately after the surgery (in a short-term) and in a long-term (several months) perspectives. We conclude that high FFR and iFR values in short-term and long-term perspectives are not necessary correlate with CFR improvement and long-term blood flow recovery in coronary arteries.
{"title":"Personalized computational estimation of relative change in coronary blood flow after percutaneous coronary intervention in short-term and long-term perspectives","authors":"S. Simakov, T. Gamilov, A. Danilov, F. Liang, P. Chomakhidze, Mariam K. Gappoeva, Alina A. Rebrova, P. Kopylov","doi":"10.1515/rnam-2022-0024","DOIUrl":"https://doi.org/10.1515/rnam-2022-0024","url":null,"abstract":"Abstract Coronary artery disease is the leading cause of mortality worldwide, accounting for 12.8% of all deaths. Although the clinical benefits of treating stenosis with percutaneous coronary intervention (PCI) have been extensively demonstrated, residual myocardial ischemia remains in about 30–50% of patients even after a formally successful PCI. We apply previously developed and validated 1D model of haemodynamics, which distributes terminal hydraulic resistance based on the diameters of the parent vessels and Murray’s law by a recursive algorithm. In our new model the terminal resistance is decreased according to a transmural perfusion ratio increase. In contrast to our previous work we calculate the transmural perfusion ratio for personally defined zones. Thus, peripheral hydraulic resistance of myocardial perfusion is personalized based on patient data, whichwere extracted from computed tomography perfusion images. The model serves as a computational tool for simulating pre- to post-PCI changes in coronary haemodynamics of four patients. We simulate fractional flow reserve (FFR), coronary flow reserve (CFR), instantaneous wave-free ratio (iFR), average flow in selected arteries in hyperemic and rest conditions before PCI and after PCI immediately after the surgery (in a short-term) and in a long-term (several months) perspectives. We conclude that high FFR and iFR values in short-term and long-term perspectives are not necessary correlate with CFR improvement and long-term blood flow recovery in coronary arteries.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":"37 1","pages":"279 - 291"},"PeriodicalIF":0.6,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46109070","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The aortic valve neocuspidization (AVNeo) procedure requires the design of patient-specific neo-cusps which can be made numerically through the neovalve closure modelling. Prior the simulation, it is required to ‘suture virtually’ the neocusps into the patient’s aortic geometry, i.e., to find such state in which the neocusps are placed in the aortic root lumen without intersections of physical surfaces and neo-valve prolapse, and the position of the suture boundary satisfies the boundary conditions. To solve this problem, we tried to mimic neocusps suturing in Ozaki’s operation. As a result, we propose a new algorithm for ‘virtual suturing’ of given neocusps, considered as thin shells. The approach is able to work with both small and large (compared to an optimal size) neocusps and to handle each cusp independently of the others.
{"title":"Computational mimicking of surgical leaflet suturing for virtual aortic valve neocuspidization","authors":"A. Liogky","doi":"10.1515/rnam-2022-0023","DOIUrl":"https://doi.org/10.1515/rnam-2022-0023","url":null,"abstract":"Abstract The aortic valve neocuspidization (AVNeo) procedure requires the design of patient-specific neo-cusps which can be made numerically through the neovalve closure modelling. Prior the simulation, it is required to ‘suture virtually’ the neocusps into the patient’s aortic geometry, i.e., to find such state in which the neocusps are placed in the aortic root lumen without intersections of physical surfaces and neo-valve prolapse, and the position of the suture boundary satisfies the boundary conditions. To solve this problem, we tried to mimic neocusps suturing in Ozaki’s operation. As a result, we propose a new algorithm for ‘virtual suturing’ of given neocusps, considered as thin shells. The approach is able to work with both small and large (compared to an optimal size) neocusps and to handle each cusp independently of the others.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":"37 1","pages":"263 - 277"},"PeriodicalIF":0.6,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42460261","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
D. Tikhvinskii, Lema R. Merzhoeva, A. Chupakhin, A. Karpenko, D. Parshin
Abstract Abdominal aortic aneurysm is a widespread disease of cardiovascular system. Predicting a moment of its rupture is an important task for modern vascular surgery. At the same time, little attention is paid to the comorbidities, which are often the causes of severe postoperative complications or even death. This work is devoted to a numerical study of the haemodynamics of the model geometry for possible localizations of abdominal aortic aneurysm: on the aortic trunk or on its bifurcation. Both rigid and FSI numerical simulations are considered and compared with the model aortic configuration without aneurysm. It is shown that in the case of localization of the aneurysm on the bifurcation, the pressure in aorta increases upstream. Moreover, only in the case of a special geometry,when the radii of the iliac arteries are equal (r1 = r2), and the angle between them is 60 degrees, there is a linear relationship between the pressure in the aorta above the aneurysm and the size of the aneurysm itself: the slope of the straight line is in the interval a ∈ (0.003; 0.857), and the coefficient of determination is R2 ⩾ 0.75. The area bounded by the curve of the ‘pressure–velocity’ diagram for the values of velocity and pressure upstream in the presence of an aneurysm decreases compared to a healthy case (a vessel without an aneurysm). The simulation results in the rigid and FSI formulations agree qualitatively with each other. The obtained results provide a better understanding of the relationship between the geometrical parameters of the aneurysm and the changing of haemodynamics in the aortic bifurcation and its effect on the cardiovascular system upstream of the aneurysm.
