Abstract A spatially distributed mathematical model is presented that simulates the growth of a non-invasive tumour undergoing treatment by fractionated proton therapy with the use of non-radioactive tumour-specific nanosensitizers. Nanosensitizers are injected intravenously before each irradiation to increase the locally deposited dose via a chain of reactions with therapeutic protons. Modelling simulations show that the use of nanosensitizers allows increasing treatment efficacy. However, their effect is restricted by the necessity of decreasing the energy deposited in tumour in order to comply to the normal damage restrictions. Normalization of tumour microvasculature that accompanies the treatment, also compromises nanosensitizers effect as it impairs their inflow in tumour. It is shown that spatial optimization of irradiation, with conservation of total dose deposited in tumour, can increase tumour cell damage for each single irradiation. However, eventually it may not lead to the overall increase of treatment efficacy, in terms of minimization of the number of remaining viable tumour cells, due to the influence of tumour cell repopulation between irradiations. It is suggested that an efficient way towards minimization of tumour cell repopulation may be the faster suppression of angiogenesis by eradication of metabolically deprived tumour cells. This method can be efficient even despite the fact that it would also cause the decrease of supply of nanosensitizers into the tumour.
{"title":"Mathematical modelling for spatial optimization of irradiation during proton radiotherapy with nanosensitizers","authors":"Maxim Kuznetsov, Andrey Kolobov","doi":"10.1515/rnam-2023-0023","DOIUrl":"https://doi.org/10.1515/rnam-2023-0023","url":null,"abstract":"Abstract A spatially distributed mathematical model is presented that simulates the growth of a non-invasive tumour undergoing treatment by fractionated proton therapy with the use of non-radioactive tumour-specific nanosensitizers. Nanosensitizers are injected intravenously before each irradiation to increase the locally deposited dose via a chain of reactions with therapeutic protons. Modelling simulations show that the use of nanosensitizers allows increasing treatment efficacy. However, their effect is restricted by the necessity of decreasing the energy deposited in tumour in order to comply to the normal damage restrictions. Normalization of tumour microvasculature that accompanies the treatment, also compromises nanosensitizers effect as it impairs their inflow in tumour. It is shown that spatial optimization of irradiation, with conservation of total dose deposited in tumour, can increase tumour cell damage for each single irradiation. However, eventually it may not lead to the overall increase of treatment efficacy, in terms of minimization of the number of remaining viable tumour cells, due to the influence of tumour cell repopulation between irradiations. It is suggested that an efficient way towards minimization of tumour cell repopulation may be the faster suppression of angiogenesis by eradication of metabolically deprived tumour cells. This method can be efficient even despite the fact that it would also cause the decrease of supply of nanosensitizers into the tumour.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136198949","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 This work is dedicated to the pressure-correction projection method for the volume-averaged Navier–Stokes system for porous media. A set of parameters controlling the presence of inertia and viscosity is introduced into the system. Switching parameters allows us to reduce the system to either the Brinkman system or the Darcy equation. Considering the jump in the parameters between mesh cells allows capturing the contact of media of different types, such as free-flow and porous media flow. We apply Chorin’s projection method to decouple the system. The splitting of the system yields a momentum conservation equation and an anisotropic pressure correction equation. We propose a combination of collocated finite-volume methods to solve the problem.
{"title":"Pressure-correction projection method for modelling the incompressible fluid flow in porous media","authors":"K. Terekhov","doi":"10.1515/rnam-2023-0019","DOIUrl":"https://doi.org/10.1515/rnam-2023-0019","url":null,"abstract":"Abstract This work is dedicated to the pressure-correction projection method for the volume-averaged Navier–Stokes system for porous media. A set of parameters controlling the presence of inertia and viscosity is introduced into the system. Switching parameters allows us to reduce the system to either the Brinkman system or the Darcy equation. Considering the jump in the parameters between mesh cells allows capturing the contact of media of different types, such as free-flow and porous media flow. We apply Chorin’s projection method to decouple the system. The splitting of the system yields a momentum conservation equation and an anisotropic pressure correction equation. We propose a combination of collocated finite-volume methods to solve the problem.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46895771","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. Ammosov, S. Stepanov, D. Spiridonov, Wenyuan Li
Abstract In the present paper, the authors rigorously derive Richards’ multicontinuum model using the multicontinuum homogenization approach. This approach is based on formulating constraint cell problems and a homogenization-like expansion. We present numerical results for the two continua case with separable coefficients. First, we explore the relationships between the effective coefficients and the hydraulic conductivity. Then, we solve test problems with different contrasts to study the developed multicontinuum model.
