Abstract In this paper, we introduce and study the concept of split monotone variational inclusion problem with multiple output sets (SMVIPMOS). We propose a new iterative scheme, which employs the viscosity approximation technique for approximating the solution of the SMVIPMOS with fixed point constraints of a nonexpansive mapping in real Hilbert spaces. The proposed method utilises the inertial technique for accelerating the speed of convergence and a self-adaptive step size so that its implementation does not require prior knowledge of the operator norm. Under mild conditions, we obtain a strong convergence result for the proposed algorithm and obtain a consequent result, which complements several existing results in the literature. Moreover, we apply our result to study the notions of split variational inequality problem with multiple output sets with fixed point constraints and split convex minimisation problem with multiple output sets with fixed point constraints in Hilbert spaces. Finally, we present some numerical experiments to demonstrate the implementability of our proposed method.
{"title":"On Split Monotone Variational Inclusion Problem with Multiple Output Sets with Fixed Point Constraints","authors":"V. A. Uzor, T. O. Alakoya, O. Mewomo","doi":"10.1515/cmam-2022-0199","DOIUrl":"https://doi.org/10.1515/cmam-2022-0199","url":null,"abstract":"Abstract In this paper, we introduce and study the concept of split monotone variational inclusion problem with multiple output sets (SMVIPMOS). We propose a new iterative scheme, which employs the viscosity approximation technique for approximating the solution of the SMVIPMOS with fixed point constraints of a nonexpansive mapping in real Hilbert spaces. The proposed method utilises the inertial technique for accelerating the speed of convergence and a self-adaptive step size so that its implementation does not require prior knowledge of the operator norm. Under mild conditions, we obtain a strong convergence result for the proposed algorithm and obtain a consequent result, which complements several existing results in the literature. Moreover, we apply our result to study the notions of split variational inequality problem with multiple output sets with fixed point constraints and split convex minimisation problem with multiple output sets with fixed point constraints in Hilbert spaces. Finally, we present some numerical experiments to demonstrate the implementability of our proposed method.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42365646","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 In this paper, the analytic solution of the delay fractional model is derived by the method of steps. The theoretical result implies that the regularity of the solution at s + {s^{+}} is better than that at 0 + {0^{+}} , where s is a constant time delay. The behavior of derivative discontinuity is also discussed. Then, improved regularity solution is obtained by the decomposition technique and a fitted L 1 {L1} numerical scheme is designed for it. For the case of initial singularity, the optimal convergence order is reached on uniform meshes when α ∈ [ 2 3 , 1 ) {alphain[frac{2}{3},1)} , α is the order of fractional derivative. Furthermore, an improved fitted L 1 {L1} method is proposed and the region of optimal convergence order is larger. For the case t > s {t>s} , stability and min { 2 α , 1 } {min{2alpha,1}} order convergence of the fitted L 1 {L1} scheme are deduced. At last, the numerical tests are carried out and confirm the theoretical result.
{"title":"The Tracking of Derivative Discontinuities for Delay Fractional Equations Based on Fitted L1 Method","authors":"Dakang Cen, Seakweng Vong","doi":"10.1515/cmam-2022-0231","DOIUrl":"https://doi.org/10.1515/cmam-2022-0231","url":null,"abstract":"Abstract In this paper, the analytic solution of the delay fractional model is derived by the method of steps. The theoretical result implies that the regularity of the solution at s + {s^{+}} is better than that at 0 + {0^{+}} , where s is a constant time delay. The behavior of derivative discontinuity is also discussed. Then, improved regularity solution is obtained by the decomposition technique and a fitted L 1 {L1} numerical scheme is designed for it. For the case of initial singularity, the optimal convergence order is reached on uniform meshes when α ∈ [ 2 3 , 1 ) {alphain[frac{2}{3},1)} , α is the order of fractional derivative. Furthermore, an improved fitted L 1 {L1} method is proposed and the region of optimal convergence order is larger. For the case t > s {t>s} , stability and min { 2 α , 1 } {min{2alpha,1}} order convergence of the fitted L 1 {L1} scheme are deduced. At last, the numerical tests are carried out and confirm the theoretical result.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45965201","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}
Ulrich Langer, Richard Löscher, Olaf Steinbach, Huidong Yang
Abstract We consider standard tracking-type, distributed elliptic optimal control problems with L2 L^{2} regularization, and their finite element discretization. We are investigating the L2 L^{2} error between the finite element approximation uϱh u_{varrho h} of the state uϱ u_{varrho} and the desired state (target) u¯ overline{u} in terms of the regularization parameter 𝜚 and the mesh size ℎ that leads to the optimal choice ϱ=h4 varrho=h^{4} . It turns out that, for this choice of the regularization parameter, we can devise simple Jacobi-like preconditioned MINRES or Bramble–Pasciak CG methods that allow us to solve the reduced discrete optimality system in asymptotically optimal complexity with respect to the arithmetical operations and memory demand. The theoretical results are confirmed by several benchmark problems with targets of various regularities including discontinuous targets.
