2,169 results on '"Scheme (mathematics)"'
Search Results
2. High order semi-implicit weighted compact nonlinear scheme for viscous Burgers’ equations
- Author
-
Xun Chen, Yanqun Jiang, Rong Fan, and Xu Zhang
- Subjects
Numerical Analysis ,General Computer Science ,Advection ,Applied Mathematics ,Courant–Friedrichs–Lewy condition ,Reynolds number ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Stability (probability) ,Theoretical Computer Science ,Viscosity ,symbols.namesake ,Nonlinear system ,Modeling and Simulation ,Scheme (mathematics) ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,Applied mathematics ,020201 artificial intelligence & image processing ,0101 mathematics ,High order ,Mathematics - Abstract
This paper develops a high order semi-implicit weighted compact nonlinear scheme (WCNS) for one- and two-dimensional viscous Burgers’ equations. This semi-implicit WCNS combines a fifth-order WCNS with a third-order implicit-explicit Runge–Kutta (IMEX-RK) time-stepping method. The advection terms of viscous Burgers’ equations are treated explicitly, while the viscosity terms are treated implicitly. Stability analysis shows that the CFL condition of the semi-implicit WCNS is controlled only by the advection terms. Compared to the explicit time-stepping method, the semi-implicit method has the advantage in terms of computational efficiency. Numerical results validate the accuracy and efficiency of the semi-implicit WCNS. This method can also solve viscous Burgers’ equations with large Reynolds numbers and has high-resolution shock-capturing ability.
- Published
- 2021
3. On the Grothendieck–Serre conjecture on principal bundles in mixed characteristic
- Author
-
Roman Fedorov
- Subjects
Large class ,Pure mathematics ,Conjecture ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Principal (computer security) ,Local ring ,Field (mathematics) ,Regular local ring ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,16. Peace & justice ,01 natural sciences ,Mathematics - Algebraic Geometry ,Scheme (mathematics) ,FOS: Mathematics ,0101 mathematics ,Algebraic Geometry (math.AG) ,Quotient ,Mathematics - Abstract
Let R be a regular local ring. Let G be a reductive R-group scheme. A conjecture of Grothendieck and Serre predicts that a principal G-bundle over R is trivial if it is trivial over the quotient field of R. The conjecture is known when R contains a field. We prove the conjecture for a large class of regular local rings not containing fields in the case when G is split., Comment: The final version to be published in Transactions of the AMS. Results about quadratic forms are strengthened. In the section on Bertini type theorems a correction in the case of a non-perfect residue field is made. Other minor corrections and improvements
- Published
- 2021
4. A simple and high accurate finite volume scheme for diffusion equations
- Author
-
Shuhong Song and Zhucui Jing
- Subjects
010101 applied mathematics ,Statistics and Probability ,Finite volume method ,Artificial Intelligence ,Simple (abstract algebra) ,Scheme (mathematics) ,Mathematical analysis ,General Engineering ,010103 numerical & computational mathematics ,0101 mathematics ,Diffusion (business) ,01 natural sciences ,Mathematics - Abstract
Most of numerical methods for diffusion equations, refer to vertex unknowns directly or indirectly, and their accuracy is ultimately determined by the approximation to vertex unknowns. Based on the “twin-fitting” method, a simple and high accurate treatment for the vertex unknowns is developed and is applied to the nine-point scheme for diffusion problem. Numerical experiments show that the resulting nine-point scheme is high accurate for diffusion problems with discontinuous diffusion coefficients on distorted meshes.
- Published
- 2021
5. Étale cohomology of arithmetic schemes and zeta values of arithmetic surfaces
- Author
-
Kanetomo Sato
- Subjects
Arithmetic surface ,Rational number ,Algebra and Number Theory ,Conjecture ,Mathematics - Number Theory ,Selmer group ,Mathematics::Number Theory ,010102 general mathematics ,Étale cohomology ,010103 numerical & computational mathematics ,01 natural sciences ,Prime (order theory) ,Riemann zeta function ,Primary 19F27, 14G10, Secondary 11R34, 14F42 ,symbols.namesake ,Mathematics::K-Theory and Homology ,Scheme (mathematics) ,Mathematics - K-Theory and Homology ,symbols ,0101 mathematics ,Arithmetic ,Mathematics - Abstract
In this paper, we give an approach to the zeta values of a (proper regular) arithmetic scheme X at the integers r>=d:=dim(X), using \'etale cohomology of X with Q_p(r) and Z_p(r)-coefficients., Comment: 62 pages. Remark 2.2 has been added. Proposition 7.3 has been modified. Some details of the proof of Lemma 7.9 have been improved, and many typos have been corrected. To appear in Journal of Number Theory
- Published
- 2021
6. Concentration on Poisson spaces via modified Φ-Sobolev inequalities
- Author
-
Holger Sambale, Christoph Thäle, and Anna Gusakova
- Subjects
Statistics and Probability ,Poisson processes ,Polytope ,Poisson distribution ,01 natural sciences ,Sobolev inequality ,010104 statistics & probability ,symbols.namesake ,inequalities ,Cylinder ,Stochastic geometry ,0101 mathematics ,Mathematics ,60D05, 60G55, 60H05 ,Modified Phi-Sobolev ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Recursion (computer science) ,L-p-estimates ,Moment (mathematics) ,Modeling and Simulation ,Scheme (mathematics) ,symbols ,Concentration inequalities ,Mathematics - Probability - Abstract
Concentration properties of functionals of general Poisson processes are studied. Using a modified Phi-Sobolev inequality a recursion scheme for moments is established, which is of independent interest. This is applied to derive moment and concentration inequalities for functionals on abstract Poisson spaces. Applications of the general results in stochastic geometry, namely Poisson cylinder models and Poisson random polytopes, are presented as well. (C) 2021 Elsevier B.V. All rights reserved.
- Published
- 2021
7. α-Robust H1-norm convergence analysis of ADI scheme for two-dimensional time-fractional diffusion equation
- Author
-
Hu Chen, Tao Sun, and Yue Wang
- Subjects
Numerical Analysis ,Diffusion equation ,Initial singularity ,Applied Mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Fractional calculus ,010101 applied mathematics ,Computational Mathematics ,Scheme (mathematics) ,Norm (mathematics) ,Gronwall's inequality ,Convergence (routing) ,Order (group theory) ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
A fully discrete ADI scheme is proposed for solving the two-dimensional time-fractional diffusion equation with weakly singular solutions, where L1 scheme on graded mesh is adopted to tackle the initial singularity. An improved discrete fractional Gronwall inequality is employed to give an α-robust H 1 -norm convergence analysis of the fully discrete ADI scheme, where the error bound does not blow up when the order of fractional derivative α → 1 − . Numerical results show that the theoretical analysis is sharp.