{"title":"Multicontinuum homogenization for Richards’ equation: The derivation and numerical experiments","authors":"D. Ammosov, S. Stepanov, D. Spiridonov, Wenyuan Li","doi":"10.1515/rnam-2023-0016","DOIUrl":"https://doi.org/10.1515/rnam-2023-0016","url":null,"abstract":"Abstract In the present paper, the authors rigorously derive Richards’ multicontinuum model using the multicontinuum homogenization approach. This approach is based on formulating constraint cell problems and a homogenization-like expansion. We present numerical results for the two continua case with separable coefficients. First, we explore the relationships between the effective coefficients and the hydraulic conductivity. Then, we solve test problems with different contrasts to study the developed multicontinuum model.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47603539","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}
Pub Date : 2023-08-01DOI: 10.1515/rnam-2023-frontmatter4
{"title":"Frontmatter","authors":"","doi":"10.1515/rnam-2023-frontmatter4","DOIUrl":"https://doi.org/10.1515/rnam-2023-frontmatter4","url":null,"abstract":"","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136106839","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 approximate solution of the Cauchy problem for second-order evolution equations is performed, first of all, using three-level time approximations. Such approximations are easily constructed and relatively uncomplicated to investigate when using uniform time grids. When solving applied problems numerically, we should focus on approximations with variable time steps. When using multilevel schemes on non-uniform grids, we should maintain accuracy by choosing appropriate approximations and ensuring stability of the approximate solution. In this paper, we construct unconditionally stable schemes of the first- and second-order accuracy on a non-uniform time grid for the approximate solution of the Cauchy problem for a second-order evolutionary equation. The novelty of the paper consists in the fact that these stability estimates are obtained without any restrictions on the magnitude of the step change and on the number of step changes. We use a special transformation of the original second-order differential-operator equation to a system of first-order equations. For the system of first-order equations, we apply standard two-level time approximations. We obtained stability estimates for the initial data and the right-hand side in finite-dimensional Hilbert space. Eliminating auxiliary variables leads to three-level schemes for the initial second-order evolution equation. Numerical experiments were performed for the test problem for a one-dimensional in space bi-parabolic equation. The accuracy and stability properties of the constructed schemes are demonstrated on non-uniform grids with randomly varying grid steps.
{"title":"Operator-difference schemes on non-uniform grids for second-order evolutionary equations","authors":"Petr N. Vabishchevich","doi":"10.1515/rnam-2023-0020","DOIUrl":"https://doi.org/10.1515/rnam-2023-0020","url":null,"abstract":"Abstract The approximate solution of the Cauchy problem for second-order evolution equations is performed, first of all, using three-level time approximations. Such approximations are easily constructed and relatively uncomplicated to investigate when using uniform time grids. When solving applied problems numerically, we should focus on approximations with variable time steps. When using multilevel schemes on non-uniform grids, we should maintain accuracy by choosing appropriate approximations and ensuring stability of the approximate solution. In this paper, we construct unconditionally stable schemes of the first- and second-order accuracy on a non-uniform time grid for the approximate solution of the Cauchy problem for a second-order evolutionary equation. The novelty of the paper consists in the fact that these stability estimates are obtained without any restrictions on the magnitude of the step change and on the number of step changes. We use a special transformation of the original second-order differential-operator equation to a system of first-order equations. For the system of first-order equations, we apply standard two-level time approximations. We obtained stability estimates for the initial data and the right-hand side in finite-dimensional Hilbert space. Eliminating auxiliary variables leads to three-level schemes for the initial second-order evolution equation. Numerical experiments were performed for the test problem for a one-dimensional in space bi-parabolic equation. The accuracy and stability properties of the constructed schemes are demonstrated on non-uniform grids with randomly varying grid steps.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136106961","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 We describe the mathematical modelling of the group behaviour of workers in the labor market. The worker receives the salary and seeks to improve his qualifications in order to receive higher wages. The worker enlarges his qualification by the investments in human capital. At a random moment of time, a vacancy appears that provides a jump in the worker’s salary. The mathematical model of the worker’s behaviour in the labor market is presented as an optimal control problem on an infinite time horizon. The paper presents the derivation of the Kolmogorov–Fokker–Planck equation for the Lévy process, which describes the behaviour of a large amount of workers within a social layer. The numerical solution of the Kolmogorov–Fokker–Planck equation and the calculation results are presented.