{"title":"Robust Finite Element Discretization and Solvers for Distributed Elliptic Optimal Control Problems","authors":"Ulrich Langer, Richard Löscher, Olaf Steinbach, Huidong Yang","doi":"10.1515/cmam-2022-0138","DOIUrl":"https://doi.org/10.1515/cmam-2022-0138","url":null,"abstract":"Abstract We consider standard tracking-type, distributed elliptic optimal control problems with <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> L^{2} regularization, and their finite element discretization. We are investigating the <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> L^{2} error between the finite element approximation <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>u</m:mi> <m:mrow> <m:mi>ϱ</m:mi> <m:mo></m:mo> <m:mi>h</m:mi> </m:mrow> </m:msub> </m:math> u_{varrho h} of the state <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>u</m:mi> <m:mi>ϱ</m:mi> </m:msub> </m:math> u_{varrho} and the desired state (target) <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mover accent=\"true\"> <m:mi>u</m:mi> <m:mo>¯</m:mo> </m:mover> </m:math> overline{u} in terms of the regularization parameter 𝜚 and the mesh size ℎ that leads to the optimal choice <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>ϱ</m:mi> <m:mo>=</m:mo> <m:msup> <m:mi>h</m:mi> <m:mn>4</m:mn> </m:msup> </m:mrow> </m:math> varrho=h^{4} . It turns out that, for this choice of the regularization parameter, we can devise simple Jacobi-like preconditioned MINRES or Bramble–Pasciak CG methods that allow us to solve the reduced discrete optimality system in asymptotically optimal complexity with respect to the arithmetical operations and memory demand. The theoretical results are confirmed by several benchmark problems with targets of various regularities including discontinuous targets.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135285323","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 construct a symmetric interior penalty method for an elliptic distributed optimal control problem with pointwise state constraints on general polygonal domains. The resulting discrete problems are quadratic programs with simple box constraints that can be solved efficiently by a primal-dual active set algorithm. Both theoretical analysis and corroborating numerical results are presented.
{"title":"A Symmetric Interior Penalty Method for an Elliptic Distributed Optimal Control Problem with Pointwise State Constraints","authors":"S. C. Brenner, J. Gedicke, L. Sung","doi":"10.1515/cmam-2022-0148","DOIUrl":"https://doi.org/10.1515/cmam-2022-0148","url":null,"abstract":"Abstract We construct a symmetric interior penalty method for an elliptic distributed optimal control problem with pointwise state constraints on general polygonal domains. The resulting discrete problems are quadratic programs with simple box constraints that can be solved efficiently by a primal-dual active set algorithm. Both theoretical analysis and corroborating numerical results are presented.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47032516","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 In this article, quadrature rules for the efficient computation of the stiffness matrix for the fractional Laplacian in three dimensions are presented. These rules are based on the Duffy transformation, which is a common tool for singularity removal. Here, this transformation is adapted to the needs of the fractional Laplacian in three dimensions. The integrals resulting from this Duffy transformation are regular integrals over less-dimensional domains. We present bounds on the number of Gauss points to guarantee error estimates which are of the same order of magnitude as the finite element error. The methods presented in this article can easily be adapted to other singular double integrals in three dimensions with algebraic singularities.
