SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 872-892, April 2024. Abstract. Partial differential equations (PDEs) with uncertain or random inputs have been considered in many studies of uncertainty quantification. In forward uncertainty quantification, one is interested in analyzing the stochastic response of the PDE subject to input uncertainty, which usually involves solving high-dimensional integrals of the PDE output over a sequence of stochastic variables. In practical computations, one typically needs to discretize the problem in several ways: approximating an infinite-dimensional input random field with a finite-dimensional random field, spatial discretization of the PDE using, e.g., finite elements, and approximating high-dimensional integrals using cubatures such as quasi–Monte Carlo methods. In this paper, we focus on the error resulting from dimension truncation of an input random field. We show how Taylor series can be used to derive theoretical dimension truncation rates for a wide class of problems and we provide a simple checklist of conditions that a parametric mathematical model needs to satisfy in order for our dimension truncation error bound to hold. Some of the novel features of our approach include that our results are applicable to nonaffine parametric operator equations, dimensionally truncated conforming finite element discretized solutions of parametric PDEs, and even compositions of PDE solutions with smooth nonlinear quantities of interest. As a specific application of our method, we derive an improved dimension truncation error bound for elliptic PDEs with lognormally parameterized diffusion coefficients. Numerical examples support our theoretical findings.
{"title":"Generalized Dimension Truncation Error Analysis for High-Dimensional Numerical Integration: Lognormal Setting and Beyond","authors":"Philipp A. Guth, Vesa Kaarnioja","doi":"10.1137/23m1593188","DOIUrl":"https://doi.org/10.1137/23m1593188","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 872-892, April 2024. <br/> Abstract. Partial differential equations (PDEs) with uncertain or random inputs have been considered in many studies of uncertainty quantification. In forward uncertainty quantification, one is interested in analyzing the stochastic response of the PDE subject to input uncertainty, which usually involves solving high-dimensional integrals of the PDE output over a sequence of stochastic variables. In practical computations, one typically needs to discretize the problem in several ways: approximating an infinite-dimensional input random field with a finite-dimensional random field, spatial discretization of the PDE using, e.g., finite elements, and approximating high-dimensional integrals using cubatures such as quasi–Monte Carlo methods. In this paper, we focus on the error resulting from dimension truncation of an input random field. We show how Taylor series can be used to derive theoretical dimension truncation rates for a wide class of problems and we provide a simple checklist of conditions that a parametric mathematical model needs to satisfy in order for our dimension truncation error bound to hold. Some of the novel features of our approach include that our results are applicable to nonaffine parametric operator equations, dimensionally truncated conforming finite element discretized solutions of parametric PDEs, and even compositions of PDE solutions with smooth nonlinear quantities of interest. As a specific application of our method, we derive an improved dimension truncation error bound for elliptic PDEs with lognormally parameterized diffusion coefficients. Numerical examples support our theoretical findings.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140310477","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 842-871, April 2024. Abstract. In this paper, we study the approximability of high-dimensional functions that appear, for example, in the context of many body expansions and high-dimensional model representation. Such functions, though high-dimensional, can be represented as finite sums of lower-dimensional functions. We will derive sampling inequalities for such functions, give explicit advice on the location of good sampling points, and show that such functions do not suffer from the curse of dimensionality.
{"title":"On the Approximability and Curse of Dimensionality of Certain Classes of High-Dimensional Functions","authors":"Christian Rieger, Holger Wendland","doi":"10.1137/22m1525193","DOIUrl":"https://doi.org/10.1137/22m1525193","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 842-871, April 2024. <br/> Abstract. In this paper, we study the approximability of high-dimensional functions that appear, for example, in the context of many body expansions and high-dimensional model representation. Such functions, though high-dimensional, can be represented as finite sums of lower-dimensional functions. We will derive sampling inequalities for such functions, give explicit advice on the location of good sampling points, and show that such functions do not suffer from the curse of dimensionality.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140192564","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 811-841, April 2024. Abstract. Physics informed neural networks (PINNs) require regularity of solutions of the underlying PDE to guarantee accurate approximation. Consequently, they may fail at approximating discontinuous solutions of PDEs such as nonlinear hyperbolic equations. To ameliorate this, we propose a novel variant of PINNs, termed as weak PINNs (wPINNs) for accurate approximation of entropy solutions of scalar conservation laws. wPINNs are based on approximating the solution of a min-max optimization problem for a residual, defined in terms of Kruzkhov entropies, to determine parameters for the neural networks approximating the entropy solution as well as test functions. We prove rigorous bounds on the error incurred by wPINNs and illustrate their performance through numerical experiments to demonstrate that wPINNs can approximate entropy solutions accurately.