{"title":"The group behaviour modelling of workers in the labor market","authors":"A. Shananin, N. Trusov","doi":"10.1515/rnam-2023-0017","DOIUrl":"https://doi.org/10.1515/rnam-2023-0017","url":null,"abstract":"Abstract We describe the mathematical modelling of the group behaviour of workers in the labor market. The worker receives the salary and seeks to improve his qualifications in order to receive higher wages. The worker enlarges his qualification by the investments in human capital. At a random moment of time, a vacancy appears that provides a jump in the worker’s salary. The mathematical model of the worker’s behaviour in the labor market is presented as an optimal control problem on an infinite time horizon. The paper presents the derivation of the Kolmogorov–Fokker–Planck equation for the Lévy process, which describes the behaviour of a large amount of workers within a social layer. The numerical solution of the Kolmogorov–Fokker–Planck equation and the calculation results are presented.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46830594","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}
E. Shcherbakova, S. Matveev, A. Smirnov, E. Tyrtyshnikov
Abstract In the present paper we compare two different iterative approaches to constructing nonnegative tensor train and Tucker decompositions. The first approach is based on idea of alternating projections and randomized sketching for factorization of tensors with nonnegative elements. This approach can be useful for both TT and Tucker formats. The second approach consists of two stages. At the first stage we find the unconstrained tensor train decomposition for the target array. At the second stage we use this initial approximation in order to fix it within moderate number of operations and obtain the factorization with nonnegative factors either in tensor train or Tucker model. We study the performance of these methods for both synthetic data and hyper-spectral image and demonstrate the clear advantage of the latter technique in terms of computational time and wider range of possible applications.
{"title":"Study of performance of low-rank nonnegative tensor factorization methods","authors":"E. Shcherbakova, S. Matveev, A. Smirnov, E. Tyrtyshnikov","doi":"10.1515/rnam-2023-0018","DOIUrl":"https://doi.org/10.1515/rnam-2023-0018","url":null,"abstract":"Abstract In the present paper we compare two different iterative approaches to constructing nonnegative tensor train and Tucker decompositions. The first approach is based on idea of alternating projections and randomized sketching for factorization of tensors with nonnegative elements. This approach can be useful for both TT and Tucker formats. The second approach consists of two stages. At the first stage we find the unconstrained tensor train decomposition for the target array. At the second stage we use this initial approximation in order to fix it within moderate number of operations and obtain the factorization with nonnegative factors either in tensor train or Tucker model. We study the performance of these methods for both synthetic data and hyper-spectral image and demonstrate the clear advantage of the latter technique in terms of computational time and wider range of possible applications.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46211072","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 particle tracking method based on face fluxes data calculated using finite volume methods is developed for unstructured three-dimensional polyhedral grids. The flow velocity field reconstruction on grid cells using a mixed finite element method is proposed. Cases of sinks and sources in cells as well as different cell partitionings are considered. Algorithms for streamlines and time of flight calculation are provided. Performance and convergence of the method are demonstrated on a set of reference problems.
{"title":"Particle tracking for face-based flux data on general polyhedral grids with applications to groundwater flow modelling","authors":"I. Kapyrin","doi":"10.1515/rnam-2023-0010","DOIUrl":"https://doi.org/10.1515/rnam-2023-0010","url":null,"abstract":"Abstract A particle tracking method based on face fluxes data calculated using finite volume methods is developed for unstructured three-dimensional polyhedral grids. The flow velocity field reconstruction on grid cells using a mixed finite element method is proposed. Cases of sinks and sources in cells as well as different cell partitionings are considered. Algorithms for streamlines and time of flight calculation are provided. Performance and convergence of the method are demonstrated on a set of reference problems.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49449819","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}
A. Liogky, A. Chernyshenko, A. Danilov, Fyodor A. Syomin
Abstract A new parallel numerical framework CarNum is presented for efficient coupling of mathematical models in multiphysics problems such as computational cardiac electromechanics. This framework is based on open source projects, which provide the core functionality of the platform. Computational cardiac electromechanics requires a complex pipeline of solving different types of ordinary and partial differential equations. Our framework allows one to implement different numerical schemes and provides more control in multiphysics coupling. This paper outlines a concept of the new platform and details of numerical modelling of cardiac electromechanics. First experiments with well-known cardiac electromechanics benchmarks show good agreement with other groups and decent parallel scalability.