{"title":"Fractional Laplacian – Quadrature Rules for Singular Double Integrals in 3D","authors":"Bernd Feist, Mario Bebendorf","doi":"10.1515/cmam-2022-0159","DOIUrl":"https://doi.org/10.1515/cmam-2022-0159","url":null,"abstract":"Abstract In this article, quadrature rules for the efficient computation of the stiffness matrix for the fractional Laplacian in three dimensions are presented. These rules are based on the Duffy transformation, which is a common tool for singularity removal. Here, this transformation is adapted to the needs of the fractional Laplacian in three dimensions. The integrals resulting from this Duffy transformation are regular integrals over less-dimensional domains. We present bounds on the number of Gauss points to guarantee error estimates which are of the same order of magnitude as the finite element error. The methods presented in this article can easily be adapted to other singular double integrals in three dimensions with algebraic singularities.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135582564","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}
Markus Faustmann, Ernst P. Stephan, David Wörgötter
Abstract For the singular integral definition of the fractional Laplacian, we consider an adaptive finite element method steered by two-level error indicators. For this algorithm, we show linear convergence in two and three space dimensions as well as convergence of the algorithm with optimal algebraic rates in 2D, when newest vertex bisection is employed for mesh refinement. A key step hereby is an equivalence of the nodal and Scott–Zhang interpolation operators in certain weighted L2 L^{2} -norms.
{"title":"Two-Level Error Estimation for the Integral Fractional Laplacian","authors":"Markus Faustmann, Ernst P. Stephan, David Wörgötter","doi":"10.1515/cmam-2022-0195","DOIUrl":"https://doi.org/10.1515/cmam-2022-0195","url":null,"abstract":"Abstract For the singular integral definition of the fractional Laplacian, we consider an adaptive finite element method steered by two-level error indicators. For this algorithm, we show linear convergence in two and three space dimensions as well as convergence of the algorithm with optimal algebraic rates in 2D, when newest vertex bisection is employed for mesh refinement. A key step hereby is an equivalence of the nodal and Scott–Zhang interpolation operators in certain weighted <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> L^{2} -norms.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135727304","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 : 2023-02-13DOI: 10.48550/arXiv.2302.06240
R. Eymard, David Maltese
Abstract We prove the convergence of an incremental projection numerical scheme for the time-dependent incompressible Navier–Stokes equations, without any regularity assumption on the weak solution. The velocity and the pressure are discretized in conforming spaces, whose compatibility is ensured by the existence of an interpolator for regular functions which preserves approximate divergence-free properties. Owing to a priori estimates, we get the existence and uniqueness of the discrete approximation. Compactness properties are then proved, relying on a Lions-like lemma for time translate estimates. It is then possible to show the convergence of the approximate solution to a weak solution of the problem. The construction of the interpolator is detailed in the case of the lowest degree Taylor–Hood finite element.
{"title":"Convergence of the Incremental Projection Method Using Conforming Approximations","authors":"R. Eymard, David Maltese","doi":"10.48550/arXiv.2302.06240","DOIUrl":"https://doi.org/10.48550/arXiv.2302.06240","url":null,"abstract":"Abstract We prove the convergence of an incremental projection numerical scheme for the time-dependent incompressible Navier–Stokes equations, without any regularity assumption on the weak solution. The velocity and the pressure are discretized in conforming spaces, whose compatibility is ensured by the existence of an interpolator for regular functions which preserves approximate divergence-free properties. Owing to a priori estimates, we get the existence and uniqueness of the discrete approximation. Compactness properties are then proved, relying on a Lions-like lemma for time translate estimates. It is then possible to show the convergence of the approximate solution to a weak solution of the problem. The construction of the interpolator is detailed in the case of the lowest degree Taylor–Hood finite element.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45629598","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 inverse problem for simultaneously identifying the space-dependent source term and the initial value in a time-fractional diffusion equation is studied in this paper. The simultaneous inversion is formulated into a system of two operator equations based on the Fourier method to the time-fractional diffusion equation. Under some suitable assumptions, the conditional stability of simultaneous inversion solutions is established, and the exponential Tikhonov regularization method is proposed to obtain the good approximations of simultaneous inversion solutions. Then the convergence estimations of inversion solutions are presented for a priori and a posteriori selections of regularization parameters. Finally, numerical experiments are conducted to illustrate effectiveness of the proposed method.