- Published
- 2021
8. A direction splitting scheme for Navier–Stokes–Boussinesq system in spherical shell geometries
- Author
-
Roman Frolovc, Peter Mineva, and Aziz Takhirovb
- Subjects
Physics ,Applied Mathematics ,Mechanical Engineering ,Mathematical analysis ,Computational Mechanics ,010103 numerical & computational mathematics ,01 natural sciences ,Spherical shell ,010305 fluids & plasmas ,Computer Science Applications ,Mechanics of Materials ,Scheme (mathematics) ,0103 physical sciences ,Navier stokes ,0101 mathematics - Published
- 2021
9. The Cahn–Hilliard equation with a nonlinear source term
- Author
-
Alain Miranville
- Subjects
Logarithm ,Applied Mathematics ,Weak solution ,010102 general mathematics ,01 natural sciences ,Term (time) ,010101 applied mathematics ,Nonlinear system ,Scheme (mathematics) ,Applied mathematics ,0101 mathematics ,Finite time ,Cahn–Hilliard equation ,Analysis ,Mathematics - Abstract
Our aim in this paper is to prove the existence of solutions to the Cahn–Hilliard equation with a general nonlinear source term. An essential difficulty is to obtain a global in time solution. Indeed, due to the presence of the source term, one cannot exclude the possibility of blow up in finite time when considering regular nonlinear terms and when considering an approximated scheme. Considering instead logarithmic nonlinear terms, we give sufficient conditions on the source term which ensure the existence of a global in time weak solution. These conditions are satisfied by several important models and applications which can be found in the literature.
- Published
- 2021
10. L1/LDG method for the generalized time-fractional Burgers equation
- Author
-
Zhen Wang, Dongxia Li, and Changpin Li
- Subjects
Numerical Analysis ,General Computer Science ,Discretization ,Applied Mathematics ,010103 numerical & computational mathematics ,02 engineering and technology ,Derivative ,Space (mathematics) ,01 natural sciences ,Mathematics::Numerical Analysis ,Theoretical Computer Science ,Fractional calculus ,Burgers' equation ,Discontinuous Galerkin method ,Modeling and Simulation ,Scheme (mathematics) ,0202 electrical engineering, electronic engineering, information engineering ,Order (group theory) ,Applied mathematics ,020201 artificial intelligence & image processing ,0101 mathematics ,Mathematics - Abstract
In this paper, we study the generalized time fractional Burgers equation, where the time fractional derivative is in the sense of Caputo with derivative order in ( 0 , 1 ) . If its solution u ( x , t ) has strong regularity, for example u ( ⋅ , t ) ∈ C 2 [ 0 , T ] for a given time T , then we use the L1 scheme on uniform meshes to approximate the Caputo time-fractional derivative, and use the local discontinuous Galerkin (LDG) method to approach the space derivative. However, the solution u ( x , t ) likely behaves a certain regularity at the starting time, i.e., ∂ u ∂ t and ∂ 2 u ∂ 2 t can blow up as t → 0 + albeit u ( ⋅ , t ) ∈ C [ 0 , T ] for a given time T . In this case, we use the L1 scheme on non-uniform meshes to approximate the Caputo time-fractional derivative, and use the LDG method to discretize the spatial derivative. The fully discrete schemes for both situations are established and analyzed. It is shown that the derived schemes are numerically stable and convergent. Finally, several numerical experiments are provided which support the theoretical analysis.
- Published
- 2021
11. Basic convergence theory for the network element method
- Author
-
Julien Coatléven and IFP Energies nouvelles (IFPEN)
- Subjects
Numerical Analysis ,Discretization ,Meshless methods ,Applied Mathematics ,010103 numerical & computational mathematics ,network element method ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,virtual element method ,Network element ,Modeling and Simulation ,Scheme (mathematics) ,Applied mathematics ,Meshfree methods ,Symbolic convergence theory ,[MATH]Mathematics [math] ,0101 mathematics ,Element (category theory) ,Analysis ,Mathematics - Abstract
International audience; A recent paper introduced the network element method (NEM) where the usual mesh was replaced by a discretization network. Using the associated network geometric coefficients and following the virtual element framework, a consistent and stable numerical scheme was proposed. The aim of the present paper is to derive a convergence theory for the NEM under mild assumptions on the exact problem. We also derive basic error estimates, which are sub-optimal in the sense that we have to assume more regularity than usual.
- Published
- 2021
12. Stabilized Gauge Uzawa scheme for an incompressible micropolar fluid flow
- Author
-
Toufic El Arwadi, Sarah Slayi, and Séréna Dib
- Subjects
Numerical Analysis ,Applied Mathematics ,010103 numerical & computational mathematics ,Gauge (firearms) ,01 natural sciences ,Stability (probability) ,Physics::Fluid Dynamics ,010101 applied mathematics ,Computational Mathematics ,Scheme (mathematics) ,Convergence (routing) ,Compressibility ,Fluid dynamics ,Applied mathematics ,A priori and a posteriori ,0101 mathematics ,Mathematics - Abstract
In this paper, a second order Gauge Uzawa scheme for the governing equations of an incompressible micropolar fluid flow is introduced and analyzed. The derivation of this scheme is obtained using the second order backward difference approximation. Later, the unconditional stability of the GUM scheme for the micropolar equations will be shown. Then, an a priori error estimate is established to prove the convergence. Finally, we present some numerical simulations that confirm the theoretical results.
- Published
- 2021
13. Three novel fifth-order iterative schemes for solving nonlinear equations
- Author
-
Essam R. El-Zahar, Chih-Wen Chang, and Chein-Shan Liu
- Subjects
Numerical Analysis ,Weight function ,Van der Waals equation ,Conjecture ,General Computer Science ,Basis (linear algebra) ,Applied Mathematics ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Theoretical Computer Science ,symbols.namesake ,Nonlinear system ,Modeling and Simulation ,Scheme (mathematics) ,Convergence (routing) ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,Applied mathematics ,020201 artificial intelligence & image processing ,0101 mathematics ,Constant (mathematics) ,Mathematics - Abstract
Kung and Traub’s conjecture indicates that a multipoint iterative scheme without memory and based on m evaluations of functions has an optimal convergence order p = 2 m − 1 . Consequently, a fifth-order iterative scheme requires at least four evaluations of functions. Herein, we derive three novel iterative schemes that have fifth-order convergence and involve four evaluations of functions, such that the efficiency index is E.I.=1.49535. On the basis of the analysis of error equations, we obtain our first iterative scheme from the constant weight combinations of three first- and second-class fourth-order iterative schemes. For the second iterative scheme, we devise a new weight function to derive another fifth-order iterative scheme. Finally, we derive our third iterative scheme from a combination of two second-class fourth-order iterative schemes. For testing the practical application of our schemes, we apply them to solve the van der Waals equation of state.
- Published
- 2021
14. Regularized splitting spectral method for space-fractional logarithmic Schrödinger equation
- Author
-
Bianru Cheng and Zhenhua Guo
- Subjects
Numerical Analysis ,Applied Mathematics ,Numerical analysis ,010103 numerical & computational mathematics ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Fourier transform ,Scheme (mathematics) ,Convergence (routing) ,symbols ,Applied mathematics ,Order (group theory) ,0101 mathematics ,Spectral method ,Laplace operator ,Mathematics - Abstract
In this paper, we study a regularized Lie-Trotter splitting spectral method for a regularized space-fractional logarithmic Schrodinger equation by introducing a small regularized parameter 0 e ≪ 1 . The regularized method can be used to avoid numerical blow-up in the space-fractional logarithmic Schrodinger equation. The regularized space-fractional logarithmic Schrodinger equation is proved to approximate the space-fractional logarithmic Schrodinger equation with linear convergence rate O ( e ) . The proposed numerical scheme can preserve the discrete mass and step-energy for regularized space-fractional logarithmic Schrodinger equation. The first order convergence in time for the numerical method is rigorous proved for the regularized space-fractional logarithmic Schrodinger equation. Due to the appearance of fractional Laplace operator with order α, we can adjust the parameters of order to make the equation show much more dynamic characteristics than the classical logarithmic Schrodinger equation. Numerical simulations for 1D case based on the Fourier spectral approximation in space are presented to validate the theoretical analysis.
