Advances in Computational Mathematics最新文献

{"title":"A generalized quasi-reversibility regularization method to determine a spatial source component in a time-fractional diffusion-wave equation","authors":"Ting Wei,&nbsp;Xinhang Li,&nbsp;Yuhua Luo","doi":"10.1007/s10444-026-10284-w","DOIUrl":"10.1007/s10444-026-10284-w","url":null,"abstract":"<div><p>This paper proposes a generalized quasi-reversibility regularization method for solving an inverse spatial source problem in a time-fractional diffusion-wave equation by using an additional final time observation. Based on the series formulation of the spatial source, we obtain an error estimate between the regularized source solution and the exact source component. For the quasi-reversibility regularized problem, we propose a fully discrete numerical algorithm by using the backward Euler convolution quadrature scheme for discreting the time-fractional Caputo derivative and the piecewise linear finite element scheme for dealing with the second-order elliptic operator. With the help of a semi-discrete regularized problem, the convergence estimate for the numerical source solution in a fully discrete regularized problem approaching to the exact source function is provided in terms of a certain reasonable assumptions to the given data and a suitable choice of the regularization parameter according to the noise level and stepsizes. The numerical results for six examples are given for one- and two-dimensional cases to show the efficiency of the proposed regularization method.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146145955","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Analysis of the geometric structure of neural networks and neural ODEs via morse functions","authors":"Christian Kuehn,&nbsp;Sara-Viola Kuntz","doi":"10.1007/s10444-025-10273-5","DOIUrl":"10.1007/s10444-025-10273-5","url":null,"abstract":"<div><p>Besides classical feed-forward neural networks such as multilayer perceptrons, also neural ordinary differential equations (neural ODEs) have gained particular interest in recent years. Neural ODEs can be interpreted as an infinite depth limit of feed-forward or residual neural networks. We study the input–output dynamics of finite and infinite depth neural networks with scalar output. In the finite-depth case, the input is a state associated with a finite number of nodes, which maps under multiple non-linear transformations to the state of one output node. In analogy, a neural ODE maps an affine linear transformation of the input to an affine linear transformation of its time-<i>T</i> map. We show that, depending on the specific structure of the network, the input–output map has different properties regarding the existence and regularity of critical points. These properties can be characterized via Morse functions, which are scalar functions where every critical point is non-degenerate. We prove that critical points cannot exist if the dimension of the hidden layer is monotonically decreasing or the dimension of the phase space is smaller than or equal to the input dimension. In the case that critical points exist, we classify their regularity depending on the specific architecture of the network. We show that, except for a Lebesgue measure zero set in the weight space, each critical point is non-degenerate if for finite-depth neural networks, the underlying graph has no bottleneck, and if for neural ODEs, the affine linear transformations used have full rank. For each type of architecture, the proven properties are comparable in the finite and infinite depth cases. The established theorems allow us to formulate results on universal embedding and universal approximation, i.e., on the exact and approximate representation of maps by neural networks and neural ODEs. Our dynamical systems viewpoint on the geometric structure of the input–output map provides a fundamental understanding of why certain architectures perform better than others.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-025-10273-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146145956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A comparative numerical study of spectral properties in isogeometric collocation and Galerkin methods for acoustic waves","authors":"Elena Zampieri","doi":"10.1007/s10444-026-10281-z","DOIUrl":"10.1007/s10444-026-10281-z","url":null,"abstract":"<div><p>We approximate the acoustic wave equation in two-dimensional regions using collocation and Galerkin isogeometric analysis (IGA) in space, coupled with implicit second-order Newmark schemes for time integration. We present a detailed numerical study that examines and compares the behavior of extreme eigenvalues and condition numbers of the mass and iteration IGA matrices, varying the polynomial degree <i>p</i>, mesh size <i>h</i>, regularity <i>k</i>, and the boundary conditions, that can be either Dirichlet or absorbing in order to simulate unbounded domains. We propose and validate numerically some conjectures related to the IGA collocation and Galerkin matrices for the wave equation with different types of boundary conditions, extending similar results that are known for the IGA Galerkin approximation, limitedly to the case of the Poisson problem with Dirichlet boundary conditions, and generalizing earlier results obtained within the framework of the collocation method. The results show that the spectral properties of the IGA collocation matrices are analogous and in most cases better than the corresponding IGA Galerkin discretization of the Poisson problem with Dirichlet or absorbing boundary conditions.