Article
Multidisciplinary Sciences
Brek Meuris, Saad Qadeer, Panos Stinis
Summary: This paper presents an approach that combines deep neural networks with spectral methods to solve partial differential equations. The Deep Operator Network (DeepONet) is used to identify candidate functions for expanding the solution of PDEs. The proposed approach advances the state of the art and promotes synergy between traditional scientific computing and machine learning.
SCIENTIFIC REPORTS
(2023)
Article
Mathematics, Applied
Chaojun Zhang, Xiaoqun Wang, Zhijian He
Summary: The paper investigates two important sampling (IS) techniques in the Randomized Quasi-Monte Carlo (RQMC) setting, namely, Optimal Drift IS (ODIS) and Laplace IS (LapIS). It is shown that LapIS can also be obtained through an approximate optimization procedure based on Laplace approximation, and efficient RQMC-based IS procedures are developed.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Mathematics
Pierre Germain, Tristan Leger
Summary: We propose a unified approach to prove the Lp-Lq boundedness of spectral projectors, the resolvent of the Laplace-Beltrami operator, and its derivative on IH[d. The dependence of the implicit constant on p is shown to be sharp when p and q are in duality for spectral projectors. Partial results on the Lp-Lq boundedness of the Fourier extension operator are also provided. As applications, we prove smoothing estimates for the free Schrödinger equation on IH[d and a limiting absorption principle for the electromagnetic Schrödinger equation with small potentials.
JOURNAL OF FUNCTIONAL ANALYSIS
(2023)
Article
Mathematics, Interdisciplinary Applications
Beilei Liu, Huajie Chen, Genevieve Dusson, Jun Fang, Xingyu Gao
Summary: In this paper, we propose an adaptive planewave method for eigenvalue problems in electronic structure calculations. The method combines a priori convergence rates and accurate a posteriori error estimates, effectively updating the energy cut-off for planewave discretizations. It is error controllable for linear eigenvalue problems and shows promising potential for reducing the cost of iterations in self-consistent algorithms for nonlinear eigenvalue problems.
MULTISCALE MODELING & SIMULATION
(2022)
Article
Mathematics, Applied
Cedric Chauviere, Hacene Djellout
Summary: This study proposes a new method for numerical approximation of SDEs driven by white noise, utilizing Lagrange polynomials and solving a system of deterministic ODEs to compute coefficients of the polynomial basis. The approach demonstrates fast convergence, in addition to its novelty, and shows spectral convergence in numerical examples.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Zhen-Zhen Tao, Bing Sun
Summary: This article focuses on the Galerkin spectral approximation of an H-1-norm state-constrained optimal control problem governed by a fourth-order partial differential equation (PDE). Optimal conditions for both the original control problem and its spectral approximation problem were established to analyze the solution properties. A priori and a posteriori error estimates of the spectral approximation problem were detailed, with two numerical examples conducted to validate the theoretical analysis.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Daniel Potts, Michael Schmischke
Summary: The proposed method utilizes multivariate ANOVA decomposition to approximate high-dimensional periodic functions, showing advantages in achieving importance ranking on dimensions and dimension interactions in scattered data or black-box approximation scenarios.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2021)
Article
Materials Science, Multidisciplinary
Arijit Sinhababu, Anirban Bhattacharya, Sathyanarayana Ayyalasomayajula
Summary: The study implemented a high-resolution RPSM scheme based on RK4 temporal scheme for simulating pure metal solidification using Phase Field equations. It compared conservative and non-conservative forms of the equations, finding that the conservative RPSM scheme performed better. The RPSM-based Phase Field model accurately computed dendritic growth in an efficient manner.
COMPUTATIONAL MATERIALS SCIENCE
(2021)
Article
Computer Science, Interdisciplinary Applications
Jacqueline Wentz, Alireza Doostan
Summary: In this study, a method for quantifying uncertainty in high-dimensional PDE systems with random parameters is proposed. The method utilizes a generative model to approximate the coefficients of the solutions. The approach outperforms sparsity promoting methods at small sample sizes in the examined high-dimensional problems.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Automation & Control Systems
Peng Guo, Yijie Wu, Zhebin Shen, Haorong Zhang, Peng Zhang, Fei Lou
Summary: The paper proposes a novel smoothing method for line segments based on the real-time transformation of interpolation points (SSTI), which effectively optimizes the continuous line-segment path for CNC machining. The method has good adaptability to different types of line-segment paths and ensures machining accuracy without special requirements on the original line-segment path.