- Published
- 2021
15. Superfast Second-Order Methods for Unconstrained Convex Optimization
- Author
-
Yurii Nesterov, Center of Operation Research and Econometrics [Louvain] (CORE), Université Catholique de Louvain = Catholic University of Louvain (UCL), Center of Operations Research and Econometrics (CORE), ANR-19-P3IA-0003,MIAI,MIAI @ Grenoble Alpes(2019), and European Project: 788368
- Subjects
Hessian matrix ,Control and Optimization ,0211 other engineering and technologies ,010103 numerical & computational mathematics ,02 engineering and technology ,Management Science and Operations Research ,Type (model theory) ,01 natural sciences ,Upper and lower bounds ,symbols.namesake ,Tensor methods ,0101 mathematics ,Mathematics ,Discrete mathematics ,021103 operations research ,Applied Mathematics ,Order (ring theory) ,Lower complexity bounds ,Convex optimization ,Rate of convergence ,Scheme (mathematics) ,Theory of computation ,symbols ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,Second-order methods - Abstract
In this paper, we present new second-order methods with convergence rate $$O\left( k^{-4}\right) $$ O k - 4 , where k is the iteration counter. This is faster than the existing lower bound for this type of schemes (Agarwal and Hazan in Proceedings of the 31st conference on learning theory, PMLR, pp. 774–792, 2018; Arjevani and Shiff in Math Program 178(1–2):327–360, 2019), which is $$O\left( k^{-7/2} \right) $$ O k - 7 / 2 . Our progress can be explained by a finer specification of the problem class. The main idea of this approach consists in implementation of the third-order scheme from Nesterov (Math Program 186:157–183, 2021) using the second-order oracle. At each iteration of our method, we solve a nontrivial auxiliary problem by a linearly convergent scheme based on the relative non-degeneracy condition (Bauschke et al. in Math Oper Res 42:330–348, 2016; Lu et al. in SIOPT 28(1):333–354, 2018). During this process, the Hessian of the objective function is computed once, and the gradient is computed $$O\left( \ln {1 \over \epsilon }\right) $$ O ln 1 ϵ times, where $$\epsilon $$ ϵ is the desired accuracy of the solution for our problem.
- Published
- 2021
16. On the Numerical Solution of the Near Field Refractor Problem
- Author
-
Henok Mawi and Cristian E. Gutiérrez
- Subjects
Control and Optimization ,FOS: Physical sciences ,Near and far field ,01 natural sciences ,Measure (mathematics) ,010309 optics ,Mathematics - Analysis of PDEs ,0103 physical sciences ,Convergence (routing) ,Arbitrary-precision arithmetic ,FOS: Mathematics ,Applied mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Finite set ,Mathematics ,Applied Mathematics ,010102 general mathematics ,Numerical Analysis (math.NA) ,Lipschitz continuity ,Physics::History of Physics ,Scheme (mathematics) ,Refracting telescope ,Physics - Optics ,Analysis of PDEs (math.AP) ,Optics (physics.optics) ,78A05, 35Q60, 97N40, 65J22 - Abstract
A numerical scheme is presented to solve the one source near field refractor problem to arbitrary precision and it is proved that the scheme terminates in a finite number of iterations. The convergence of the algorithm depends upon proving appropriate Lipschitz estimates for the refractor measure. The algorithm is presented in general terms and has independent interest., Comment: 23 pages, 1 figure, a reference was corrected
- Published
- 2021
17. Error estimates at low regularity of splitting schemes for NLS
- Author
-
Alexander Ostermann, Frédéric Rousset, and Katharina Schratz
- Subjects
Algebra and Number Theory ,Applied Mathematics ,Order (ring theory) ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Scheme (mathematics) ,Convergence (routing) ,FOS: Mathematics ,symbols ,Applied mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Nonlinear Schrödinger equation ,Mathematics - Abstract
We study a filtered Lie splitting scheme for the cubic nonlinear Schrödinger equation. We establish error estimates at low regularity by using discrete Bourgain spaces. This allows us to handle data in H s H^s with 0 > s > 1 0>s>1 overcoming the standard stability restriction to smooth Sobolev spaces with index s > 1 / 2 s>1/2 . More precisely, we prove convergence rates of order τ s / 2 \tau ^{s/2} in L 2 L^2 at this level of regularity.
- Published
- 2021
18. The discrete maximum principle and energy stability of a new second-order difference scheme for Allen-Cahn equations
- Author
-
Zengqiang Tan and Chengjian Zhang
- Subjects
Numerical Analysis ,Spacetime ,Applied Mathematics ,Finite difference ,Order (ring theory) ,010103 numerical & computational mathematics ,Sense (electronics) ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Maximum principle ,Energy stability ,Scheme (mathematics) ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
This paper deals with the discrete maximum principle and energy stability of a new difference scheme for solving Allen-Cahn equations. By combining the second-order central difference approximation in space and the Crank-Nicolson method with Newton linearized technique in time, a two-level linearized difference scheme for Allen-Cahn equations is derived, which can yield accuracy of order two both in time and space. Under appropriate conditions, the scheme is proved to be uniquely solvable and able to preserve the maximum principle and energy stability of the equations in the discrete sense. With some numerical experiments, the theoretical results and computational effectiveness of the scheme are further illustrated.
- Published
- 2021
19. COMPARISON OF KUMMER LOGARITHMIC TOPOLOGIES WITH CLASSICAL TOPOLOGIES
- Author
-
Heer Zhao
- Subjects
Pure mathematics ,Multiplicative group ,Group (mathematics) ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Flat topology ,Base (topology) ,01 natural sciences ,Cohomology ,14F20 (primary), 14A21 (secondary) ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Mathematics::K-Theory and Homology ,Scheme (mathematics) ,Mathematik ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Abelian group ,Algebraic number ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
We compare the Kummer flat (resp. Kummer etale) cohomology with the flat (resp. etale) cohomology with coefficients in smooth commutative group schemes, finite flat group schemes and the logarithmic multiplicative group of Kato. We will be particularly interested in the case of algebraic tori in the Kummer flat topology. We also make some computations for certain special cases of the base log scheme., Comment: Introduction has been rewritten. Theorem 1.8 has been slightly modified. The original Lemma 1.15 has been removed. The original Theorem 1.16 has been changed to Corollary 1.10. The original Theorem 1.24 (now Theorem 1.23) has been slightly modified. Errors and typos have been corrected. 30 pages. Accepted by J. Inst. Math. Jussieu. Might be slightly different from the journal version
- Published
- 2021
20. Optimal convergence of a second-order low-regularity integrator for the KdV equation
- Author
-
Xiaofei Zhao and Yifei Wu
- Subjects
Applied Mathematics ,General Mathematics ,Order (ring theory) ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Strang splitting ,Error analysis ,Scheme (mathematics) ,Integrator ,Convergence (routing) ,Applied mathematics ,0101 mathematics ,Korteweg–de Vries equation ,Mathematics - Abstract
In this paper, we establish the optimal convergence for a second-order exponential-type integrator from Hofmanová & Schratz (2017, An exponential-type integrator for the KdV equation. Numer. Math., 136, 1117–1137) for solving the Korteweg–de Vries equation with rough initial data. The scheme is explicit and efficient to implement. By rigorous error analysis, we show that the scheme provides second-order accuracy in $H^\gamma $ for initial data in $H^{\gamma +4}$ for any $\gamma \geq 0$, where the regularity requirement is lower than for classical methods. The result is confirmed by numerical experiments, and comparisons are made with the Strang splitting scheme.