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-026-10281-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146145954","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the asymptotically compatible energies of L1 and L1-SAV schemes for time-fractional Allen-Cahn model","authors":"Nan Liu,&nbsp;Haiqing Zhang,&nbsp;Hong-lin Liao","doi":"10.1007/s10444-026-10283-x","DOIUrl":"10.1007/s10444-026-10283-x","url":null,"abstract":"<div><p>Two types of asymptotically compatible energies are constructed for the variable-step L1 scheme for the time-fractional Allen-Cahn model with Caputo’s fractional derivative of order <span>(alpha in (0,1))</span>. An energy of the time-fractional Allen-Cahn model is called asymptotic compatibility if it approaches that of the classical Allen-Cahn model when the fractional order <span>(alpha rightarrow 1^-)</span>. By splitting the L1 formula into a local part and a nonlocal part, we construct two discrete gradient structures by exploring the logarithmic convexity and algebraic convexity of associated kernels, respectively. The nonlinear implicit L1 and linearly implicit L1-SAV schemes are then investigated for the time-fractional Allen-Cahn model, and new discrete energy dissipation laws are established under some mild step-ratio constraints. Extensive numerical tests are provided to examine the accuracy, energy behaviors, and solution behaviors of our numerical solvers in the long-time simulations. They suggest that the asymptotically compatible energy constructed from the algebraic convexity approximates the original energy faster than that constructed from the logarithmic convexity. It seems that both methods monotonically converge to the correct steady-state solution for any initial data.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2026-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146082991","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stable least-squares space-time boundary element methods for the wave equation","authors":"Daniel Hoonhout,&nbsp;Richard Löscher,&nbsp;Olaf Steinbach,&nbsp;Carolina Urzúa–Torres","doi":"10.1007/s10444-026-10282-y","DOIUrl":"10.1007/s10444-026-10282-y","url":null,"abstract":"<div><p>In this paper, we recast the variational formulation corresponding to the single layer boundary integral operator <span>(varvec{{text {textsf{V}}}})</span> for the wave equation as a minimization problem in <span>(varvec{L^2(Sigma )})</span>, where <span>(varvec{Sigma := partial Omega times (0,T)})</span> is the lateral boundary of the space-time domain <span>(varvec{Q:= Omega times (0,T)})</span>. For discretization, the minimization problem is restated as a mixed saddle point formulation. Unique solvability is established by combining conforming nested boundary element spaces for the mixed formulation such that the related bilinear form is discrete inf-sup stable. We analyze under which conditions the discrete inf-sup stability is satisfied, and moreover, we show that the mixed formulation provides a simple error indicator, which can be used for adaptivity. We present several numerical experiments showing the applicability of the method to different time-domain boundary integral formulations used in the literature.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2026-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-026-10282-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146056256","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An improved SNE with its applications in classification and visualization","authors":"Peilin Sun,&nbsp;Xu Qin","doi":"10.1007/s10444-025-10279-z","DOIUrl":"10.1007/s10444-025-10279-z","url":null,"abstract":"<div><p>Data embedding, as one of the dimension reduction methods in visualization and classification proposed in recent years, aims at maintaining the complete information of original data so that the difference between the original and the embedded data is imperceptible. Stochastic neighbor embedding(SNE) as a nonlinear manifold learning algorithm has received extensive attention. Considering the multimodality of actual data and the crowding problems in SNE, we propose an improved stochastic neighbor embedding based on spherical logistic distribution on three-dimensional Euclidean space, <span>(mathbf {SL_3})</span>-SNE. The technique is a variation of SNE that produces better clusterings by introducing spherical logistic distribution, which is more heavy-tailed than the normal distribution and is able to characterize the multimodality of data. Simulated and real experiment results show that the problems of crowding have been significantly alleviated, and the classification accuracy can be increased using the proposed algorithm in comparison to the existing t-SNE and vMF-SNE.