Advances in Computational Mathematics最新文献

Efficient algorithms for Tucker decomposition via approximate matrix multiplication 基于近似矩阵乘法的高效塔克分解算法

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2025-04-22 DOI: 10.1007/s10444-025-10232-0

Maolin Che, Yimin Wei, Hong Yan

引用次数: 0

Computing the action of the matrix generating function of Bernoulli polynomials on a vector with an application to non-local boundary value problems 计算伯努利多项式的矩阵生成函数对向量的作用，并应用于非局部边值问题

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2025-04-10 DOI: 10.1007/s10444-025-10231-1

Lidia Aceto, Luca Gemignani

引用次数: 0

A discontinuous plane wave neural network method for Helmholtz equation and time-harmonic Maxwell’s equations 求解Helmholtz方程和时谐Maxwell方程的不连续平面波神经网络方法

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2025-04-07 DOI: 10.1007/s10444-025-10229-9

Long Yuan, Qiya Hu

{"title":"A discontinuous plane wave neural network method for Helmholtz equation and time-harmonic Maxwell’s equations","authors":"Long Yuan, Qiya Hu","doi":"10.1007/s10444-025-10229-9","DOIUrl":"10.1007/s10444-025-10229-9","url":null,"abstract":"<div>In this paper, we propose a discontinuous plane wave neural network (DPWNN) method with (hp-)refinement for approximately solving Helmholtz equation and time-harmonic Maxwell equations. In this method, we define a quadratic functional as in the plane wave least square (PWLS) method with (h-)refinement and introduce new discretization sets spanned by element-wise neural network functions with a single hidden layer, where the activation function on each element is chosen as a complex-valued exponential function like the plane wave function. The desired approximate solution is recursively generated by iteratively solving a quasi-minimization problem associated with the functional and the sets described above, which is defined by a sequence of approximate minimizers of the underlying residual functionals, where plane wave direction angles and activation coefficients are alternatively computed by iterative algorithms. For the proposed DPWNN method, the plane wave directions are adaptively determined in the iterative process, which is different from that in the standard PWLS method (where the plane wave directions are preliminarily given). Numerical experiments will confirm that this DPWNN method can generate approximate solutions with higher accuracy than the PWLS method.</div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 2","pages":""},"PeriodicalIF":1.7,"publicationDate":"2025-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143786662","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}

引用次数: 0

Low-rank exponential integrators for stiff differential Riccati equations 刚性Riccati微分方程的低秩指数积分器

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2025-04-02 DOI: 10.1007/s10444-025-10228-w

Hao Chen, Alfio Borzì

引用次数: 0

A quasi-boundary-value method for solving a nonlinear space-fractional backward diffusion problem 求解非线性空间分数阶后向扩散问题的拟边值方法

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2025-03-31 DOI: 10.1007/s10444-025-10230-2

Xiaoli Feng, Xiaoyu Yuan, Yun Zhang

引用次数: 0

Strong convergence of a fully discrete scheme for stochastic Burgers equation with fractional-type noise 具有分数型噪声的随机Burgers方程的完全离散格式的强收敛性

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2025-03-24 DOI: 10.1007/s10444-025-10227-x

Yibo Wang, Wanrong Cao

引用次数: 0

Product kernels are efficient and flexible tools for high-dimensional scattered data interpolation 积核是一种高效、灵活的高维离散数据插值工具

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2025-03-20 DOI: 10.1007/s10444-025-10226-y

Kristof Albrecht, Juliane Entzian, Armin Iske

{"title":"Product kernels are efficient and flexible tools for high-dimensional scattered data interpolation","authors":"Kristof Albrecht, Juliane Entzian, Armin Iske","doi":"10.1007/s10444-025-10226-y","DOIUrl":"10.1007/s10444-025-10226-y","url":null,"abstract":"<div>This work concerns the construction and characterization of product kernels for multivariate approximation from a finite set of discrete samples. To this end, we consider composing different component kernels, each acting on a low-dimensional Euclidean space. Due to Aronszajn (Trans. Am. Math. Soc. 68, 337–404 1950), the product of positive semi-definite kernel functions is again positive semi-definite, where, moreover, the corresponding native space is a particular instance of a tensor product, referred to as Hilbert tensor product. We first analyze the general problem of multivariate interpolation by product kernels. Then, we further investigate the tensor product structure, in particular for grid-like samples. We use this case to show that the product of positive definite kernel functions is again positive definite. Moreover, we develop an efficient computation scheme for the well-known Newton basis. Supporting numerical examples show the good performance of product kernels, especially for their flexibility.</div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 2","pages":""},"PeriodicalIF":1.7,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-025-10226-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143655244","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}