- Published
- 2021
21. Some remarks on blueprints and $${\pmb {{\mathbb {F}}}_1}$$-schemes
- Author
-
Claudio Bartocci, Jean-Jacques Szczeciniarz, and Andrea Gentili
- Subjects
General Mathematics ,010102 general mathematics ,Symmetric monoidal category ,01 natural sciences ,Semiring ,Combinatorics ,Computational Theory and Mathematics ,Cover (topology) ,Scheme (mathematics) ,Monoid (category theory) ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
Over the past two decades several different approaches to defining a geometry over$${{\mathbb F}_1}$$F1have been proposed. In this paper, relying on Toën and Vaquié’s formalism (J.K-Theory 3: 437–500, 2009), we investigate a new category$${\mathsf {Sch}}_{\widetilde{{\mathsf B}}}$$SchB~of schemes admitting a Zariski cover by affine schemes relative to the category of blueprints introduced by Lorscheid (Adv. Math. 229: 1804–1846, 2012). A blueprint, which may be thought of as a pair consisting of a monoidMand a relation on the semiring$$M\otimes _{{{\mathbb F}_1}} {\mathbb N}$$M⊗F1N, is a monoid object in a certain symmetric monoidal category$${\mathsf B}$$B, which is shown to be complete, cocomplete, and closed. We prove that every$${\widetilde{{\mathsf B}}}$$B~-scheme$$\Sigma $$Σcan be associated, through adjunctions, with both a classical scheme$$\Sigma _{\mathbb Z}$$ΣZand a scheme$$\underline{\Sigma }$$Σ̲over$${{\mathbb F}_1}$$F1in the sense of Deitmar (in van der Geer G., Moonen B., Schoof R. (eds.) Progress in mathematics 239, Birkhäuser, Boston, 87–100, 2005), together with a natural transformation$$\Lambda :\Sigma _{\mathbb Z}\rightarrow \underline{\Sigma }\otimes _{{{\mathbb F}_1}}{\mathbb Z}$$Λ:ΣZ→Σ̲⊗F1Z. Furthermore, as an application, we show that the category of “$${{\mathbb F}_1}$$F1-schemes” defined by Connes and Consani in (Compos. Math. 146: 1383–1415, 2010) can be naturally merged with that of$${\widetilde{{\mathsf B}}}$$B~-schemes to obtain a larger category, whose objects we call “$${{\mathbb F}_1}$$F1-schemes with relations”.
- Published
- 2021
22. A third-order WENO scheme based on exponential polynomials for Hamilton-Jacobi equations
- Author
-
Hyoseon Yang, Chang Ho Kim, Youngsoo Ha, and Jungho Yoon
- Subjects
Numerical Analysis ,Smoothness (probability theory) ,Applied Mathematics ,Order of accuracy ,010103 numerical & computational mathematics ,01 natural sciences ,Hamilton–Jacobi equation ,Exponential polynomial ,010101 applied mathematics ,Computational Mathematics ,Third order ,Scheme (mathematics) ,Applied mathematics ,Finite difference operator ,0101 mathematics ,Interpolation ,Mathematics - Abstract
In this study, we provide a novel third-order weighted essentially non-oscillatory (WENO) method to solve Hamilton-Jacobi equations. The key idea is to incorporate exponential polynomials to construct numerical fluxes and smoothness indicators. First, the new smoothness indicators are designed by using the finite difference operator annihilating exponential polynomials such that singular regions can be distinguished from smooth regions more efficiently. Moreover, to construct numerical flux, we employ an interpolation method based on exponential polynomials which yields improved results around steep gradients. The proposed scheme retains the optimal order of accuracy (i.e., three) in smooth areas, even near the critical points. To illustrate the ability of the new scheme, some numerical results are provided along with comparisons with other WENO schemes.
- Published
- 2021
23. Goal-oriented mesh adaptation method for nonlinear problems including algebraic errors
- Author
-
Vít Dolejší, Filip Roskovec, and Ondřej Bartoš
- Subjects
Partial differential equation ,Adaptive algorithm ,Discretization ,MathematicsofComputing_NUMERICALANALYSIS ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Nonlinear system ,Consistency (database systems) ,Computational Theory and Mathematics ,Linearization ,Modeling and Simulation ,Scheme (mathematics) ,Applied mathematics ,0101 mathematics ,Algebraic number ,Mathematics - Abstract
We deal with the goal-oriented error estimates and mesh adaptation for nonlinear partial differential equations. The setting of the adjoint problem and the resulting estimates are not based on a differentiation of the primal problem but on a suitable linearization which guarantees the adjoint consistency of the numerical scheme. Furthermore, we develop an efficient adaptive algorithm which balances the errors arising from the discretization and the use of nonlinear as well as linear iterative solvers. Several numerical examples demonstrate the efficiency of this algorithm.
- Published
- 2021
24. On high-order schemes for tempered fractional partial differential equations
- Author
-
Linlin Bu and Cornelis W. Oosterlee
- Subjects
Numerical Analysis ,Partial differential equation ,Applied Mathematics ,010103 numerical & computational mathematics ,Space (mathematics) ,01 natural sciences ,Stability (probability) ,010101 applied mathematics ,Computational Mathematics ,Scheme (mathematics) ,Convergence (routing) ,Fractional diffusion ,Applied mathematics ,0101 mathematics ,High order ,Mathematics - Abstract
In this paper, we propose third-order semi-discretized schemes in space based on the tempered weighted and shifted Grunwald difference (tempered-WSGD) operators for the tempered fractional diffusion equation. We also show stability and convergence analysis for the fully discrete scheme based a Crank–Nicolson scheme in time. A third-order scheme for the tempered Black–Scholes equation is also proposed and tested numerically. Some numerical experiments are carried out to confirm accuracy and effectiveness of these proposed methods.
- Published
- 2021
25. Two-grid methods for nonlinear time fractional diffusion equations by L1-Galerkin FEM
- Author
-
Qingfeng Li, Yunqing Huang, Yang Wang, and Yanping Chen
- Subjects
Numerical Analysis ,General Computer Science ,Applied Mathematics ,010103 numerical & computational mathematics ,02 engineering and technology ,Grid ,01 natural sciences ,Finite element method ,Theoretical Computer Science ,Nonlinear system ,symbols.namesake ,Asymptotically optimal algorithm ,Modeling and Simulation ,Scheme (mathematics) ,0202 electrical engineering, electronic engineering, information engineering ,Fractional diffusion ,symbols ,Applied mathematics ,020201 artificial intelligence & image processing ,0101 mathematics ,Newton's method ,Linear equation ,Mathematics - Abstract
In this paper, two efficient two-grid algorithms with L 1 scheme are presented for solving two-dimensional nonlinear time fractional diffusion equations. The classical L 1 scheme is considered in the time direction, and the two-grid FE method is used to approximate spatial direction. To linearize the discrete equations, the Newton iteration approach and correction technique are applied. The two-grid algorithms reduce the solution of the nonlinear fractional problem on a fine grid to one linear equation on the same fine grid and an original nonlinear problem on a much coarser grid. As a result, our algorithms save total computational cost. Theoretical analysis shows that the two-grid algorithms maintain asymptotically optimal accuracy. Moreover, the numerical experiment presented further confirms the theoretical results.
