Advances in Computational Mathematics最新文献_第9页

Multi-fidelity learning of reduced order models for parabolic PDE constrained optimization 抛物型PDE约束优化降阶模型的多保真度学习

IF 2.1 3区数学

Advances in Computational Mathematics Pub Date : 2026-03-03 DOI: 10.1007/s10444-026-10296-6

Benedikt Klein, Mario Ohlberger

引用次数: 0

Structure-preserving Lift & Learn: Scientific machine learning for nonlinear conservative partial differential equations 保持结构的提升与学习：非线性保守偏微分方程的科学机器学习

IF 2.1 3区数学

Advances in Computational Mathematics Pub Date : 2026-03-03 DOI: 10.1007/s10444-026-10294-8

Harsh Sharma, Juan Diego Draxl Giannoni, Boris Kramer

{"title":"Structure-preserving Lift & Learn: Scientific machine learning for nonlinear conservative partial differential equations","authors":"Harsh Sharma, Juan Diego Draxl Giannoni, Boris Kramer","doi":"10.1007/s10444-026-10294-8","DOIUrl":"10.1007/s10444-026-10294-8","url":null,"abstract":"<div><p>This work presents <i>structure-preserving Lift & Learn</i>, a scientific machine learning method that employs lifting variable transformations to learn structure-preserving reduced-order models for nonlinear partial differential equations (PDEs) with conservation laws. We propose a hybrid learning approach based on a recently developed energy-quadratization strategy that uses knowledge of the nonlinearity at the PDE level to derive an equivalent quadratic lifted system with quadratic system energy. The lifted dynamics obtained via energy quadratization are linear in the old variables, making model learning very effective in the lifted setting. Based on the lifted quadratic PDE model form, the proposed method derives quadratic reduced terms analytically and then uses those derived terms to formulate a constrained optimization problem to learn the remaining linear reduced operators in a structure-preserving way. The proposed hybrid learning approach yields computationally efficient quadratic reduced-order models that respect the underlying physics of the high-dimensional problem. We demonstrate the generalizability of quadratic models learned via the proposed structure-preserving Lift & Learn method through three numerical examples: the one-dimensional wave equation with exponential nonlinearity, the two-dimensional sine-Gordon equation, and the two-dimensional Klein-Gordon-Zakharov equations. The numerical results show that the proposed learning approach is competitive with the state-of-the-art structure-preserving data-driven model reduction method in terms of both accuracy and computational efficiency.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 2","pages":""},"PeriodicalIF":2.1,"publicationDate":"2026-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147336373","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

Optimal compactly supported functions in Sobolev spaces Sobolev空间中的最优紧支持函数

IF 2.1 3区数学

Advances in Computational Mathematics Pub Date : 2026-03-03 DOI: 10.1007/s10444-026-10293-9

Robert Schaback

{"title":"Optimal compactly supported functions in Sobolev spaces","authors":"Robert Schaback","doi":"10.1007/s10444-026-10293-9","DOIUrl":"10.1007/s10444-026-10293-9","url":null,"abstract":"<div><p>This paper constructs unique compactly supported functions in Sobolev spaces that have minimal norm, maximal support, and maximal central value, under certain renormalizations. They may serve as optimized basis functions in interpolation or approximation, or as shape functions in meshless methods for PDE solving. Their norm is useful for proving upper bounds for convergence rates of interpolation in Sobolev spaces <span>(H_2^m(mathbb {R}^d))</span>, and this paper gives the correct rate <span>(m-d/2)</span> that arises as convergence like <span>(h^{m-d/2})</span> for interpolation at meshwidth <span>(hrightarrow 0)</span> or a blow-up like <span>(r^{-(m-d/2)})</span> for norms of compactly supported functions with support radius <span>(rrightarrow 0)</span>. In Hilbert spaces with infinitely smooth reproducing kernels, like Gaussians or inverse multiquadrics, there are no compactly supported functions at all, but in spaces with limited smoothness, compactly supported functions exist and can be optimized in the above way. The construction is described in Hilbert space via projections, and analytically via trace operators. Numerical examples are provided.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 2","pages":""},"PeriodicalIF":2.1,"publicationDate":"2026-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-026-10293-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147336375","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

Curve fitting on a quantum annealer for an advanced navigation method 一种先进导航方法的量子退火炉曲线拟合

IF 2.1 3区数学

Advances in Computational Mathematics Pub Date : 2026-02-23 DOI: 10.1007/s10444-026-10292-w

Philipp Isserstedt, Daniel Jaroszewski, Wolfgang Mergenthaler, Felix Paul, Bastian Harrach