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2026-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146048528","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Strong convergence of linear implicit virtual element methods for the nonlinear stochastic parabolic equation with multiplicative noise","authors":"Zhixin Liu,&nbsp;Minghui Song,&nbsp;Yuhang Zhang","doi":"10.1007/s10444-025-10280-6","DOIUrl":"10.1007/s10444-025-10280-6","url":null,"abstract":"<div><p>In this paper, we propose and analyze two novel fully discrete schemes for solving nonlinear stochastic parabolic equation with multiplicative noise. The conforming virtual element method is used for the spatial direction, and the semi-implicit Euler-Maruyama and two-step backward differentiation formula (BDF2)-Maruyama methods are used for the temporal direction, respectively. The proposed schemes offer flexibility in mesh processing and are capable of using general polygonal meshes. Additionally, both schemes are linear implicit methods that only require solving a linear system at each time step, significantly improving computational efficiency. We prove the mean-square stability of the two fully discrete schemes and derive strong approximation errors with optimal convergence rates in both time and space. As far as we know, this is the first attempt to solve time-dependent stochastic partial differential equations using the virtual element method. Finally, some numerical results are presented to validate the theoretical results and to demonstrate the efficiency of the numerical methods.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2026-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146048529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An optimal preconditioned discontinuous Galerkin method for biharmonic equation with (C^0)-reconstructed approximation","authors":"Ruo Li,&nbsp;Qicheng Liu,&nbsp;Fanyi Yang","doi":"10.1007/s10444-025-10277-1","DOIUrl":"10.1007/s10444-025-10277-1","url":null,"abstract":"<div><p>We present a high-order interior penalty discontinuous Galerkin method based on a reconstructed approximation to the biharmonic equation. The first contribution is that the approximation space is reconstructed from nodal values by solving a local least squares fitting problem per element. The numerical solution converges with optimal rates under error measurements. The second contribution is that an optimal preconditioned solver is proposed to solve the linear system efficiently that not only the condition number of the preconditioned system admits a uniform upper bound independent of the mesh size, but also the solver for the preconditioning matrix is of the optimal convergence rate. Such advantages for solvers to linear systems from penalty methods are seldom attained before.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2026-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146027173","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A low-rank neural representation of entropy solutions","authors":"Donsub Rim,&nbsp;Gerrit Welper","doi":"10.1007/s10444-025-10274-4","DOIUrl":"10.1007/s10444-025-10274-4","url":null,"abstract":"<div><p>We construct a new representation of entropy solutions to nonlinear scalar conservation laws with a smooth convex flux function in a single spatial dimension. The representation is a generalization of the method of characteristics and possesses a compositional form. While it is a nonlinear representation, the embedded dynamics of the solution in the time variable is linear. This representation is then discretized as a manifold of implicit neural representations where the feedforward neural network architecture has a low-rank structure. Finally, we show that the low-rank neural representation with a fixed number of layers and a small number of coefficients can approximate any entropy solution regardless of the complexity of the shock topology, while retaining the linearity of the embedded dynamics.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2026-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-025-10274-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145955150","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Two-grid domain decomposition methods for the coupled Navier–Stokes–Darcy model","authors":"Xinxin Sun,&nbsp;Guangzhi Du,&nbsp;Yuhong Zhang,&nbsp;Liyun Zuo","doi":"10.1007/s10444-025-10278-0","DOIUrl":"10.1007/s10444-025-10278-0","url":null,"abstract":"<div><p>In this paper, by combining the two-grid decoupled strategy and an existing domain decomposition method, two novel two-grid domain decomposition methods are constructed and analyzed for the coupled Navier–Stokes–Darcy model with Beavers–Joseph–Saffman interface condition. The proposed algorithms can decouple the Navier–Stokes–Darcy model into two independent Navier–Stokes and Darcy subsystems on both the coarse and fine grids, respectively, which can be solved in parallel with existing code and efficient solvers; hence, they could significantly enhance the computational efficiency. Numerical analysis indicates that both algorithms could reach the same convergence order as that of the standard Galerkin method with a proper configuration between the coarse grid size and the fine mesh size. Some numerical results are reported to show the main features of the two proposed algorithms.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2026-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145961683","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}