- Published
- 2021
26. Framed sheaves on projective space and Quot schemes
- Author
-
Andrea T. Ricolfi, Alberto Cazzaniga, Cazzaniga A., and Ricolfi A.T.
- Subjects
General Mathematics ,010102 general mathematics ,01 natural sciences ,Tangent-obstruction theories ,Moduli space ,Deformation theory ,Combinatorics ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Character (mathematics) ,Quot scheme ,Hyperplane ,Scheme (mathematics) ,0103 physical sciences ,FOS: Mathematics ,Torsion (algebra) ,Projective space ,Sheaf ,010307 mathematical physics ,0101 mathematics ,Framed sheave ,Algebraic Geometry (math.AG) ,Moduli of sheave ,Quotient ,Mathematics - Abstract
We prove that, given integers $m\geq 3$, $r\geq 1$ and $n\geq 0$, the moduli space of torsion free sheaves on $\mathbb P^m$ with Chern character $(r,0,\ldots,0,-n)$ that are trivial along a hyperplane $D \subset \mathbb P^m$ is isomorphic to the Quot scheme $\mathrm{Quot}_{\mathbb A^m}(\mathscr O^{\oplus r},n)$ of $0$-dimensional length $n$ quotients of the free sheaf $\mathscr O^{\oplus r}$ on $\mathbb A^m$., Minor improvements. Final version
- Published
- 2021
27. A note on numerical solution of classical Darboux problem
- Author
-
Ram Jiwari and Kotapally Harish Kumar
- Subjects
Grid size ,General Mathematics ,Numerical technique ,010102 general mathematics ,General Engineering ,Integral form ,01 natural sciences ,Integral equation ,Chebyshev filter ,010101 applied mathematics ,Nonlinear system ,Wavelet ,Scheme (mathematics) ,Collocation method ,Applied mathematics ,0101 mathematics ,Hyperbolic partial differential equation ,Mathematics - Abstract
Recently, many authors studied the numerical solution of the classical Darboux problem in its integral form via a two-dimensional nonlinear Volterra-Fredholm integral equation. In the present article, a numerical technique based on the Chebyshev wavelet is proposed to solve the Darboux problem directly without converting into a nonlinear Volterra-Fredholm integral equation. The proposed technique is different from the techniques discussed in [1, 2, 4, 7, 11, 16, 17, 18, 19]. The proposed approach produces higher accuracy than its counterpart techniques. The proposed scheme illustrated with suitable examples to show the advantages in terms of its accuracy with lesser grid size.
- Published
- 2021
28. Convergence of SP-iteration for generalized nonexpansive mapping in Banach spaces
- Author
-
Javid Ali and Izhar Uddin
- Subjects
010101 applied mathematics ,Pure mathematics ,Class (set theory) ,Weak convergence ,Scheme (mathematics) ,Convergence (routing) ,Regular polygon ,Banach space ,010103 numerical & computational mathematics ,0101 mathematics ,Fixed point ,01 natural sciences ,Mathematics - Abstract
UDC 517.9 Phuengrattana and Suantai [J. Comput. and Appl. Math., 235, 3006 – 3014 (2011)] introduced an iteration scheme and they named this iteration as SP-iteration. In this paper, we study the convergence behaviour of SP-iteration scheme for the class of generalized nonexpansive mappings. One weak convergence theorem and two strong convergence theorems in uniformly convex Banach spaces are obtained. We also furnish a numerical example in support of our main result. In process, our results generalize and improve many existing results in the literature.
- Published
- 2021
29. A Global Newton Method for the Nonsmooth Vector Fields on Riemannian Manifolds
- Author
-
Fabrícia R. Oliveira and Fabiana R. de Oliveira
- Subjects
021103 operations research ,Control and Optimization ,Line search ,Applied Mathematics ,0211 other engineering and technologies ,010103 numerical & computational mathematics ,02 engineering and technology ,Management Science and Operations Research ,Riemannian manifold ,01 natural sciences ,symbols.namesake ,Singularity ,Scheme (mathematics) ,Theory of computation ,Convergence (routing) ,symbols ,Applied mathematics ,Vector field ,0101 mathematics ,Newton's method ,Mathematics - Abstract
This paper proposes and analyzes a globalized version of the Newton method for finding a singularity of the nonsmooth vector fields. Basically, the new method combines a version of nonsmooth Newton method with a nonmonotone line search strategy. The global convergence analysis of the proposed method as well as results on its rate are established under mild assumptions. Finally, numerical experiments illustrating the practical advantages of the proposed scheme are reported.
- Published
- 2021
30. Generalized Ratio-cum-Product Estimator for Finite Population Mean under Two-Phase Sampling Scheme
- Author
-
Gajendra K. Vishwakarma and Sayed Mohammed Zeeshan
- Subjects
Statistics and Probability ,Two phase sampling ,021103 operations research ,Population mean ,0211 other engineering and technologies ,Estimator ,02 engineering and technology ,01 natural sciences ,010104 statistics & probability ,Scheme (mathematics) ,Product (mathematics) ,Applied mathematics ,Point estimation ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
A method to lower the MSE of a proposed estimator relative to the MSE of the linear regression estimator under two-phase sampling scheme is developed. Estimators are developed to estimate the mean of the variate under study with the help of auxiliary variate (which are unknown but it can be accessed conveniently and economically). The mean square errors equations are obtained for the proposed estimators. In addition, optimal sample sizes are obtained under the given cost function. The comparison study has been done to set up conditions for which developed estimators are more effective than other estimators with novelty. The empirical study is also performed to supplement the claim that the developed estimators are more efficient.
- Published
- 2021
31. The Sub-Riemannian Limit of Curvatures for Curves and Surfaces and a Gauss–Bonnet Theorem in the Rototranslation Group
- Author
-
Wanzhen Li, Haiming Liu, Jiajing Miao, and Jianyun Guan
- Subjects
Article Subject ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Lie group ,02 engineering and technology ,01 natural sciences ,symbols.namesake ,Gauss–Bonnet theorem ,Scheme (mathematics) ,Euclidean geometry ,QA1-939 ,0202 electrical engineering, electronic engineering, information engineering ,Gaussian curvature ,symbols ,020201 artificial intelligence & image processing ,Mathematics::Differential Geometry ,Limit (mathematics) ,0101 mathematics ,Mathematics ,Geodesic curvature - Abstract
The rototranslation group ℛ T is the group comprising rotations and translations of the Euclidean plane which is a 3-dimensional Lie group. In this paper, we use the Riemannian approximation scheme to compute sub-Riemannian limits of the Gaussian curvature for a Euclidean C 2 -smooth surface in the rototranslation group away from characteristic points and signed geodesic curvature for Euclidean C 2 -smooth curves on surfaces. Based on these results, we obtain a Gauss–Bonnet theorem in the rototranslation group.
