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Generating high-quality geometry problems is both an important and challenging task in education. Compared to math word problems, geometry problems further emphasize multi-modal formats and the translation between informal and formal…

Artificial Intelligence · Computer Science 2025-06-04 Zhuoxuan Jiang , Tianyang Zhang , Peiyan Peng , Jing Chen , Yinong Xun , Haotian Zhang , Lichi Li , Yong Li , Shaohua Zhang

Trajectory optimization and posture generation are hard problems in robot locomotion, which can be non-convex and have multiple local optima. Progress on these problems is further hindered by a lack of open benchmarks, since comparisons of…

Robotics · Computer Science 2017-10-10 Martim Brandao , Kenji Hashimoto , Atsuo Takanishi

In this paper, a local-global model reduction method is presented to solve stochastic optimal control problems governed by partial differential equations (PDEs). If the optimal control problems involve uncertainty, we need to use a few…

Numerical Analysis · