Advances in Computational Mathematics最新文献

{"title":"Quasi-Monte Carlo methods for mixture distributions and approximated distributions via piecewise linear interpolation","authors":"Tiangang Cui,&nbsp;Josef Dick,&nbsp;Friedrich Pillichshammer","doi":"10.1007/s10444-025-10223-1","DOIUrl":"10.1007/s10444-025-10223-1","url":null,"abstract":"<div><p>We study numerical integration over bounded regions in <span>(mathbb {R}^s)</span>, <span>(s ge 1)</span>, with respect to some probability measure. We replace random sampling with quasi-Monte Carlo methods, where the underlying point set is derived from deterministic constructions which aim to fill the space more evenly than random points. Ordinarily, such quasi-Monte Carlo point sets are designed for the uniform measure, and the theory only works for product measures when a coordinate-wise transformation is applied. Going beyond this setting, we first consider the case where the target density is a mixture distribution where each term in the mixture comes from a product distribution. Next, we consider target densities which can be approximated with such mixture distributions. In order to be able to use an approximation of the target density, we require the approximation to be a sum of coordinate-wise products and that the approximation is positive everywhere (so that they can be re-scaled to probability density functions). We use tensor product hat function approximations for this purpose here, since a hat function approximation of a positive function is itself positive. We also study more complex algorithms, where we first approximate the target density with a general Gaussian mixture distribution and approximate this mixture distribution with an adaptive hat function approximation on rotated intervals. The Gaussian mixture approximation allows us (at least to some degree) to locate the essential parts of the target density, whereas the adaptive hat function approximation allows us to approximate the finer structure of the target density. We prove convergence rates for each of the integration techniques based on quasi-Monte Carlo sampling for integrands with bounded partial mixed derivatives. The employed algorithms are based on digital (<i>t</i>, <i>s</i>)-sequences over the finite field <span>(mathbb {F}_2)</span> and an inversion method. Numerical examples illustrate the performance of the algorithms for some target densities and integrands.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-025-10223-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143184707","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":"Parametric model order reduction for a wildland fire model via the shifted POD-based deep learning method","authors":"Shubhaditya Burela,&nbsp;Philipp Krah,&nbsp;Julius Reiss","doi":"10.1007/s10444-025-10220-4","DOIUrl":"10.1007/s10444-025-10220-4","url":null,"abstract":"<div><p>Parametric model order reduction techniques often struggle to accurately represent transport-dominated phenomena due to a slowly decaying Kolmogorov <i>n</i>-width. To address this challenge, we propose a non-intrusive, data-driven methodology that combines the shifted proper orthogonal decomposition (POD) with deep learning. Specifically, the shifted POD technique is utilized to derive a high-fidelity, low-dimensional model of the flow, which is subsequently utilized as input to a deep learning framework to forecast the flow dynamics under various temporal and parameter conditions. The efficacy of the proposed approach is demonstrated through the analysis of one- and two-dimensional wildland fire models with varying reaction rates, and its error is compared with the error of other similar methods. The results indicate that the proposed approach yields reliable results within the percent range, while also enabling rapid prediction of system states within seconds.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2025-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-025-10220-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143077609","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 scaling fractional asymptotical regularization method for linear inverse problems","authors":"Lele Yuan,&nbsp;Ye Zhang","doi":"10.1007/s10444-025-10222-2","DOIUrl":"10.1007/s10444-025-10222-2","url":null,"abstract":"<div><p>In this paper, we propose a Scaling Fractional Asymptotical Regularization (S-FAR) method for solving linear ill-posed operator equations in Hilbert spaces, inspired by the work of (2019 <i>Fract. Calc. Appl. Anal.</i> 22(3) 699-721). Our method is incorporated into the general framework of linear regularization and demonstrates that, under both Hölder and logarithmic source conditions, the S-FAR with fractional orders in the range (1, 2] offers accelerated convergence compared to comparable order optimal regularization methods. Additionally, we introduce a de-biasing strategy that significantly outperforms previous approaches, alongside a thresholding technique for achieving sparse solutions, which greatly enhances the accuracy of approximations. A variety of numerical examples, including one- and two-dimensional model problems, are provided to validate the accuracy and acceleration benefits of the FAR method.