- Published
- 2021
32. A new linesearch iterative scheme for finding a common solution of split equilibrium and fixed point problems
- Author
-
Kanokwan Sitthithakerngkiet, Somyot Plubtieng, and Thidaporn Seangwattana
- Subjects
021103 operations research ,Applied Mathematics ,General Mathematics ,Numerical analysis ,0211 other engineering and technologies ,Hilbert space ,02 engineering and technology ,Fixed point ,01 natural sciences ,Dual (category theory) ,010101 applied mathematics ,symbols.namesake ,Scheme (mathematics) ,Convergence (routing) ,symbols ,Applied mathematics ,Equilibrium problem ,0101 mathematics ,Mathematics - Abstract
In this paper, we propose a new linesearch iterative scheme for finding a common solution of split equilibrium and fixed point problems without pseudomonotonicity of the bifunction f in a real Hilbert space. When setting the solution of dual equilibrium problem is nonempty, we obtain a strong convergence theorem which is generated by the iterative scheme. Moreover, we also receive a new linesearch iterative scheme for finding a solution of the split equilibrium problem in suitable assumptions, and report some numerical results to illustrate the convergence of the proposed scheme.
- Published
- 2021
33. An implicit scheme for simulation of free surface non-Newtonian fluid flows on dynamically adapted grids
- Author
-
Yuri V. Vassilevski, Kirill Nikitin, and Ruslan M. Yanbarisov
- Subjects
Numerical Analysis ,Materials science ,Viscoelastic fluid ,Mechanics ,01 natural sciences ,Non-Newtonian fluid ,010305 fluids & plasmas ,Physics::Fluid Dynamics ,010101 applied mathematics ,Modeling and Simulation ,Scheme (mathematics) ,Free surface ,0103 physical sciences ,Navier stokes ,0101 mathematics ,Mesh adaptation - Abstract
This work presents a new approach to modelling of free surface non-Newtonian (viscoplastic or viscoelastic) fluid flows on dynamically adapted octree grids. The numerical model is based on the implicit formulation and the staggered location of governing variables. We verify our model by comparing simulations with experimental and numerical results known from the literature.
- Published
- 2021
34. An Unfitted dG Scheme for Coupled Bulk-Surface PDEs on Complex Geometries
- Author
-
Sebastian Westerheide and Christian Engwer
- Subjects
010101 applied mathematics ,Surface (mathematics) ,Computational Mathematics ,Numerical Analysis ,Conservation law ,Computer science ,Applied Mathematics ,Scheme (mathematics) ,Mathematical analysis ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences - Abstract
The unfitted discontinuous Galerkin (UDG) method allows for conservative dG discretizations of partial differential equations (PDEs) based on cut cell meshes. It is hence particularly suitable for solving continuity equations on complex-shaped bulk domains. In this paper based on and extending the PhD thesis of the second author, we show how the method can be transferred to PDEs on curved surfaces. Motivated by a class of biological model problems comprising continuity equations on a static bulk domain and its surface, we propose a new UDG scheme for bulk-surface models. The method combines ideas of extending surface PDEs to higher-dimensional bulk domains with concepts of trace finite element methods. A particular focus is given to the necessary steps to retain discrete analogues to conservation laws of the discretized PDEs. A high degree of geometric flexibility is achieved by using a level set representation of the geometry. We present theoretical results to prove stability of the method and to investigate its conservation properties. Convergence is shown in an energy norm and numerical results show optimal convergence order in bulk/surface H 1 {H^{1}} - and L 2 {L^{2}} -norms.
- Published
- 2021
35. Unconditionally positivity preserving and energy dissipative schemes for Poisson–Nernst–Planck equations
- Author
-
Jie Shen and Jie Xu
- Subjects
Applied Mathematics ,Numerical analysis ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,Poisson distribution ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Maximum principle ,Scheme (mathematics) ,FOS: Mathematics ,symbols ,Dissipative system ,Applied mathematics ,Nernst equation ,Mathematics - Numerical Analysis ,0101 mathematics ,Planck ,65M12, 35K61, 35K55, 65Z05, 70F99 ,Energy (signal processing) ,Mathematics - Abstract
We develop a set of numerical schemes for the Poisson--Nernst--Planck equations. We prove that our schemes are mass conservative, uniquely solvable and keep positivity unconditionally. Furthermore, the first-order scheme is proven to be unconditionally energy dissipative. These properties hold for various spatial discretizations. Numerical results are presented to validate these properties. Moreover, numerical results indicate that the second-order scheme is also energy dissipative, and both the first- and second-order schemes preserve the maximum principle for cases where the equation satisfies the maximum principle., Comment: 24 pages, 10 figures
- Published
- 2021
36. A dual self-monitored reconstruction scheme on theTV-regularized inverse conductivity problem
- Author
-
Antonios Charalambopoulos, Drosos Kourounis, and Vanessa Markaki
- Subjects
Applied Mathematics ,Scheme (mathematics) ,010102 general mathematics ,Inverse ,010103 numerical & computational mathematics ,0101 mathematics ,Conductivity ,Topology ,01 natural sciences ,Mathematics ,Dual (category theory) - Abstract
Recently in Charalambopoulos et al. (2020), we presented a methodology aiming at reconstructing bounded total variation ($TV$) conductivities via a technique simulating the so-called half-quadratic minimization approach, encountered in Aubert & Kornprobst (2002, Mathematical Problems in Image Processing. New York, NY: Springer). The method belongs to a duality framework, in which the auxiliary function $\omega (x)$ was introduced, offering a tool for smoothing the members of the admissible set of conductivity profiles. The dual variable $\omega (x)$, in that approach, after every external update, served in the formation of an intermediate optimization scheme, concerning exclusively the sought conductivity $\alpha (x)$. In this work, we develop a novel investigation stemming from the previous approach, having though two different fundamental components. First, we do not detour herein the $BV$-assumption on the conductivity profile, which means that the functional under optimization contains the $TV$ of $\alpha (x)$ itself. Secondly, the auxiliary dual variable $\omega (x)$ and the conductivity $\alpha (x)$ acquire an equivalent role and concurrently, a parallel pacing in the minimization process. A common characteristic between these two approaches is that the function $\omega (x)$ is an indicator of the conductivity’s ‘jump’ set. A fortiori, this crucial property has been ameliorated herein, since the reciprocal role of the elements of the pair $(\alpha ,\omega )$ offers a self-monitoring structure very efficient to the minimization descent.
- Published
- 2021
37. Semistable abelian varieties and maximal torsion 1-crystalline submodules
- Author
-
Cody Gunton
- Subjects
Complement (group theory) ,Algebra and Number Theory ,Mathematics - Number Theory ,Group (mathematics) ,Mathematics::Number Theory ,010102 general mathematics ,Torsion 1-crystalline representation ,01 natural sciences ,Prime (order theory) ,Combinatorics ,Mathematics - Algebraic Geometry ,Monodromy ,Scheme (mathematics) ,0103 physical sciences ,FOS: Mathematics ,Torsion (algebra) ,Component (group theory) ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Abelian group ,Néron component group ,Algebraic Geometry (math.AG) ,Log 1-motive ,Mathematics - Abstract
Let $p$ be a prime, let $K$ be a discretely valued extension of $\mathbb{Q}_p$, and let $A_{K}$ be an abelian $K$-variety with semistable reduction. Extending work by Kim and Marshall from the case where $p>2$ and $K/\mathbb{Q}_p$ is unramified, we prove an $l=p$ complement of a Galois cohomological formula of Grothendieck for the $l$-primary part of the N��ron component group of $A_{K}$. Our proof involves constructing, for each $m\in \mathbb{Z}_{\geq 0}$, a finite flat $\mathscr{O}_K$-group scheme with generic fiber equal to the maximal 1-crystalline submodule of $A_{K}[p^{m}]$. As a corollary, we have a new proof of the Coleman-Iovita monodromy criterion for good reduction of abelian $K$-varieties., Final version
- Published
- 2021
38. A uniformly convergent scheme for two-parameter problems having layer behaviour
- Author
-
Devendra Kumar
- Subjects
Two parameter ,Applied Mathematics ,Uniform convergence ,Mathematical analysis ,010103 numerical & computational mathematics ,Sense (electronics) ,01 natural sciences ,Computer Science Applications ,010101 applied mathematics ,Boundary layer ,Computational Theory and Mathematics ,Scheme (mathematics) ,Collocation method ,0101 mathematics ,Layer (object-oriented design) ,Mathematics - Abstract
We present a numerical scheme for the solution of two-parameter singularly perturbed problems whose solution has multi-scale behaviour in the sense that there are small regions where the solution c...