{"title":"wPINNs: Weak Physics Informed Neural Networks for Approximating Entropy Solutions of Hyperbolic Conservation Laws","authors":"Tim De Ryck, Siddhartha Mishra, Roberto Molinaro","doi":"10.1137/22m1522504","DOIUrl":"https://doi.org/10.1137/22m1522504","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 811-841, April 2024. <br/> Abstract. Physics informed neural networks (PINNs) require regularity of solutions of the underlying PDE to guarantee accurate approximation. Consequently, they may fail at approximating discontinuous solutions of PDEs such as nonlinear hyperbolic equations. To ameliorate this, we propose a novel variant of PINNs, termed as weak PINNs (wPINNs) for accurate approximation of entropy solutions of scalar conservation laws. wPINNs are based on approximating the solution of a min-max optimization problem for a residual, defined in terms of Kruzkhov entropies, to determine parameters for the neural networks approximating the entropy solution as well as test functions. We prove rigorous bounds on the error incurred by wPINNs and illustrate their performance through numerical experiments to demonstrate that wPINNs can approximate entropy solutions accurately.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140135959","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 775-810, April 2024. Abstract. Cell average decomposition (CAD) plays a critical role in constructing bound-preserving (BP) high-order discontinuous Galerkin and finite volume methods for hyperbolic conservation laws. Seeking optimal CAD (OCAD) that attains the mildest BP Courant–Friedrichs–Lewy (CFL) condition is a fundamentally important yet difficult problem. The classic CAD, proposed in 2010 by Zhang and Shu using the Gauss–Lobatto quadrature, has been widely used over the past decade. Zhang and Shu only checked for the 1D [math] and [math] spaces that their classic CAD is optimal. However, we recently discovered that the classic CAD is generally not optimal for the multidimensional [math] and [math] spaces. Yet, it remained unknown for a decade what CAD is optimal for higher-degree polynomial spaces, especially in multiple dimensions. This paper presents the first systematical analysis and establishes the general theory on the OCAD problem, which lays a foundation for designing more efficient BP schemes. The analysis is very nontrivial and involves novel techniques from several branches of mathematics, including Carathéodory’s theorem from convex geometry, and the invariant theory of symmetric group in abstract algebra. Most notably, we discover that the OCAD problem is closely related to polynomial optimization of a positive linear functional on the positive polynomial cone, thereby establishing four useful criteria for examining the optimality of a feasible CAD. Using the established theory, we rigorously prove that the classic CAD is optimal for general 1D [math] spaces and general 2D [math] spaces of an arbitrary [math]. For the widely used 2D [math] spaces, the classic CAD is, however, not optimal, and we develop a generic approach to find out the genuine OCAD and propose a more practical quasi-optimal CAD, both of which provide much milder BP CFL conditions than the classic CAD yet require much fewer nodes. These findings notably improve the efficiency of general high-order BP methods for a large class of hyperbolic equations while requiring only a minor adjustment of the implementation code. The notable advantages in efficiency are further confirmed by numerical results.