INTERNATIONAL JOURNAL OF ADVANCED MANUFACTURING TECHNOLOGY
(2022)
Article
Mathematics
Siddiqui Saima, Bingzhao Li, Samad Muhammad Adnan
Summary: This article highlights the importance of formulating Generalized Sampling Expansions (GSE) in the realm of quaternions, especially in the context of one-dimensional quaternion Fourier transform, and demonstrates its significant role in image processing.
Article
Mathematics, Applied
Juan Zhang, Lixiu Dong, Zhengru Zhang
Summary: In this paper, a second-order accurate numerical scheme using the scalar auxiliary variable (SAV) method and Fourier-spectral method in space is proposed and analyzed for the droplet liquid film coarsening model. The scheme is linear and efficient, and it exhibits unconditional energy stability due to the application of SAV approach. A rigorous error estimate is provided, showing that the scheme with Fourier-spectral method converges with order O (t² + hm), where t and h are time and space step sizes, respectively. Numerical experiments confirm the efficiency and accuracy of the proposed scheme, including tests of convergence, mass conservation, and energy decrease. The simulation of coarsening process with time is also observed.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics
Xiulian Shi
Summary: In this paper, a spectral approximation method is proposed and analyzed for the numerical solutions of fractional integro-differential equations with weakly kernels. The method transforms the original equations into an equivalent weakly singular Volterra integral equation and introduces smoothing transformations to eliminate singularity. The proposed method is investigated for spectral accuracy and validated with numerical examples.
JOURNAL OF MATHEMATICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Jingfu Peng, Pengsheng Huang, Ye Ding, Han Ding
Summary: This paper presents an analytical decoupled C3 continuous local path smoothing method for industrial robots, which smooths the tool position path in the reference frame and tool orientation in the rotation parametric space using quintic B-splines. By optimizing transition lengths, the method ensures smoother angular motion on the remaining linear segments, improving motion smoothness and tracking accuracy. The effectiveness of the method is validated through simulation and experiments.
ROBOTICS AND COMPUTER-INTEGRATED MANUFACTURING
(2021)
Article
Mathematics, Applied
Tom Chou, Sihong Shao, Mingtao Xia
Summary: This article introduces a novel adaptive spectral method for numerically solving partial differential equations in unbounded domains. It provides a numerical analysis of the method using generalized Hermite functions and explores parameter tuning for improving accuracy and efficiency.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Interdisciplinary Applications
Sergey Dolgov, Robert Scheichl
SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION
(2019)
Article
Mathematics, Applied
A. D. Gilbert, I. G. Graham, F. Y. Kuo, R. Scheichl, I. H. Sloan
NUMERISCHE MATHEMATIK
(2019)
Article
Computer Science, Interdisciplinary Applications
R. Butler, T. Dodwell, A. Reinarz, A. Sandhu, R. Scheichl, L. Seelinger
COMPUTER PHYSICS COMMUNICATIONS
(2020)
Article
Chemistry, Physical
Hideki Kobayashi, Paul B. Rohrbach, Robert Scheichl, Nigel B. Wilding, Robert L. Jack
JOURNAL OF CHEMICAL PHYSICS
(2019)
Article
Computer Science, Theory & Methods
Sergey Dolgov, Karim Anaya-Izquierdo, Colin Fox, Robert Scheichl
STATISTICS AND COMPUTING
(2020)
Article
Mathematics, Applied
Ivan G. Graham, Matthew J. Parkinson, Robert Scheichl
Summary: The study presents an analysis of multilevel Monte Carlo (MLMC) techniques for uncertainty quantification in the radiative transport equation with heterogeneous random fields as coefficients. Error analysis for the deterministic case is provided, along with error estimates explicit in coefficients and applicable to low regularity and jumps. The expected cost for computing a typical quantity of interest remains consistent with single sample estimates, and the multilevel version of the approach shows significant improvement over Monte Carlo in certain scenarios.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2021)
Article
Computer Science, Interdisciplinary Applications
J. Lang, R. Scheichl, D. Silvester
JOURNAL OF COMPUTATIONAL PHYSICS
(2020)
Article
Engineering, Mechanical
T. J. Dodwell, S. Kynaston, R. Butler, R. T. Haftka, Nam H. Kim, R. Scheichl
Summary: By adopting the MLMC framework, this research demonstrates that only a small number of expensive fine-scale computations are needed to accurately estimate the failure statistics of a composite structure. The results show significant computational gains with the introduction of the MLMC method and selective refinement for efficiently calculating structural failure probabilities.
