Pub Date : 2024-03-04DOI: 10.4310/cms.2024.v22.n3.a2
Rodolfo Ruben Rosales, Benjamin Seibold, David Shirokoff, Dong Zhou
This paper studies the spatial manifestations of order reduction that occur when timestepping initial-boundary-value problems (IBVPs) with high-order Runge–Kutta methods. For such IBVPs, geometric structures arise that do not have an analog in ODE IVPs: boundary layers appear, induced by a mismatch between the approximation error in the interior and at the boundaries. To understand those boundary layers, an analysis of the modes of the numerical scheme is conducted, which explains under which circumstances boundary layers persist over many time steps. Based on this, two remedies to order reduction are studied: first, a new condition on the Butcher tableau, called weak stage order, that is compatible with diagonally implicit Runge–Kutta schemes; and second, the impact of modified boundary conditions on the boundary layer theory is analyzed.
{"title":"Spatial manifestations of order reduction in Runge–Kutta methods for initial boundary value problems","authors":"Rodolfo Ruben Rosales, Benjamin Seibold, David Shirokoff, Dong Zhou","doi":"10.4310/cms.2024.v22.n3.a2","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n3.a2","url":null,"abstract":"This paper studies the spatial manifestations of order reduction that occur when timestepping initial-boundary-value problems (IBVPs) with high-order Runge–Kutta methods. For such IBVPs, geometric structures arise that do not have an analog in ODE IVPs: boundary layers appear, induced by a mismatch between the approximation error in the interior and at the boundaries. To understand those boundary layers, an analysis of the modes of the numerical scheme is conducted, which explains under which circumstances boundary layers persist over many time steps. Based on this, two remedies to order reduction are studied: first, a new condition on the Butcher tableau, called weak stage order, that is compatible with diagonally implicit Runge–Kutta schemes; and second, the impact of modified boundary conditions on the boundary layer theory is analyzed.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"3 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140035306","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 : 2024-03-04DOI: 10.4310/cms.2024.v22.n3.a6
Yunrui Zheng
We consider the evolution of contact lines for thermal convection of viscous fluids in a two-dimensional open-top vessel. The domain is bounded above by a free moving boundary and otherwise by the solid wall of a vessel. The dynamics of the fluid are governed by the incompressible Boussinesq approximation under the influence of gravity, and the interface between fluid and air is under the effect of capillary forces. Here we develop global well posedness theory in the framework of nonlinear energy methods for the initial data sufficiently close to equilibrium. Moreover, the solutions decay to equilibrium at an exponential rate. Our methods are mainly based on the elliptic analysis near corners and a priori estimates of a geometric formulation of the Boussinesq equations.
{"title":"Stability of contact lines in 2D stationary Benard convection","authors":"Yunrui Zheng","doi":"10.4310/cms.2024.v22.n3.a6","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n3.a6","url":null,"abstract":"We consider the evolution of contact lines for thermal convection of viscous fluids in a two-dimensional open-top vessel. The domain is bounded above by a free moving boundary and otherwise by the solid wall of a vessel. The dynamics of the fluid are governed by the incompressible Boussinesq approximation under the influence of gravity, and the interface between fluid and air is under the effect of capillary forces. Here we develop global well posedness theory in the framework of nonlinear energy methods for the initial data sufficiently close to equilibrium. Moreover, the solutions decay to equilibrium at an exponential rate. Our methods are mainly based on the elliptic analysis near corners and <i>a priori</i> estimates of a geometric formulation of the Boussinesq equations.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"93 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140034481","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 : 2024-03-04DOI: 10.4310/cms.2024.v22.n3.a4
Yazhou Chen, Bin Huang, Xiaoding Shi
In this paper we study the initial-boundary-value problem for the barotropic compressible magnetohydrodynamic system with slip boundary conditions in three-dimensional exterior domain. We establish the global existence and uniqueness of classical solutions to the exterior domain problem with the regular initial data that are of small energy but possibly large oscillations with constant state as far field which could be either vacuum or nonvacuum. In particular, the initial density of such a classical solution is allowed to have large oscillations and can contain vacuum states. Moreover, the large-time behavior of the solution is also shown.