{"title":"On Optimal Cell Average Decomposition for High-Order Bound-Preserving Schemes of Hyperbolic Conservation Laws","authors":"Shumo Cui, Shengrong Ding, Kailiang Wu","doi":"10.1137/23m1549365","DOIUrl":"https://doi.org/10.1137/23m1549365","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 775-810, April 2024. <br/> Abstract. Cell average decomposition (CAD) plays a critical role in constructing bound-preserving (BP) high-order discontinuous Galerkin and finite volume methods for hyperbolic conservation laws. Seeking optimal CAD (OCAD) that attains the mildest BP Courant–Friedrichs–Lewy (CFL) condition is a fundamentally important yet difficult problem. The classic CAD, proposed in 2010 by Zhang and Shu using the Gauss–Lobatto quadrature, has been widely used over the past decade. Zhang and Shu only checked for the 1D [math] and [math] spaces that their classic CAD is optimal. However, we recently discovered that the classic CAD is generally not optimal for the multidimensional [math] and [math] spaces. Yet, it remained unknown for a decade what CAD is optimal for higher-degree polynomial spaces, especially in multiple dimensions. This paper presents the first systematical analysis and establishes the general theory on the OCAD problem, which lays a foundation for designing more efficient BP schemes. The analysis is very nontrivial and involves novel techniques from several branches of mathematics, including Carathéodory’s theorem from convex geometry, and the invariant theory of symmetric group in abstract algebra. Most notably, we discover that the OCAD problem is closely related to polynomial optimization of a positive linear functional on the positive polynomial cone, thereby establishing four useful criteria for examining the optimality of a feasible CAD. Using the established theory, we rigorously prove that the classic CAD is optimal for general 1D [math] spaces and general 2D [math] spaces of an arbitrary [math]. For the widely used 2D [math] spaces, the classic CAD is, however, not optimal, and we develop a generic approach to find out the genuine OCAD and propose a more practical quasi-optimal CAD, both of which provide much milder BP CFL conditions than the classic CAD yet require much fewer nodes. These findings notably improve the efficiency of general high-order BP methods for a large class of hyperbolic equations while requiring only a minor adjustment of the implementation code. The notable advantages in efficiency are further confirmed by numerical results.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140096937","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 749-774, April 2024. Abstract. We consider a Beckmann formulation of an unbalanced optimal transport (UOT) problem. The [math]-convergence of this formulation of UOT to the corresponding optimal transport (OT) problem is established as the balancing parameter [math] goes to infinity. The discretization of the problem is further shown to be asymptotic preserving regarding the same limit, which ensures that a numerical method can be applied uniformly and the solutions converge to the one of the OT problem automatically. Particularly, there exists a critical value, which is independent of the mesh size, such that the discrete problem reduces to the discrete OT problem for [math] being larger than this critical value. The discrete problem is solved by a convergent primal-dual hybrid algorithm and the iterates for UOT are also shown to converge to that for OT. Finally, numerical experiments on shape deformation and partial color transfer are implemented to validate the theoretical convergence and the proposed numerical algorithm.
SIAM 数值分析期刊》第 62 卷第 2 期第 749-774 页，2024 年 4 月。 摘要我们考虑了不平衡最优输运（UOT）问题的贝克曼公式。当平衡参数[math]达到无穷大时，UOT 的[math]-收敛性被确定为相应的最优传输（OT）问题。进一步证明了问题的离散化对同一极限具有渐近保全性，这确保了数值方法可以均匀地应用，并且解自动收敛到 OT 问题的解。特别是存在一个与网格大小无关的临界值，当[math]大于该临界值时，离散问题会简化为离散加时赛问题。离散问题通过收敛的初等-二元混合算法求解，UOT 的迭代也证明收敛于 OT 的迭代。最后，对形状变形和部分颜色转移进行了数值实验，以验证理论收敛性和所提出的数值算法。
{"title":"On the Convergence of Continuous and Discrete Unbalanced Optimal Transport Models for 1-Wasserstein Distance","authors":"Zhe Xiong, Lei Li, Ya-Nan Zhu, Xiaoqun Zhang","doi":"10.1137/22m1520748","DOIUrl":"https://doi.org/10.1137/22m1520748","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 749-774, April 2024. <br/> Abstract. We consider a Beckmann formulation of an unbalanced optimal transport (UOT) problem. The [math]-convergence of this formulation of UOT to the corresponding optimal transport (OT) problem is established as the balancing parameter [math] goes to infinity. The discretization of the problem is further shown to be asymptotic preserving regarding the same limit, which ensures that a numerical method can be applied uniformly and the solutions converge to the one of the OT problem automatically. Particularly, there exists a critical value, which is independent of the mesh size, such that the discrete problem reduces to the discrete OT problem for [math] being larger than this critical value. The discrete problem is solved by a convergent primal-dual hybrid algorithm and the iterates for UOT are also shown to converge to that for OT. Finally, numerical experiments on shape deformation and partial color transfer are implemented to validate the theoretical convergence and the proposed numerical algorithm.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140043463","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 718-748, April 2024. Abstract. At the fully discrete setting, stability of the discontinuous Petrov–Galerkin (DPG) method with optimal test functions requires local test spaces that ensure the existence of Fortin operators. We construct such operators for [math] and [math] on simplices in any space dimension and arbitrary polynomial degree. The resulting test spaces are smaller than previously analyzed cases. For parameter-dependent norms, we achieve uniform boundedness by the inclusion of face bubble functions that are polynomials on faces and decay exponentially in the interior. As an example, we consider a canonical DPG setting for reaction-dominated diffusion. Our test spaces guarantee uniform stability and quasi-optimal convergence of the scheme. We present numerical experiments that illustrate the loss of stability and error control by the residual for small diffusion coefficient when using standard polynomial test spaces, whereas we observe uniform stability and error control with our construction.