- Published
- 2021
39. Modified Crank–Nicolson Scheme with Richardson Extrapolation for One-Dimensional Heat Equation
- Author
-
Feyisa Edosa Merga and Hailu Muleta Chemeda
- Subjects
Spacetime ,General Mathematics ,General Physics and Astronomy ,Richardson extrapolation ,010103 numerical & computational mathematics ,General Chemistry ,01 natural sciences ,010101 applied mathematics ,Scheme (mathematics) ,Convergence (routing) ,General Earth and Planetary Sciences ,Order (group theory) ,Crank–Nicolson method ,Applied mathematics ,Heat equation ,0101 mathematics ,General Agricultural and Biological Sciences ,Mathematics - Abstract
In this paper, Modified Crank–Nicolson method is combined with Richardson extrapolation to solve the 1D heat equation. The method is found to be unconditionally stable, consistent and hence the convergence of the method is guaranteed. The method is also found to be second-order convergent both in space and time variables. When combined with the Richardson extrapolation, the order of the method is improved from second-to fourth-order. To validate the proposed method, two model examples are considered and solved for different values of spatial and temporal step lengths.
- Published
- 2021
40. Dual finite frames for vector spaces over an arbitrary field with applications
- Author
-
Patricia Mariela Morillas
- Subjects
Fields ,Dual Frames ,Computer science ,Vector Spaces ,General Mathematics ,010102 general mathematics ,Matrix representation ,Hilbert space ,Field (mathematics) ,010103 numerical & computational mathematics ,Ultrametric Normed Vector Spaces ,Metric Vector Spaces ,Hilbert Spaces ,01 natural sciences ,Algebra ,symbols.namesake ,Scheme (mathematics) ,Metric (mathematics) ,QA1-939 ,symbols ,0101 mathematics ,Complex number ,Ultrametric space ,Mathematics ,Vector space - Abstract
In the present paper, we study frames for finite-dimensional vector spaces over an arbitrary field. We develop a theory of dual frames in order to obtain and study the different representations of the elements of the vector space provided by a frame. We relate the introduced theory with the classical one of dual frames for Hilbert spaces and apply it to study dual frames for three types of vector spaces: for vector spaces over conjugate closed subfields of the complex numbers (in particular, for cyclotomic fields), for metric vector spaces, and for ultrametric normed vector spaces over complete non-archimedean valued fields. Finally, we consider the matrix representation of operators using dual frames and its application to the solution of operators equations in a Petrov-Galerkin scheme.
- Published
- 2021
41. Approximation of functions of Lipschitz class and solution of Fokker-Planck equation by two-dimensional Legendre wavelet operational matrix
- Author
-
Shyam Lal and Priya Kumari
- Subjects
010304 chemical physics ,Lipschitz class ,Legendre wavelet ,Applied Mathematics ,010102 general mathematics ,Estimator ,General Chemistry ,01 natural sciences ,Operational matrix ,Planck's law ,Simple (abstract algebra) ,Scheme (mathematics) ,0103 physical sciences ,Applied mathematics ,Fokker–Planck equation ,0101 mathematics ,Mathematics - Abstract
In this paper, Lipschitz class of two-variables is considered. This is the genralization of well-known Lipschitz class of functions. A new estimator of functions belonging to generalized Lipschitz class has been obtained. Also, the solutions for the Fokker-Planck equations have been obtained for two different cases by two-dimensional Legendre wavelet operational matrix method. The approximated solutions of the time-and space-Fokker Planck equation have been compared with the exact solutions and the solutions obtained by homotopy perturbation method. The proposed scheme is simple, effective and suitable for the solution of Fokker-Planck equation.
- Published
- 2021
42. An Inertial Iterative Algorithm with Strong Convergence for Solving Modified Split Feasibility Problem in Banach Spaces
- Author
-
Yazheng Dang, Shufen Liu, and Huijuan Jia
- Subjects
Sequence ,021103 operations research ,Inertial frame of reference ,Article Subject ,Iterative method ,Spectral radius ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,Banach space ,Monotonic function ,02 engineering and technology ,01 natural sciences ,Scheme (mathematics) ,Convergence (routing) ,QA1-939 ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In this paper, we propose an iterative scheme for a special split feasibility problem with the maximal monotone operator and fixed-point problem in Banach spaces. The algorithm implements Halpern’s iteration with an inertial technique for the problem. Under some mild assumption of the monotonicity of the related mapping, we establish the strong convergence of the sequence generated by the algorithm which does not require the spectral radius of A T A. Finally, the numerical example is presented to demonstrate the efficiency of the algorithm.
- Published
- 2021
43. Alternating stationary iterative methods based on double splittings
- Author
-
Debasisha Mishra, Ashish Kumar Nandi, Nachiketa Mishra, and Vaibhav Shekhar
- Subjects
Iterative method ,Linear system ,010103 numerical & computational mathematics ,Mutually exclusive events ,01 natural sciences ,010101 applied mathematics ,Set (abstract data type) ,Computational Mathematics ,Matrix (mathematics) ,Computational Theory and Mathematics ,Simple (abstract algebra) ,Modeling and Simulation ,Scheme (mathematics) ,Convergence (routing) ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
Matrix double splitting iterations are simple in implementation while solving real non-singular (rectangular) linear systems. In this paper, we present two Alternating Double Splitting (ADS) schemes formulated by two double splittings and then alternating the respective iterations. The convergence conditions are then discussed along with comparative analysis. The set of double splittings used in each ADS scheme induces a preconditioned system which helps in showing the convergence of the ADS schemes. We also show that the classes of matrices for which one ADS scheme is better than the other, are mutually exclusive. Numerical experiments confirm that the proposed ADS schemes have several computational advantages over the existing methods. Though the problems are considered in the rectangular matrix settings, the same problems are even new in non-singular matrix settings.