引用次数: 0

The Kolmogorov N-width for linear transport: exact representation and the influence of the data 线性输运的Kolmogorov n -宽度：精确表示和数据的影响

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2025-03-05 DOI: 10.1007/s10444-025-10224-0

Florian Arbes, Constantin Greif, Karsten Urban

{"title":"The Kolmogorov N-width for linear transport: exact representation and the influence of the data","authors":"Florian Arbes, Constantin Greif, Karsten Urban","doi":"10.1007/s10444-025-10224-0","DOIUrl":"10.1007/s10444-025-10224-0","url":null,"abstract":"<div>The Kolmogorov N-width describes the best possible error one can achieve by elements of an N-dimensional linear space. Its decay has extensively been studied in approximation theory and for the solution of partial differential equations (PDEs). Particular interest has occurred within model order reduction (MOR) of parameterized PDEs, e.g., by the reduced basis method (RBM). While it is known that the N-width decays exponentially fast (and thus admits efficient MOR) for certain problems, there are examples of the linear transport and the wave equation, where the decay rate deteriorates to (N^{-1/2}). On the other hand, it is widely accepted that a smooth parameter dependence admits a fast decay of the N-width. However, a detailed analysis of the influence of properties of the data (such as regularity or slope) on the rate of the N-width seems to be lacking. In this paper, we state that the optimal linear space is a direct sum of shift-isometric eigenspaces corresponding to the largest eigenvalues, yielding an exact representation of the N-width as their sum. For the linear transport problem, which is modeled by half-wave symmetric initial and boundary conditions g, we obtain such an optimal decomposition by sorted trigonometric functions with eigenvalues that match the Fourier coefficients of g. Further, for normalized g in the Sobolev space (H^r) of broken order (r>0), the sorted eigenfunctions give the sharp upper bound of the N-width, which is a reciprocal of a certain power sum. Yet, for ease, we also provide the decay ((pi N)^{-r}), obtained by the non-optimal space of ordering the trigonometric functions by frequency rather than by eigenvalue. Our theoretical investigations are complemented by numerical experiments which confirm the sharpness of our bounds and give additional quantitative insight.</div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 2","pages":""},"PeriodicalIF":1.7,"publicationDate":"2025-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-025-10224-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143546172","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}

引用次数: 0

On the recovery of two function-valued coefficients in the Helmholtz equation for inverse scattering problems via neural networks 反散射问题中Helmholtz方程中两个函数值系数的神经网络恢复

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2025-02-11 DOI: 10.1007/s10444-025-10225-z

Zehui Zhou

{"title":"On the recovery of two function-valued coefficients in the Helmholtz equation for inverse scattering problems via neural networks","authors":"Zehui Zhou","doi":"10.1007/s10444-025-10225-z","DOIUrl":"10.1007/s10444-025-10225-z","url":null,"abstract":"<div>Recently, deep neural networks (DNNs) have become powerful tools for solving inverse scattering problems. However, the approximation and generalization rates of DNNs for solving these problems remain largely under-explored. In this work, we introduce two types of combined DNNs (uncompressed and compressed) to reconstruct two function-valued coefficients in the Helmholtz equation for inverse scattering problems from the scattering data at two different frequencies. An analysis of the approximation and generalization capabilities of the proposed neural networks for simulating the regularized pseudo-inverses of the linearized forward operators in direct scattering problems is provided. The results show that, with sufficient training data and parameters, the proposed neural networks can effectively approximate the inverse process with desirable generalization. Preliminary numerical results show the feasibility of the proposed neural networks for recovering two types of isotropic inhomogeneous media. Furthermore, the trained neural network is capable of reconstructing the isotropic representation of certain types of anisotropic media.</div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-025-10225-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143379751","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}

引用次数: 0

On a non-uniform (alpha )-robust IMEX-L1 mixed FEM for time-fractional PIDEs 时间分数型PIDEs的非均匀(alpha ) -鲁棒IMEX-L1混合有限元分析

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2025-02-10 DOI: 10.1007/s10444-025-10221-3

Lok Pati Tripathi, Aditi Tomar, Amiya K. Pani

引用次数: 0