{"title":"Robust DPG Test Spaces and Fortin Operators—The [math] and [math] Cases","authors":"Thomas Führer, Norbert Heuer","doi":"10.1137/23m1550360","DOIUrl":"https://doi.org/10.1137/23m1550360","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 718-748, April 2024. <br/> Abstract. At the fully discrete setting, stability of the discontinuous Petrov–Galerkin (DPG) method with optimal test functions requires local test spaces that ensure the existence of Fortin operators. We construct such operators for [math] and [math] on simplices in any space dimension and arbitrary polynomial degree. The resulting test spaces are smaller than previously analyzed cases. For parameter-dependent norms, we achieve uniform boundedness by the inclusion of face bubble functions that are polynomials on faces and decay exponentially in the interior. As an example, we consider a canonical DPG setting for reaction-dominated diffusion. Our test spaces guarantee uniform stability and quasi-optimal convergence of the scheme. We present numerical experiments that illustrate the loss of stability and error control by the residual for small diffusion coefficient when using standard polynomial test spaces, whereas we observe uniform stability and error control with our construction.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140043449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 667-691, April 2024. Abstract. We study the convergences of three projected Sobolev gradient flows to the ground state of the Gross–Pitaevskii eigenvalue problem. They are constructed as the gradient flows of the Gross–Pitaevskii energy functional with respect to the [math]-metric and two other equivalent metrics on [math], including the iterate-independent [math]-metric and the iterate-dependent [math]-metric. We first prove the energy dissipation property and the global convergence to a critical point of the Gross–Pitaevskii energy for the discrete-time [math] and [math]-gradient flow. We also prove local exponential convergence of all three schemes to the ground state.
{"title":"On the Convergence of Sobolev Gradient Flow for the Gross–Pitaevskii Eigenvalue Problem","authors":"Ziang Chen, Jianfeng Lu, Yulong Lu, Xiangxiong Zhang","doi":"10.1137/23m1552553","DOIUrl":"https://doi.org/10.1137/23m1552553","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 667-691, April 2024. <br/> Abstract. We study the convergences of three projected Sobolev gradient flows to the ground state of the Gross–Pitaevskii eigenvalue problem. They are constructed as the gradient flows of the Gross–Pitaevskii energy functional with respect to the [math]-metric and two other equivalent metrics on [math], including the iterate-independent [math]-metric and the iterate-dependent [math]-metric. We first prove the energy dissipation property and the global convergence to a critical point of the Gross–Pitaevskii energy for the discrete-time [math] and [math]-gradient flow. We also prove local exponential convergence of all three schemes to the ground state.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140032044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 692-717, April 2024. Abstract. We construct a right inverse of the trace operator [math] on the reference triangle [math] that maps suitable piecewise polynomial data on [math] into polynomials of the same degree and is bounded in all [math] norms with [math] and [math]. The analysis relies on new stability estimates for three classes of single edge operators. We then generalize the construction for [math]th-order normal derivatives, [math].
{"title":"Stable Lifting of Polynomial Traces on Triangles","authors":"Charles Parker, Endre Süli","doi":"10.1137/23m1564948","DOIUrl":"https://doi.org/10.1137/23m1564948","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 692-717, April 2024. <br/> Abstract. We construct a right inverse of the trace operator [math] on the reference triangle [math] that maps suitable piecewise polynomial data on [math] into polynomials of the same degree and is bounded in all [math] norms with [math] and [math]. The analysis relies on new stability estimates for three classes of single edge operators. We then generalize the construction for [math]th-order normal derivatives, [math].","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140032046","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 593-621, April 2024. Abstract. The aim of this paper is to carry out convergence analysis and algorithm implementation of a novel sample-wise backpropagation method for training a class of stochastic neural networks (SNNs). The preliminary discussion on such an SNN framework was first introduced in [Archibald et al., Discrete Contin. Dyn. Syst. Ser. S, 15 (2022), pp. 2807–2835]. The structure of the SNN is formulated as a discretization of a stochastic differential equation (SDE). A stochastic optimal control framework is introduced to model the training procedure, and a sample-wise approximation scheme for the adjoint backward SDE is applied to improve the efficiency of the stochastic optimal control solver, which is equivalent to the backpropagation for training the SNN. The convergence analysis is derived by introducing a novel joint conditional expectation for the gradient process. Under the convexity assumption, our result indicates that the number of SNN training steps should be proportional to the square of the number of layers in the convex optimization case. In the implementation of the sample-based SNN algorithm with the benchmark MNIST dataset, we adopt the convolution neural network (CNN) architecture and demonstrate that our sample-based SNN algorithm is more robust than the conventional CNN.