{"title":"Curve fitting on a quantum annealer for an advanced navigation method","authors":"Philipp Isserstedt, Daniel Jaroszewski, Wolfgang Mergenthaler, Felix Paul, Bastian Harrach","doi":"10.1007/s10444-026-10292-w","DOIUrl":"10.1007/s10444-026-10292-w","url":null,"abstract":"<div><p>We explore the applicability of quantum annealing to the approximation task of curve fitting. To this end, we consider a function that shall approximate a given set of data points and is written as a finite linear combination of standardized functions, e.g., orthogonal polynomials. Consequently, the decision variables subject to optimization are the coefficients of that expansion. Although this task can be accomplished classically, it can also be formulated as a quadratic unconstrained binary optimization problem, which is suited to be solved with quantum annealing. Given that the size of the problem stays below a certain threshold, we find that quantum annealing yields comparable results to the classical solution. Regarding a real-world use case, we discuss the problem to find an optimized speed profile for a vessel using the framework of dynamic programming and outline how the aforementioned approximation task can be put into play. Similar to the curve fitting task, our findings indicate that quantum annealing is currently only feasible if the routing problem is modeled sufficiently small and sparse.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 2","pages":""},"PeriodicalIF":2.1,"publicationDate":"2026-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147287097","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

A posteriori error estimates for a (C^1) virtual element method applied to the thin plate vibration problem. 应用于薄板振动问题的$$C^1$$虚元法的后验误差估计。

IF 2.1 3区数学

Advances in Computational Mathematics Pub Date : 2026-02-23 DOI: 10.1007/s10444-026-10288-6

Franco Dassi, Andrés E. Rubiano, Iván Velásquez

引用次数: 0

Spectral approximation of a class of stochastic time-fractional evolution equations 一类随机时间分数进化方程的谱近似

IF 2.1 3区数学

Advances in Computational Mathematics Pub Date : 2026-02-19 DOI: 10.1007/s10444-026-10291-x

Simen Knutsen Furset

引用次数: 0

A direct parallel-in-time finite difference solver for quenching combustion problem 淬火燃烧问题的直接并行时域有限差分求解器

IF 2.1 3区数学

Advances in Computational Mathematics Pub Date : 2026-02-18 DOI: 10.1007/s10444-026-10286-8

Yufeng Xu, Ying Zhu, Desong Kong, Zhoushun Zheng

{"title":"A direct parallel-in-time finite difference solver for quenching combustion problem","authors":"Yufeng Xu, Ying Zhu, Desong Kong, Zhoushun Zheng","doi":"10.1007/s10444-026-10286-8","DOIUrl":"10.1007/s10444-026-10286-8","url":null,"abstract":"<div><p>In this paper, we explore a direct parallel-in-time method based on the diagonalization of the time-stepping matrix to solve the Kawarada equation, which arises from the quenching combustion process. Unlike the traditional time-marching finite difference method, where a nonlinear system may need to be solved at each time step once implicit schemes are employed, here, we form a large sparse linear system involving all unknown variables on the time-space domain, so that numerical solution can be handled all at once. Thanks to the non-uniform step sizes, the diagonalization technique can be introduced into a parallel-in-time process. Therefore, the dynamics near quenching time and quenching point can be described more accurately. We use a time window technique to release the deficiency of diagonalization, leading the accuracy and efficiency of the algorithm to be well-balanced. In numerical analysis, we propose a practical <i>PAMS</i>-framework to verify the effectiveness of the aforementioned method from four aspects including <i>Positivity, Asymptoticity, Monotonicity, and Stability</i>. Finally, several numerical experiments are conducted on 1D and 2D Kawarada equations, which demonstrate that the studied parallel-in-time method is reliable and highly efficient for quenching-type reaction diffusion equations. Meanwhile, computational time is reduced satisfactorily compared to the time-marching method.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 2","pages":""},"PeriodicalIF":2.1,"publicationDate":"2026-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146230774","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

Minimum curvature method for surface reconstruction 曲面重建的最小曲率法

IF 2.1 3区数学

Advances in Computational Mathematics Pub Date : 2026-02-18 DOI: 10.1007/s10444-026-10290-y

Hyeona Lim, Seongjai Kim

引用次数: 0

Chebyshev accelerating technique for solving generalized non-symmetric eigenvalue problems 求解广义非对称特征值问题的Chebyshev加速技术

IF 2.1 3区数学

Advances in Computational Mathematics Pub Date : 2026-02-12 DOI: 10.1007/s10444-026-10287-7

Lan Cheng, Cun-Qiang Miao, Yue-Yang Zhang

引用次数: 0

Space-time isogeometric method for a nonlocal parabolic problem 非局部抛物问题的时空等几何方法