- Published
- 2021
44. A leap-frog finite element method for wave propagation of Maxwell–Schrödinger equations with nonlocal effect in metamaterials
- Author
-
Y.M. Zhao, Z.Y. Wang, and Changhui Yao
- Subjects
Wave propagation ,Group (mathematics) ,Mathematical analysis ,Metamaterial ,010103 numerical & computational mathematics ,01 natural sciences ,Drude model ,Finite element method ,Schrödinger equation ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Computational Theory and Mathematics ,Modeling and Simulation ,Norm (mathematics) ,Scheme (mathematics) ,symbols ,0101 mathematics ,Mathematics - Abstract
In this paper, a novel system of Maxwell–Schrodinger equations with nonlocal effect in metamaterials is derived from the Drude model, hydrodynamical model and Schrodinger equation. A leap-frog finite element scheme, which can be solved one by one efficiently, is constructed by presenting a group of initial values. This scheme is proved to be stable conditionally in energy norm. It is confirmed that the error convergent rate is O ( τ 2 + h r ) by splitting the proof into three parts, where τ is the time step, h is the mesh size and r is the maximum total degree of polynomials in finite element spaces. Finally, some numerical results are given to verify the theories.
- Published
- 2021
45. On a novel full decoupling, linear, second‐order accurate, and unconditionally energy stable numerical scheme for the anisotropic phase‐field dendritic crystal growth model
- Author
-
Xiaofeng Yang
- Subjects
Physics ,Numerical Analysis ,Field (physics) ,Applied Mathematics ,Mathematical analysis ,General Engineering ,Phase (waves) ,Order (ring theory) ,Crystal growth ,010103 numerical & computational mathematics ,Decoupling (cosmology) ,01 natural sciences ,010101 applied mathematics ,Scheme (mathematics) ,0101 mathematics ,Anisotropy ,Energy (signal processing) - Published
- 2021
46. Approximation of fixed points for a class of mappings satisfying property (CSC) in Banach spaces
- Author
-
Junaid Ahmad, Muhammad Arshad, and Kifayat Ullah
- Subjects
Statistics and Probability ,Numerical Analysis ,Iterative and incremental development ,Pure mathematics ,Class (set theory) ,Property (philosophy) ,Current (mathematics) ,Applied Mathematics ,010102 general mathematics ,Regular polygon ,Banach space ,Fixed point ,01 natural sciences ,Computer Science Applications ,010101 applied mathematics ,Scheme (mathematics) ,Signal Processing ,0101 mathematics ,Analysis ,Information Systems ,Mathematics - Abstract
In this research article, we connect the $$K^{*}$$ iterative process with the class of mappings having property (CSC). We provide some weak and strong convergence theorems regarding the iterative scheme for mappings endowed with property (CSC) in uniformly convex Banach spaces. An example of mappings endowed with property (CSC) is provided which does not satisfy property (C). The $$K^{*}$$ iteration process and many other iterative processes are connected with this example to support the theoretical outcome. Our results improve and extend the corresponding well-known results of the current literature.
- Published
- 2021
47. On a Monte Carlo scheme for some linear stochastic partial differential equations
- Author
-
Akihiro Tanaka and Takuya Nakagawa
- Subjects
Statistics and Probability ,Stochastic partial differential equation ,Applied Mathematics ,Scheme (mathematics) ,010102 general mathematics ,Monte Carlo method ,Applied mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
The aim of this paper is to study the simulation of the expectation for the solution of linear stochastic partial differential equation driven by the space-time white noise with the bounded measurable coefficient and different boundary conditions. We first propose a Monte Carlo type method for the expectation of the solution of a linear stochastic partial differential equation and prove an upper bound for its weak rate error. In addition, we prove the central limit theorem for the proposed method in order to obtain confidence intervals for it. As an application, the Monte Carlo scheme applies to the stochastic heat equation with various boundary conditions, and we provide the result of numerical experiments which confirm the theoretical results in this paper.
- Published
- 2021
48. Stable Pair Invariants of Local Calabi–Yau 4-folds
- Author
-
Cao, Yalong, Kool, Martijn, Monavari, Sergej, Sub Fundamental Mathematics, Fundamental mathematics, Sub Fundamental Mathematics, and Fundamental mathematics
- Subjects
High Energy Physics - Theory ,Surface (mathematics) ,Pure mathematics ,General Mathematics ,FOS: Physical sciences ,Type (model theory) ,01 natural sciences ,Section (fiber bundle) ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Genus (mathematics) ,0103 physical sciences ,FOS: Mathematics ,Calabi–Yau manifold ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Mathematics ,Degree (graph theory) ,010102 general mathematics ,Mathematical Physics (math-ph) ,High Energy Physics - Theory (hep-th) ,Scheme (mathematics) ,Vertex (curve) ,010307 mathematical physics - Abstract
In 2008, Klemm-Pandharipande defined Gopakumar-Vafa type invariants of a Calabi-Yau 4-fold $X$ using Gromov-Witten theory. Recently, Cao-Maulik-Toda proposed a conjectural description of these invariants in terms of stable pair theory. When $X$ is the total space of the sum of two line bundles over a surface $S$, and all stable pairs are scheme theoretically supported on the zero section, we express stable pair invariants in terms of intersection numbers on Hilbert schemes of points on $S$. As an application, we obtain new verifications of the Cao-Maulik-Toda conjectures for low degree curve classes and find connections to Carlsson-Okounkov numbers. Some of our verifications involve genus zero Gopakumar-Vafa type invariants recently determined in the context of the log-local principle by Bousseau-Brini-van Garrel. Finally, using the vertex formalism, we provide a few more verifications of the Cao-Maulik-Toda conjectures when thickened curves contribute and also for the case of local $\mathbb{P}^3$., Published version, 23 pages
- Published
- 2021
49. Error Analysis of an Unconditionally Energy Stable Local Discontinuous Galerkin Scheme for the Cahn–Hilliard Equation with Concentration-Dependent Mobility
- Author
-
Fengna Yan and Yan Xu
- Subjects
Numerical Analysis ,Applied Mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Concentration dependent ,Discontinuous Galerkin method ,Error analysis ,Scheme (mathematics) ,Applied mathematics ,0101 mathematics ,Cahn–Hilliard equation ,Energy (signal processing) ,Mathematics - Abstract
In this paper, we mainly study the error analysis of an unconditionally energy stable local discontinuous Galerkin (LDG) scheme for the Cahn–Hilliard equation with concentration-dependent mobility. The time discretization is based on the invariant energy quadratization (IEQ) method. The fully discrete scheme leads to a linear algebraic system to solve at each time step. The main difficulty in the error estimates is the lack of control on some jump terms at cell boundaries in the LDG discretization. Special treatments are needed for the initial condition and the non-constant mobility term of the Cahn–Hilliard equation. For the analysis of the non-constant mobility term, we take full advantage of the semi-implicit time-discrete method and bound some numerical variables in L ∞ L^{\infty} -norm by the mathematical induction method. The optimal error results are obtained for the fully discrete scheme.
- Published
- 2021
50. A dual scheme for solving linear countable semi-infinite fractional programming problems
- Author
-
Do Sang Kim and Ta Quang Son
- Subjects
021103 operations research ,Control and Optimization ,Semi-infinite ,0211 other engineering and technologies ,Duality (optimization) ,Computational intelligence ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Dual (category theory) ,Linear-fractional programming ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Fractional programming ,Scheme (mathematics) ,Applied mathematics ,Countable set ,0101 mathematics ,Mathematics - Abstract
Using a dual method for solving linear fractional programming problems, we propose an approach to find optimal solutions of linear countable semi-infinite fractional programming problems via optimizing sequences. Duality theorems are established. A dual scheme for solving linear countable semi-infinite fractional problems is proposed. Examples are provided.
- Published
- 2021
Catalog
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.