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2025-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143071908","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 difference finite element method based on nonconforming finite element methods for 3D elliptic problems","authors":"Jianjian Song,&nbsp;Dongwoo Sheen,&nbsp;Xinlong Feng,&nbsp;Yinnian He","doi":"10.1007/s10444-025-10219-x","DOIUrl":"10.1007/s10444-025-10219-x","url":null,"abstract":"<div><p>In this paper, a class of 3D elliptic equations is solved by using the combination of the finite difference method in one direction and nonconforming finite element methods in the other two directions. A finite-difference (FD) discretization based on <span>(P_1)</span>-element in the <i>z</i>-direction and a finite-element (FE) discretization based on <span>(P_1^{NC})</span>-nonconforming element in the (<i>x</i>, <i>y</i>)-plane are used to convert the 3D equation into a series of 2D ones. This paper analyzes the convergence of <span>(P_1^{NC})</span>-nonconforming finite element methods in the 2D elliptic equation and the error estimation of the <span>({H^1})</span>-norm of the DFE method. Finally, in this paper, the DFE method is tested on the 3D elliptic equation with the FD method based on the <span>(P_1)</span> element in the <i>z</i>-direction and the FE method based on the Crouzeix-Raviart element, the <span>(P_1)</span> linear element, the Park-Sheen element, and the <span>(Q_1)</span> bilinear element, respectively, in the (<i>x</i>, <i>y</i>)-plane.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2025-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-025-10219-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143027255","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 all-frequency stable integral system for Maxwell’s equations in 3-D penetrable media: continuous and discrete model analysis","authors":"Mahadevan Ganesh,&nbsp;Stuart C. Hawkins,&nbsp;Darko Volkov","doi":"10.1007/s10444-024-10218-4","DOIUrl":"10.1007/s10444-024-10218-4","url":null,"abstract":"<div><p>We introduce a new system of surface integral equations for Maxwell’s transmission problem in three dimensions (3-D). This system has two remarkable features, both of which we prove. First, it is well-posed at all frequencies. Second, the underlying linear operator has a uniformly bounded inverse as the frequency approaches zero, ensuring that there is no low-frequency breakdown. The system is derived from a formulation we introduced in our previous work, which required additional integral constraints to ensure well-posedness across all frequencies. In this study, we eliminate those constraints and demonstrate that our new self-adjoint, constraints-free linear system—expressed in the desirable form of an identity plus a compact weakly-singular operator—is stable for all frequencies. Furthermore, we propose and analyze a fully discrete numerical method for these systems and provide a proof of spectrally accurate convergence for the computational method. We also computationally demonstrate the high-order accuracy of the algorithm using benchmark scatterers with curved surfaces.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142986729","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 reduced-order model for advection-dominated problems based on the Radon Cumulative Distribution Transform","authors":"Tobias Long,&nbsp;Robert Barnett,&nbsp;Richard Jefferson-Loveday,&nbsp;Giovanni Stabile,&nbsp;Matteo Icardi","doi":"10.1007/s10444-024-10209-5","DOIUrl":"10.1007/s10444-024-10209-5","url":null,"abstract":"<div><p>Problems with dominant advection, discontinuities, travelling features, or shape variations are widespread in computational mechanics. However, classical linear model reduction and interpolation methods typically fail to reproduce even relatively small parameter variations, making the reduced models inefficient and inaccurate. This work proposes a model order reduction approach based on the Radon Cumulative Distribution Transform (RCDT). We demonstrate numerically that this non-linear transformation can overcome some limitations of standard proper orthogonal decomposition (POD) reconstructions and is capable of interpolating accurately some advection-dominated phenomena, although it may introduce artefacts due to the discrete forward and inverse transform. The method is tested on various test cases coming from both manufactured examples and fluid dynamics problems.