{"title":"Computational analysis of the impact of aortic bifurcation geometry to AAA haemodynamics","authors":"D. Tikhvinskii, Lema R. Merzhoeva, A. Chupakhin, A. Karpenko, D. Parshin","doi":"10.1515/rnam-2022-0026","DOIUrl":"https://doi.org/10.1515/rnam-2022-0026","url":null,"abstract":"Abstract Abdominal aortic aneurysm is a widespread disease of cardiovascular system. Predicting a moment of its rupture is an important task for modern vascular surgery. At the same time, little attention is paid to the comorbidities, which are often the causes of severe postoperative complications or even death. This work is devoted to a numerical study of the haemodynamics of the model geometry for possible localizations of abdominal aortic aneurysm: on the aortic trunk or on its bifurcation. Both rigid and FSI numerical simulations are considered and compared with the model aortic configuration without aneurysm. It is shown that in the case of localization of the aneurysm on the bifurcation, the pressure in aorta increases upstream. Moreover, only in the case of a special geometry,when the radii of the iliac arteries are equal (r1 = r2), and the angle between them is 60 degrees, there is a linear relationship between the pressure in the aorta above the aneurysm and the size of the aneurysm itself: the slope of the straight line is in the interval a ∈ (0.003; 0.857), and the coefficient of determination is R2 ⩾ 0.75. The area bounded by the curve of the ‘pressure–velocity’ diagram for the values of velocity and pressure upstream in the presence of an aneurysm decreases compared to a healthy case (a vessel without an aneurysm). The simulation results in the rigid and FSI formulations agree qualitatively with each other. The obtained results provide a better understanding of the relationship between the geometrical parameters of the aneurysm and the changing of haemodynamics in the aortic bifurcation and its effect on the cardiovascular system upstream of the aneurysm.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":"37 1","pages":"311 - 329"},"PeriodicalIF":0.6,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41901159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Quantitative systems pharmacology (QSP) is a relatively new modelling discipline, formed within the ever-growing domain of model-informed drug development and actively evolving throughout the last decade. This modelling technique is based on the systems analysis and is used to get a quantitative rather than qualitative understanding of systems dynamics and explore the mechanisms of action of a drug. However, there is no well-defined methodology for the QSP model development, which significantly complicates the practical application of these models. In the current work, we overview the existing mathematical models of antidiabetic therapies and propose a modelling method, which overcomes common limitations and is able to produce a physiologically based mechanistic model describing gliflozin action in type 2 diabetes mellitus. From the practical standpoint, sensitivity analysis preformed in this work helped to reveal subpopulation of patients with better response to gliflozin therapy.
{"title":"Algorithms and methodological challenges in the development and application of quantitative systems pharmacology models: a case study in type 2 diabetes","authors":"V. Sokolov","doi":"10.1515/rnam-2022-0025","DOIUrl":"https://doi.org/10.1515/rnam-2022-0025","url":null,"abstract":"Abstract Quantitative systems pharmacology (QSP) is a relatively new modelling discipline, formed within the ever-growing domain of model-informed drug development and actively evolving throughout the last decade. This modelling technique is based on the systems analysis and is used to get a quantitative rather than qualitative understanding of systems dynamics and explore the mechanisms of action of a drug. However, there is no well-defined methodology for the QSP model development, which significantly complicates the practical application of these models. In the current work, we overview the existing mathematical models of antidiabetic therapies and propose a modelling method, which overcomes common limitations and is able to produce a physiologically based mechanistic model describing gliflozin action in type 2 diabetes mellitus. From the practical standpoint, sensitivity analysis preformed in this work helped to reveal subpopulation of patients with better response to gliflozin therapy.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":"37 1","pages":"293 - 309"},"PeriodicalIF":0.6,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45577366","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A concept of optimal disturbances of periodic solutions for a system of time-delay differential equations is defined. An algorithm for computing the optimal disturbances is proposed and justified. This algorithm is tested on the known system of four nonlinear time-delay differential equations modelling the dynamics of the experimental infection caused by the lymphocytic choriomeningitis virus. The results of numerical experiments are discussed.