{"title":"Global strong solutions to the compressible magnetohydrodynamic equations with slip boundary conditions in a 3D exterior domain","authors":"Yazhou Chen, Bin Huang, Xiaoding Shi","doi":"10.4310/cms.2024.v22.n3.a4","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n3.a4","url":null,"abstract":"In this paper we study the initial-boundary-value problem for the barotropic compressible magnetohydrodynamic system with slip boundary conditions in three-dimensional exterior domain. We establish the global existence and uniqueness of classical solutions to the exterior domain problem with the regular initial data that are of small energy but possibly large oscillations with constant state as far field which could be either vacuum or nonvacuum. In particular, the initial density of such a classical solution is allowed to have large oscillations and can contain vacuum states. Moreover, the large-time behavior of the solution is also shown.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"2 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140035271","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 : 2024-03-04DOI: 10.4310/cms.2024.v22.n3.a3
Ziruo Cai, Jinglai Li, Xiaoqun Zhang
The Langevin algorithms are frequently used to sample the posterior distributions in Bayesian inference. In many practical problems, however, the posterior distributions often consist of non-differentiable components, posing challenges for the standard Langevin algorithms, as they require to evaluate the gradient of the energy function in each iteration. To this end, a popular remedy is to utilize the proximity operator, and as a result one needs to solve a proximity subproblem in each iteration. The conventional practice is to solve the subproblems accurately, which can be exceedingly expensive, as the subproblem needs to be solved in each iteration. We propose an approximate primal-dual fixed-point algorithm for solving the subproblem, which only seeks an approximate solution of the subproblem and therefore reduces the computational cost considerably. We provide theoretical analysis of the proposed method and also demonstrate its performance with numerical examples.
{"title":"Approximate primal-dual fixed-point based langevin algorithms for non-smooth convex potentials","authors":"Ziruo Cai, Jinglai Li, Xiaoqun Zhang","doi":"10.4310/cms.2024.v22.n3.a3","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n3.a3","url":null,"abstract":"The Langevin algorithms are frequently used to sample the posterior distributions in Bayesian inference. In many practical problems, however, the posterior distributions often consist of non-differentiable components, posing challenges for the standard Langevin algorithms, as they require to evaluate the gradient of the energy function in each iteration. To this end, a popular remedy is to utilize the proximity operator, and as a result one needs to solve a proximity subproblem in each iteration. The conventional practice is to solve the subproblems accurately, which can be exceedingly expensive, as the subproblem needs to be solved in each iteration. We propose an approximate primal-dual fixed-point algorithm for solving the subproblem, which only seeks an approximate solution of the subproblem and therefore reduces the computational cost considerably. We provide theoretical analysis of the proposed method and also demonstrate its performance with numerical examples.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"62 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140035265","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 : 2024-02-01DOI: 10.4310/cms.2024.v22.n2.a3
Hua Zhong
For the three-dimensional gas flow in vibrational nonequilibrium, the linearized stability of the planar rarefaction waves is obtained in this paper in terms of the rarefaction wave strength is small enough. The main feature of the problem is that the $L^2$-norm of the perturbations may grow in time.
{"title":"Linearized stability of planar rarefaction wave for 3D gas dynamics in thermal nonequilibrium","authors":"Hua Zhong","doi":"10.4310/cms.2024.v22.n2.a3","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n2.a3","url":null,"abstract":"For the three-dimensional gas flow in vibrational nonequilibrium, the linearized stability of the planar rarefaction waves is obtained in this paper in terms of the rarefaction wave strength is small enough. The main feature of the problem is that the $L^2$-norm of the perturbations may grow in time.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"73 2 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139668487","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 : 2024-02-01DOI: 10.4310/cms.2024.v22.n2.a7
Sirui Li, Chenchen Wang, Jie Xu
We consider the hydrodynamics for biaxial nematic phases described by a field of orthonormal frame, which can be derived from a molecular-theory-based tensor model. We prove the uniqueness of global weak solutions to the Cauchy problem of the frame hydrodynamics in dimension two. The proof is mainly based on the suitable weaker energy estimates within the Littlewood–Paley analysis. We take full advantage of the estimates of nonlinear terms with rotational derivatives on $SO(3)$, together with cancellation relations and dissipative structures of the biaxial frame system.