PROBABILISTIC ENGINEERING MECHANICS
(2021)
Article
Multidisciplinary Sciences
T. J. Dodwell, L. R. Fleming, C. Buchanan, P. Kyvelou, G. Detommaso, P. D. Gosling, R. Scheichl, W. S. Kendall, L. Gardner, M. A. Girolami, C. J. Oates
Summary: The emergence of additive manufacture for metallic material enables components of near arbitrary complexity to be produced, but these components exhibit greater levels of variation in geometric and mechanical properties compared to standard components, posing a barrier to potential users. Researchers demonstrate that intrinsic variation in AM steel can be well described by a generative statistical model, allowing for prediction of design quality before manufacture by combining probabilistic mechanics and uncertainty quantification.
PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
(2021)
Article
Mathematics, Applied
Chupeng Ma, Robert Scheichl, Tim Dodwell
Summary: This paper studies the generalized finite element method (GFEM) for solving second order elliptic equations with rough coefficients. New optimal local approximation spaces for GFEMs are proposed, and the advantages of these new spaces are discussed. An efficient and easy-to-implement technique for generating the discrete A-harmonic spaces is also proposed, and numerical experiments are conducted to confirm the effectiveness of the new method.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2022)
Article
Physics, Fluids & Plasmas
Hideki Kobayashi, Paul B. Rohrbach, Robert Scheichl, Nigel B. Wilding, Robert L. Jack
Summary: The study used a two-level simulation method to analyze the critical point associated with demixing of binary hard sphere mixtures. The results showed a strong and unexpected dependence of the critical point on the size ratio between large and small particles, which is related to three-body effective interactions and the geometry of the underlying hard-sphere packings.
Proceedings Paper
Computer Science, Artificial Intelligence
Jakob Kruse, Gianluca Detommaso, Ullrich Koethe, Robert Scheichl
Summary: This work introduces a method for invertible neural architectures using coupling block designs, achieving an efficiently invertible block with dense, triangular Jacobian by recursively subdividing and coupling within resulting subsets. Through a hierarchical architecture, the method allows sampling from joint distributions and corresponding posteriors using a single invertible network, demonstrating its effectiveness in density estimation and Bayesian inference on various data sets.
THIRTY-FIFTH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE, THIRTY-THIRD CONFERENCE ON INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE AND THE ELEVENTH SYMPOSIUM ON EDUCATIONAL ADVANCES IN ARTIFICIAL INTELLIGENCE
(2021)
Article
Astronomy & Astrophysics
Karl Jansen, Eike H. Muller, Robert Scheichl
Article
Mathematics, Applied
Markus Bachmayr, Ivan G. Graham, Van Kien Nguyen, Robert Scheichl
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2020)
Article
Mathematics, Applied
T. J. Dodwell, C. Ketelsen, R. Scheichl, A. L. Teckentrup
Article
Mathematics, Applied
Xuesong Wang, Yao Cheng
Summary: The local discontinuous Galerkin (LDG) method on a Shishkin mesh is investigated for a one-dimensional singularly perturbed reaction-diffusion problem. Improved pointwise error estimates are derived based on the discrete Green's function in the regular and layer regions. The convergence rates of the pointwise error for both the LDG approximation to the solution and its derivative are analyzed, showing optimal rates in different domains. Moreover, optimal pointwise error estimates are established when the regular component of the exact solution belongs to the finite element space. Numerical experiments are conducted to validate the theoretical findings.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Arash Goligerdian, Mahmood Khaksar-e Oshagh
Summary: This paper presents a computational method for simulating more accurate models for population growth with immigration, using integral equations with a delay parameter. The method utilizes Legendre wavelets within the Galerkin scheme as an orthonormal basis and employs the composite Gauss-Legendre quadrature rule for computing integrals. An error bound analysis demonstrates the convergence rate of the method, and various numerical examples are provided to validate the efficiency and accuracy of the technique as well as the theoretical error estimate.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
A. Sreelakshmi, V. P. Shyaman, Ashish Awasthi
Summary: This paper focuses on constructing a lucid and utilitarian approach to solve linear and non-linear two-dimensional partial differential equations. Through testing, it is found that the proposed method is highly applicable and accurate, showing excellent performance in terms of cost-cutting and time efficiency.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Shujiang Tang
Summary: This paper investigates the impact of the structure of local smoothness indicators on the computational performance of the WENO-Z scheme. A new class of two-parameter local smoothness indicators is proposed, which combines the classical WENO-JS and WENO-UD5 schemes and appends the coefficients of higher-order terms. A new WENO scheme, WENO-NSLI, is constructed using the global smoothness indicators of WENO-UD5. Numerical experiments show that the new scheme achieves optimal accuracy and has higher resolution compared to WENO-JS, WENO-Z, and WENO-UD5.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Xue-Feng Duan, Yong-Shen Zhang, Qing-Wen Wang