{"title":"CarNum: parallel numerical framework for computational cardiac electromechanics","authors":"A. Liogky, A. Chernyshenko, A. Danilov, Fyodor A. Syomin","doi":"10.1515/rnam-2023-0011","DOIUrl":"https://doi.org/10.1515/rnam-2023-0011","url":null,"abstract":"Abstract A new parallel numerical framework CarNum is presented for efficient coupling of mathematical models in multiphysics problems such as computational cardiac electromechanics. This framework is based on open source projects, which provide the core functionality of the platform. Computational cardiac electromechanics requires a complex pipeline of solving different types of ordinary and partial differential equations. Our framework allows one to implement different numerical schemes and provides more control in multiphysics coupling. This paper outlines a concept of the new platform and details of numerical modelling of cardiac electromechanics. First experiments with well-known cardiac electromechanics benchmarks show good agreement with other groups and decent parallel scalability.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47658149","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 presents the dynamical core of the new sea ice model SIMUG (Sea Ice Model on Unstructured Grid) on the A- and CD-types of unstructured triangular grids in the local-element basis on sphere. Three standardized box tests to reproduce the Linear Kinematic Features (LKFs), and the short-term forecast in the real Arctic Ocean geometry with the realistic atmosphere and ocean forcing demonstrate the model quality compared to other sea ice models like CICE, FESOM, MITgcm, and ICON-O. The distinctive features of the model presented are a wide choice of transport schemes, and the new numerical implementation with the serial and parallel C++ coding and INMOST, Ani2D, and Ani3D packages to deal with unstructured grids. Code profiling and scalability assessment are carried out. In general, the A-version of the ice drift model works faster, but has fewer degrees of freedom on the same grid. Due to the increase in the degrees of freedom, the model on the CD grid gives ultra-resolution of LKFs, but requires more strict conditions for stability.
提出了基于球面局部元的A型和cd型非结构化三角形网格海冰模型SIMUG (sea ice model on Unstructured Grid)的动力学核心。与CICE、FESOM、MITgcm和ICON-O等其他海冰模式相比，模拟线性运动特征(LKFs)的三个标准化箱试验和具有真实大气和海洋强迫的真实北冰洋几何形状的短期预报证明了模式的质量。该模型的显著特点是具有广泛的传输方案选择，以及采用串行和并行c++编码和INMOST、Ani2D和Ani3D软件包处理非结构化网格的新数值实现。进行了代码分析和可伸缩性评估。一般来说，a版本的冰漂移模型工作得更快，但在同一网格上的自由度更少。由于自由度的增加，CD网格上的模型给出了LKFs的超分辨率，但对稳定性的要求更严格。
{"title":"SIMUG – finite element model of sea ice dynamics on triangular grid in local Cartesian basis","authors":"Sergey S. Petrov, N. Iakovlev","doi":"10.1515/rnam-2023-0012","DOIUrl":"https://doi.org/10.1515/rnam-2023-0012","url":null,"abstract":"Abstract The paper presents the dynamical core of the new sea ice model SIMUG (Sea Ice Model on Unstructured Grid) on the A- and CD-types of unstructured triangular grids in the local-element basis on sphere. Three standardized box tests to reproduce the Linear Kinematic Features (LKFs), and the short-term forecast in the real Arctic Ocean geometry with the realistic atmosphere and ocean forcing demonstrate the model quality compared to other sea ice models like CICE, FESOM, MITgcm, and ICON-O. The distinctive features of the model presented are a wide choice of transport schemes, and the new numerical implementation with the serial and parallel C++ coding and INMOST, Ani2D, and Ani3D packages to deal with unstructured grids. Code profiling and scalability assessment are carried out. In general, the A-version of the ice drift model works faster, but has fewer degrees of freedom on the same grid. Due to the increase in the degrees of freedom, the model on the CD grid gives ultra-resolution of LKFs, but requires more strict conditions for stability.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47707361","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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