{"title":"Simultaneous Inversion of the Space-Dependent Source Term and the Initial Value in a Time-Fractional Diffusion Equation","authors":"Shuang Yu, Zewen Wang, Hongqi Yang","doi":"10.1515/cmam-2022-0058","DOIUrl":"https://doi.org/10.1515/cmam-2022-0058","url":null,"abstract":"Abstract The inverse problem for simultaneously identifying the space-dependent source term and the initial value in a time-fractional diffusion equation is studied in this paper. The simultaneous inversion is formulated into a system of two operator equations based on the Fourier method to the time-fractional diffusion equation. Under some suitable assumptions, the conditional stability of simultaneous inversion solutions is established, and the exponential Tikhonov regularization method is proposed to obtain the good approximations of simultaneous inversion solutions. Then the convergence estimations of inversion solutions are presented for a priori and a posteriori selections of regularization parameters. Finally, numerical experiments are conducted to illustrate effectiveness of the proposed method.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45910681","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 main objective of this article is to represent an efficient numerical approach based on shifted Jacobi polynomials to solve nonlinear stochastic differential equations driven by fractional Brownian motion. In this method, function approximation and operational matrices based on shifted Jacobi polynomials have been studied, which are further used with appropriate collocation points to reduce nonlinear stochastic differential equations driven by fractional Brownian motion into a system of algebraic equations. Newton’s method has been used to solve this nonlinear system of equations, and the desired approximate solution is achieved. Moreover, the error and convergence analysis of the presented method are also established in detail. Additionally, the applicability of the proposed method is demonstrated by solving some numerical examples.
{"title":"A Novel Study Based on Shifted Jacobi Polynomials to Find the Numerical Solutions of Nonlinear Stochastic Differential Equations Driven by Fractional Brownian Motion","authors":"P. K. Singh, S. Saha Ray","doi":"10.1515/cmam-2022-0187","DOIUrl":"https://doi.org/10.1515/cmam-2022-0187","url":null,"abstract":"Abstract The main objective of this article is to represent an efficient numerical approach based on shifted Jacobi polynomials to solve nonlinear stochastic differential equations driven by fractional Brownian motion. In this method, function approximation and operational matrices based on shifted Jacobi polynomials have been studied, which are further used with appropriate collocation points to reduce nonlinear stochastic differential equations driven by fractional Brownian motion into a system of algebraic equations. Newton’s method has been used to solve this nonlinear system of equations, and the desired approximate solution is achieved. Moreover, the error and convergence analysis of the presented method are also established in detail. Additionally, the applicability of the proposed method is demonstrated by solving some numerical examples.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46681375","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 : 2023-01-17DOI: 10.48550/arXiv.2301.06686
Hongxia Guo, G. Hu
Abstract We aim to analyze and calculate time-dependent acoustic wave scattering by a bounded obstacle and a locally perturbed non-selfintersecting curve. The scattering problem is equivalently reformulated as an initial-boundary value problem of the wave equation in a truncated bounded domain through a well-defined transparent boundary condition. Well-posedness and stability of the reduced problem are established. Numerically, we adopt the perfect matched layer (PML) scheme for simulating the propagation of perturbed waves. By designing a special absorbing medium in a semi-circular PML, we show the well-posedness and stability of the truncated initial-boundary value problem. Finally, we prove that the PML solution converges exponentially to the exact solution in the physical domain. Numerical results are reported to verify the exponential convergence with respect to absorbing medium parameters and thickness of the PML.
{"title":"Well-Posedness and Convergence Analysis of PML Method for Time-Dependent Acoustic Scattering Problems Over a Locally Rough Surface","authors":"Hongxia Guo, G. Hu","doi":"10.48550/arXiv.2301.06686","DOIUrl":"https://doi.org/10.48550/arXiv.2301.06686","url":null,"abstract":"Abstract We aim to analyze and calculate time-dependent acoustic wave scattering by a bounded obstacle and a locally perturbed non-selfintersecting curve. The scattering problem is equivalently reformulated as an initial-boundary value problem of the wave equation in a truncated bounded domain through a well-defined transparent boundary condition. Well-posedness and stability of the reduced problem are established. Numerically, we adopt the perfect matched layer (PML) scheme for simulating the propagation of perturbed waves. By designing a special absorbing medium in a semi-circular PML, we show the well-posedness and stability of the truncated initial-boundary value problem. Finally, we prove that the PML solution converges exponentially to the exact solution in the physical domain. Numerical results are reported to verify the exponential convergence with respect to absorbing medium parameters and thickness of the PML.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2023-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44600549","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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