{"title":"Optimal disturbances for periodic solutions of time-delay differential equations","authors":"M. Y. Khristichenko, Y. Nechepurenko","doi":"10.1515/rnam-2022-0017","DOIUrl":"https://doi.org/10.1515/rnam-2022-0017","url":null,"abstract":"Abstract A concept of optimal disturbances of periodic solutions for a system of time-delay differential equations is defined. An algorithm for computing the optimal disturbances is proposed and justified. This algorithm is tested on the known system of four nonlinear time-delay differential equations modelling the dynamics of the experimental infection caused by the lymphocytic choriomeningitis virus. The results of numerical experiments are discussed.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":"37 1","pages":"203 - 212"},"PeriodicalIF":0.6,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42778205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A homogeneous Dirichlet initial-boundary value problem for a quasilinear parabolic equation with a time-fractional derivative and coefficients at the elliptic part that depend on the gradient of the solution is considered. Conditions on the coefficients ensure the monotonicity and Lipschitz property of the elliptic operator on the set of functions whose gradients in space variables are uniformly bounded. For this problem, a linear regularized mesh scheme is constructed and investigated. A sufficient condition is derived for the regularization parameter that ensures the so-called local correctness of the mesh scheme. On the basis of correctness and approximation estimates for model problems with time-fractional Caputo or Caputo–Fabrizio derivatives, accuracy estimates are given in terms of mesh and regularization parameters under the assumption of the existence of a smooth solution to the differential problem. The presented results of the numerical experiments confirm the obtained asymptotic accuracy estimates.
{"title":"Linear regularized finite difference scheme for the quasilinear subdiffusion equation","authors":"A. Lapin, E. Laitinen","doi":"10.1515/rnam-2022-0019","DOIUrl":"https://doi.org/10.1515/rnam-2022-0019","url":null,"abstract":"Abstract A homogeneous Dirichlet initial-boundary value problem for a quasilinear parabolic equation with a time-fractional derivative and coefficients at the elliptic part that depend on the gradient of the solution is considered. Conditions on the coefficients ensure the monotonicity and Lipschitz property of the elliptic operator on the set of functions whose gradients in space variables are uniformly bounded. For this problem, a linear regularized mesh scheme is constructed and investigated. A sufficient condition is derived for the regularization parameter that ensures the so-called local correctness of the mesh scheme. On the basis of correctness and approximation estimates for model problems with time-fractional Caputo or Caputo–Fabrizio derivatives, accuracy estimates are given in terms of mesh and regularization parameters under the assumption of the existence of a smooth solution to the differential problem. The presented results of the numerical experiments confirm the obtained asymptotic accuracy estimates.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":"37 1","pages":"221 - 229"},"PeriodicalIF":0.6,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46827088","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
R. Fadeev, K. Alipova, Anna S. Koshkina, Timofey E. Lapin, N. Ozerova, Alina E. Pereladova, Andrey V. Sakhno, M. Tolstykh
Abstract In the present paper, we describe a one-dimensional glacier parameterization for use in the numerical weather prediction models. The proposed scheme is implemented into the global atmospheric model SLAV. To avoid inconsistency of surface temperature and turbulent heat fluxes in the lower troposphere, glacier parameterization has been iteratively coupled with both planetary boundary layer and land surface schemes. First results from numerical experiments with the SLAV model show that the introduction of a simplified description of the glacier heat capacity can significantly improve the 2-meter temperature long-range weather forecast skill.