{"title":"Uniqueness of global weak solutions to the frame hydrodynamics for biaxial nematic phases in $mathbb{R}^2$","authors":"Sirui Li, Chenchen Wang, Jie Xu","doi":"10.4310/cms.2024.v22.n2.a7","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n2.a7","url":null,"abstract":"We consider the hydrodynamics for biaxial nematic phases described by a field of orthonormal frame, which can be derived from a molecular-theory-based tensor model. We prove the uniqueness of global weak solutions to the Cauchy problem of the frame hydrodynamics in dimension two. The proof is mainly based on the suitable weaker energy estimates within the Littlewood–Paley analysis. We take full advantage of the estimates of nonlinear terms with rotational derivatives on $SO(3)$, together with cancellation relations and dissipative structures of the biaxial frame system.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"11 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139668592","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 : 2024-02-01DOI: 10.4310/cms.2024.v22.n2.a6
Bijun Zuo
In this paper, we study the initial-boundary value problem for the full inhomogeneous, incompressible Navier–Stokes equations with temperature-dependent viscosity and heat conductivity coefficients. The viscosity coefficient may be degenerate in the sense that it may vanish in the region of absolutely zero temperature. Our main result is to prove the global existence of large weak solutions to such a system. The proof is based on a three-level approximate scheme, the Galerkin method, De Giorgi’s method, and appropriate compactness arguments.
{"title":"On global solutions to the inhomogeneous, incompressible Navier–Stokes equations with temperature-dependent coefficients","authors":"Bijun Zuo","doi":"10.4310/cms.2024.v22.n2.a6","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n2.a6","url":null,"abstract":"In this paper, we study the initial-boundary value problem for the full inhomogeneous, incompressible Navier–Stokes equations with temperature-dependent viscosity and heat conductivity coefficients. The viscosity coefficient may be degenerate in the sense that it may vanish in the region of absolutely zero temperature. Our main result is to prove the global existence of large weak solutions to such a system. The proof is based on a three-level approximate scheme, the Galerkin method, De Giorgi’s method, and appropriate compactness arguments.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"16 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139668607","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 : 2024-02-01DOI: 10.4310/cms.2024.v22.n2.a1
Senwei Liang, Liyao Lyu, Chunmei Wang, Haizhao Yang
We propose reproducing activation functions (RAFs) motivated by applied and computational harmonic analysis to improve deep learning accuracy for various applications ranging from computer vision to scientific computing. The idea is to employ several basic functions and their learnable linear combination to construct neuron-wise data-driven activation functions for each neuron. Armed with RAFs, neural networks (NNs) can reproduce traditional approximation tools and, therefore, approximate target functions with a smaller number of parameters than traditional NNs. As demonstrated by extensive numerical tests, the proposed RAFs can facilitate the convergence of deep learning optimization for a solution with higher accuracy than existing deep learning solvers for audio/image/video reconstruction, PDEs, and eigenvalue problems. With RAFs, the errors of audio/video reconstruction, PDEs, and eigenvalue problems are decreased by over 14%, 73%, 99%, respectively, compared with baseline, while the performance of image reconstruction increases by 58%. Numerically, in the NN training, RAFs can generate neural tangent kernels with better condition numbers than traditional activation functions, which provides a prospective for understanding the improved optimization convergence using the theory of neural tangent kernel. The code is available $href{https://github.com/LeungSamWai/Reproducing-Activation-Function}{https://github.com/LeungSamWai/Reproducing-Activation-Function}$.