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2025-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142913015","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":"On convergence of the generalized Lanczos trust-region method for trust-region subproblems","authors":"Bo Feng,&nbsp;Gang Wu","doi":"10.1007/s10444-024-10217-5","DOIUrl":"10.1007/s10444-024-10217-5","url":null,"abstract":"<div><p>The generalized Lanczos trust-region (GLTR) method is one of the most popular approaches for solving large-scale trust-region subproblem (TRS). In Jia and Wang, <i>SIAM J. Optim., 31, 887–914</i> 2021. Z. Jia et al. considered the convergence of this method and established some <i>a priori</i> error bounds on the residual and the Lagrange multiplier. In this paper, we revisit the convergence of the GLTR method and try to improve these bounds. First, we establish a sharper upper bound on the residual. Second, we present a <i>non-asymptotic</i> bound for the convergence of the Lagrange multiplier and define a factor that plays an important role in the convergence of the Lagrange multiplier. Third, we revisit the convergence of the Krylov subspace method for the cubic regularization variant of the trust-region subproblem and substantially improve the convergence result established in Jia et al., <i>SIAM J. Matrix Anal. Appl. 43 (2022), pp. 812–839</i> 2022 on the multiplier. Numerical experiments demonstrate the effectiveness of our theoretical results.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2025-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142912941","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":"Unfitted finite element method for the quad-curl interface problem","authors":"Hailong Guo,&nbsp;Mingyan Zhang,&nbsp;Qian Zhang,&nbsp;Zhimin Zhang","doi":"10.1007/s10444-024-10213-9","DOIUrl":"10.1007/s10444-024-10213-9","url":null,"abstract":"<div><p>In this paper, we introduce a novel unfitted finite element method to solve the quad-curl interface problem. We adapt Nitsche’s method for <span>({operatorname {curl}}{operatorname {curl}})</span>-conforming elements and double the degrees of freedom on interface elements. To ensure stability, we incorporate ghost penalty terms and a discrete divergence-free term. We establish the well-posedness of our method and demonstrate an optimal error bound in the discrete energy norm. We also analyze the stiffness matrix’s condition number. Our numerical tests back up our theory on convergence rates and condition numbers.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142888189","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 nonsingular-kernel Dirichlet-to-Dirichlet mapping method for the exterior Stokes problem","authors":"Xiaojuan Liu,&nbsp;Maojun Li,&nbsp;Tao Yin,&nbsp;Shangyou Zhang","doi":"10.1007/s10444-024-10216-6","DOIUrl":"10.1007/s10444-024-10216-6","url":null,"abstract":"<div><p>This paper studies the finite element method for solving the exterior Stokes problem in two dimensions. A nonlocal boundary condition is defined using a nonsingular-kernel Dirichlet-to-Dirichlet (DtD) mapping, which maps the Dirichlet data on an interior circle to the Dirichlet data on another circular artificial boundary based on the Poisson integral formula of the Stokes problem. The truncated problem is then solved using the MINI-element method and a simple DtD iteration strategy, resulting into a sequence of unique and geometrically (<i>h</i>- independent) convergent solutions. The quasi-optimal error estimate is proved for the iterative solution at the end of the iteration process. Numerical experiments are presented to demonstrate the accuracy and efficiency of the proposed method.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142841447","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":"Discretisation of an Oldroyd-B viscoelastic fluid flow using a Lie derivative formulation","authors":"Ben S. Ashby,&nbsp;Tristan Pryer","doi":"10.1007/s10444-024-10211-x","DOIUrl":"10.1007/s10444-024-10211-x","url":null,"abstract":"<div><p>In this article, we present a numerical method for the Stokes flow of an Oldroyd-B fluid. The viscoelastic stress evolves according to a constitutive law formulated in terms of the upper convected time derivative. A finite difference method is used to discretise along fluid trajectories to approximate the advection and deformation terms of the upper convected derivative in a simple, cheap and cohesive manner, as well as ensuring that the discrete conformation tensor is positive definite. A full implementation with coupling to the fluid flow is presented, along with a detailed discussion of the issues that arise with such schemes. We demonstrate the performance of this method with detailed numerical experiments in a lid-driven cavity setup. Numerical results are benchmarked against published data, and the method is shown to perform well in this challenging case.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-024-10211-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142826394","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}