{"title":"Glacier parameterization in SLAV numerical weather prediction model","authors":"R. Fadeev, K. Alipova, Anna S. Koshkina, Timofey E. Lapin, N. Ozerova, Alina E. Pereladova, Andrey V. Sakhno, M. Tolstykh","doi":"10.1515/rnam-2022-0016","DOIUrl":"https://doi.org/10.1515/rnam-2022-0016","url":null,"abstract":"Abstract In the present paper, we describe a one-dimensional glacier parameterization for use in the numerical weather prediction models. The proposed scheme is implemented into the global atmospheric model SLAV. To avoid inconsistency of surface temperature and turbulent heat fluxes in the lower troposphere, glacier parameterization has been iteratively coupled with both planetary boundary layer and land surface schemes. First results from numerical experiments with the SLAV model show that the introduction of a simplified description of the glacier heat capacity can significantly improve the 2-meter temperature long-range weather forecast skill.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":"37 1","pages":"189 - 201"},"PeriodicalIF":0.6,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44943943","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract To solve problems of radiation balance, optical sounding, and tomography, it may be necessary to take into account multiple scattering of radiation in a stochastically inhomogeneous medium. In real radiation models, for this purpose, the numerical-statistical ‘majorant cross-section method’ (MCM, delta-Woodcock tracking) is used based on the alignment of the optical density field by adding an artificial ‘delta scattering’ event. However, the computation cost of the corresponding unbiased estimate of the averaged problem solution infinitely increases as the correlation scale (correlation radius L) of standard mosaic models for a random medium density decreases. Previously, we constructed the MCM randomization providing asymptotically (for L → 0) unbiased estimates of the required functionals, in which the value of the physical attenuation coefficient is randomly chosen at the end of the particle free path l under condition l > L. Otherwise the value of the physical attenuation coefficient is the same as at the starting point of the particle (CR algorithm). In a more accurate functional correlative randomized algorithm (FCR algorithm), the coefficient remains the same with a probability determined by the correlation function. These correlative randomized algorithms were implemented for a mixture of homogeneous substance (water) and a Poisson ensemble of ‘empty’ balls. In the present paper, we construct correlative randomized algorithms for problems related to transfer through a ‘thick’ layer containing a water and a Poisson ensemble of ‘empty’ layers. A detailed comparative analysis of the results obtained by exact direct simulation (MCM) and approximate algorithms (CR, FCR) for the problems of gamma radiation transfer through a ‘thick’ water layer containing a Poisson ensemble of ‘empty’ layers or balls is presented.
{"title":"On the efficiency of using correlative randomized algorithms for solving problems of gamma radiation transfer in stochastic medium","authors":"I. N. Medvedev","doi":"10.1515/rnam-2022-0020","DOIUrl":"https://doi.org/10.1515/rnam-2022-0020","url":null,"abstract":"Abstract To solve problems of radiation balance, optical sounding, and tomography, it may be necessary to take into account multiple scattering of radiation in a stochastically inhomogeneous medium. In real radiation models, for this purpose, the numerical-statistical ‘majorant cross-section method’ (MCM, delta-Woodcock tracking) is used based on the alignment of the optical density field by adding an artificial ‘delta scattering’ event. However, the computation cost of the corresponding unbiased estimate of the averaged problem solution infinitely increases as the correlation scale (correlation radius L) of standard mosaic models for a random medium density decreases. Previously, we constructed the MCM randomization providing asymptotically (for L → 0) unbiased estimates of the required functionals, in which the value of the physical attenuation coefficient is randomly chosen at the end of the particle free path l under condition l > L. Otherwise the value of the physical attenuation coefficient is the same as at the starting point of the particle (CR algorithm). In a more accurate functional correlative randomized algorithm (FCR algorithm), the coefficient remains the same with a probability determined by the correlation function. These correlative randomized algorithms were implemented for a mixture of homogeneous substance (water) and a Poisson ensemble of ‘empty’ balls. In the present paper, we construct correlative randomized algorithms for problems related to transfer through a ‘thick’ layer containing a water and a Poisson ensemble of ‘empty’ layers. A detailed comparative analysis of the results obtained by exact direct simulation (MCM) and approximate algorithms (CR, FCR) for the problems of gamma radiation transfer through a ‘thick’ water layer containing a Poisson ensemble of ‘empty’ layers or balls is presented.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":"37 1","pages":"231 - 240"},"PeriodicalIF":0.6,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45510400","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The paper is devoted to a posteriori error identities for the stationary reaction–convection–diffusion problem with mixed Dirichlét–Neumann boundary conditions. They reflect the most general relations between deviations of approximations from the exact solutions and those values that can be observed in a numerical experiment. The identities contain no mesh dependent constants and are valid for any function in the admissible (energy) class. Therefore, the identities and the estimates that follow from them generate universal and fully reliable tools of a posteriori error control.
{"title":"Error identities for the reaction–convection–diffusion problem and applications to a posteriori error control","authors":"S. Repin","doi":"10.1515/rnam-2022-0021","DOIUrl":"https://doi.org/10.1515/rnam-2022-0021","url":null,"abstract":"Abstract The paper is devoted to a posteriori error identities for the stationary reaction–convection–diffusion problem with mixed Dirichlét–Neumann boundary conditions. They reflect the most general relations between deviations of approximations from the exact solutions and those values that can be observed in a numerical experiment. The identities contain no mesh dependent constants and are valid for any function in the admissible (energy) class. Therefore, the identities and the estimates that follow from them generate universal and fully reliable tools of a posteriori error control.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":"695 23","pages":"241 - 252"},"PeriodicalIF":0.6,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41281430","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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