{"title":"Reproducing activation function for deep learning","authors":"Senwei Liang, Liyao Lyu, Chunmei Wang, Haizhao Yang","doi":"10.4310/cms.2024.v22.n2.a1","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n2.a1","url":null,"abstract":"We propose reproducing activation functions (RAFs) motivated by applied and computational harmonic analysis to improve deep learning accuracy for various applications ranging from computer vision to scientific computing. The idea is to employ several basic functions and their learnable linear combination to construct neuron-wise data-driven activation functions for each neuron. Armed with RAFs, neural networks (NNs) can reproduce traditional approximation tools and, therefore, approximate target functions with a smaller number of parameters than traditional NNs. As demonstrated by extensive numerical tests, the proposed RAFs can facilitate the convergence of deep learning optimization for a solution with higher accuracy than existing deep learning solvers for audio/image/video reconstruction, PDEs, and eigenvalue problems. With RAFs, the errors of audio/video reconstruction, PDEs, and eigenvalue problems are decreased by over 14%, 73%, 99%, respectively, compared with baseline, while the performance of image reconstruction increases by 58%. Numerically, in the NN training, RAFs can generate neural tangent kernels with better condition numbers than traditional activation functions, which provides a prospective for understanding the improved optimization convergence using the theory of neural tangent kernel. The code is available $href{https://github.com/LeungSamWai/Reproducing-Activation-Function}{https://github.com/LeungSamWai/Reproducing-Activation-Function}$.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"12 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139668493","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 : 2024-02-01DOI: 10.4310/cms.2024.v22.n2.a8
Xiaohua Niu, Yang Xiang, Xiaodong Yan
In this paper, we study the weak solutions of a modified degenerate Cahn–Hilliard type model for surface diffusion. With degenerate phase-dependent diffusion mobility and additional stabilizing function, this model is able to give the correct sharp interface limit. We introduce a notion of weak solutions for the nonlinear model. The existence of such solutions is obtained by approximations of the proposed model with non-degenerate mobilities. We also employ this method to prove the existence of weak solutions to a related model where the chemical potential contains a nonlocal term originating from self-climb of dislocations in crystalline materials.
{"title":"Weak solutions for a modified degenerate Cahn–Hilliard model for surface diffusion","authors":"Xiaohua Niu, Yang Xiang, Xiaodong Yan","doi":"10.4310/cms.2024.v22.n2.a8","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n2.a8","url":null,"abstract":"In this paper, we study the weak solutions of a modified degenerate Cahn–Hilliard type model for surface diffusion. With degenerate phase-dependent diffusion mobility and additional stabilizing function, this model is able to give the correct sharp interface limit. We introduce a notion of weak solutions for the nonlinear model. The existence of such solutions is obtained by approximations of the proposed model with non-degenerate mobilities. We also employ this method to prove the existence of weak solutions to a related model where the chemical potential contains a nonlocal term originating from self-climb of dislocations in crystalline materials.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"71 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139668596","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 : 2024-02-01DOI: 10.4310/cms.2024.v22.n2.a9
Azzah Alshekhi, Jiu Ding, Noah Rhee
We propose a quadratic spline projection method that computes stationary densities of random maps with position-dependent probabilities. Using a key variation inequality for the corresponding Markov operator, we prove the norm convergence of the numerical scheme for a family of random maps consisting of the Lasota–Yorke class of interval maps. The numerical experimental results show that the new method improves the $L^1$-norm errors and increases the convergence rate greatly, compared with the previous operator-approximation-based numerical methods for random maps.
{"title":"A quadratic spline projection method for computing stationary densities of random maps","authors":"Azzah Alshekhi, Jiu Ding, Noah Rhee","doi":"10.4310/cms.2024.v22.n2.a9","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n2.a9","url":null,"abstract":"We propose a quadratic spline projection method that computes stationary densities of random maps with position-dependent probabilities. Using a key variation inequality for the corresponding Markov operator, we prove the norm convergence of the numerical scheme for a family of random maps consisting of the Lasota–Yorke class of interval maps. The numerical experimental results show that the new method improves the $L^1$-norm errors and increases the convergence rate greatly, compared with the previous operator-approximation-based numerical methods for random maps.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"183 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139668816","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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