{"title":"Numerical Analysis for Convergence of a Sample-Wise Backpropagation Method for Training Stochastic Neural Networks","authors":"Richard Archibald, Feng Bao, Yanzhao Cao, Hui Sun","doi":"10.1137/22m1523765","DOIUrl":"https://doi.org/10.1137/22m1523765","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 593-621, April 2024. <br/> Abstract. The aim of this paper is to carry out convergence analysis and algorithm implementation of a novel sample-wise backpropagation method for training a class of stochastic neural networks (SNNs). The preliminary discussion on such an SNN framework was first introduced in [Archibald et al., Discrete Contin. Dyn. Syst. Ser. S, 15 (2022), pp. 2807–2835]. The structure of the SNN is formulated as a discretization of a stochastic differential equation (SDE). A stochastic optimal control framework is introduced to model the training procedure, and a sample-wise approximation scheme for the adjoint backward SDE is applied to improve the efficiency of the stochastic optimal control solver, which is equivalent to the backpropagation for training the SNN. The convergence analysis is derived by introducing a novel joint conditional expectation for the gradient process. Under the convexity assumption, our result indicates that the number of SNN training steps should be proportional to the square of the number of layers in the convex optimization case. In the implementation of the sample-based SNN algorithm with the benchmark MNIST dataset, we adopt the convolution neural network (CNN) architecture and demonstrate that our sample-based SNN algorithm is more robust than the conventional CNN.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140001008","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Alessandro Coclite, Giuseppe M. Coclite, Francesco Maddalena, Tiziano Politi
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 622-645, April 2024. Abstract. In this manuscript, an original numerical procedure for the nonlinear peridynamics on arbitrarily shaped two-dimensional (2D) closed manifolds is proposed. When dealing with non-parameterized 2D manifolds at the discrete scale, the problem of computing geodesic distances between two non-adjacent points arise. Here, a routing procedure is implemented for computing geodesic distances by reinterpreting the triangular computational mesh as a non-oriented graph, thus returning a suitable and general method. Moreover, the time integration of the peridynamics equation is demanded to a P-(EC)[math] formulation of the implicit [math]-Newmark scheme. The convergence of the overall proposed procedure is questioned and rigorously proved. Its abilities and limitations are analyzed by simulating the evolution of a 2D sphere. The performed numerical investigations are mainly motivated by the issues related to the insurgence of singularities in the evolution problem. The obtained results return an interesting picture of the role played by the nonlocal character of the integrodifferential equation in the intricate processes leading to the spontaneous formation of singularities in real materials.
{"title":"A Numerical Framework for Nonlinear Peridynamics on Two-Dimensional Manifolds Based on Implicit P-(EC)[math] Schemes","authors":"Alessandro Coclite, Giuseppe M. Coclite, Francesco Maddalena, Tiziano Politi","doi":"10.1137/22m1498942","DOIUrl":"https://doi.org/10.1137/22m1498942","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 622-645, April 2024. <br/> Abstract. In this manuscript, an original numerical procedure for the nonlinear peridynamics on arbitrarily shaped two-dimensional (2D) closed manifolds is proposed. When dealing with non-parameterized 2D manifolds at the discrete scale, the problem of computing geodesic distances between two non-adjacent points arise. Here, a routing procedure is implemented for computing geodesic distances by reinterpreting the triangular computational mesh as a non-oriented graph, thus returning a suitable and general method. Moreover, the time integration of the peridynamics equation is demanded to a P-(EC)[math] formulation of the implicit [math]-Newmark scheme. The convergence of the overall proposed procedure is questioned and rigorously proved. Its abilities and limitations are analyzed by simulating the evolution of a 2D sphere. The performed numerical investigations are mainly motivated by the issues related to the insurgence of singularities in the evolution problem. The obtained results return an interesting picture of the role played by the nonlocal character of the integrodifferential equation in the intricate processes leading to the spontaneous formation of singularities in real materials.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140015424","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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