Summary: This paper addresses a class of constrained tensor least squares problems in image restoration and proposes the alternating direction multiplier method (ADMM) to solve them. The convergence analysis of this method is presented. Numerical experiments show the feasibility and effectiveness of the ADMM method for solving constrained tensor least squares problems, and simulation experiments on image restoration are also conducted.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Wanying Mao, Qifeng Zhang, Dinghua Xu, Yinghong Xu
Summary: In this paper, we derive, analyze, and extensively test fourth-order compact difference schemes for the Rosenau equations in one and two dimensions. These schemes are applied under spatial periodic boundary conditions using the double reduction order method and bilinear compact operator. Our results show that these schemes satisfy mass and energy conservation laws and have unique solvability, unconditional convergence, and stability. The convergence order is four in space and two in time under the D infinity-norm. Several numerical examples are provided to support the theoretical findings.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Jeremy Chouchoulis, Jochen Schutz
Summary: This work presents an approximate family of implicit multiderivative Runge-Kutta time integrators for stiff initial value problems and investigates two different methods for computing higher order derivatives. Numerical results demonstrate that adding separate formulas yields better performance in dealing with stiff problems.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Hui Yang, Shengfeng Zhu
Summary: In this paper, shape optimization in incompressible Stokes flows is investigated based on the penalty method for the divergence free constraint at continuous level. Shape sensitivity analysis is performed, and numerical algorithms are introduced. An iterative penalty method is used for solving the penalized state and possible adjoint numerically, and it is shown to be more efficient than the standard mixed finite element method in 2D. Asymptotic convergence analysis and error estimates for finite element discretizations of both state and adjoint are provided, and numerical results demonstrate the effectiveness of the optimization algorithms.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Reza MohammadiArani, Mehdi Dehghan, Mostafa Abbaszadeh
Summary: Lattice Boltzmann method is a powerful solver for fluid flow, but it is challenging to use it to solve other partial differential equations. This paper challenges the LBM to solve the two-dimensional DKS equation by finding a suitable local equilibrium distribution function and proposes a modification for implementing boundary conditions in complex geometries.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Arijit Das, Prakrati Kushwah, Jitraj Saha, Mehakpreet Singh
Summary: A new volume and number consistent finite volume scheme is introduced for the numerical solution of a collisional nonlinear breakage problem. The scheme achieves number consistency by introducing a single weight function in the flux formulation. The proposed scheme is efficient and robust, allowing easy coupling with computational fluid dynamics softwares.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
H. Ait el Bhira, M. Kzaz, F. Maach, J. Zerouaoui
Summary: We present an asymptotic method for efficiently computing second-order telegraph equations with high-frequency extrinsic oscillations. The method uses asymptotic expansions in inverse powers of the oscillatory parameter and derives coefficients through either recursion or solving non-oscillatory problems, leading to improved performance as the oscillation frequency increases.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Hanen Boujlida, Kaouther Ismail, Khaled Omrani
Summary: This study investigates a high-order accuracy finite difference scheme for solving the one-dimensional extended Fisher-Kolmogorov (EFK) equation. A new compact difference scheme is proposed and the a priori estimates and unique solvability are discussed using the discrete energy method. The unconditional stability and convergence of the difference solution are proved. Numerical experiments demonstrate the accuracy and efficiency of the proposed technique.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Alexander Zlotnik, Timofey Lomonosov
Summary: This paper studies a three-level explicit in time higher-order vector compact scheme for solving initial-boundary value problems for the n-dimensional wave equation and acoustic wave equation with variable speed of sound. By using additional sought functions to approximate second order non-mixed spatial derivatives of the solution, new stability bounds and error bounds of orders 4 and 3.5 are rigorously proved. Generalizations to nonuniform meshes in space and time are also discussed.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Fengli Yin, Yayun Fu
Summary: This paper develops an explicit energy-preserving scheme for solving the coupled nonlinear Schrodinger equation by combining the Lie-group method and GSAV approaches. The proposed scheme is efficient, accurate, and can preserve the modified energy of the system.
APPLIED NUMERICAL MATHEMATICS
(2024)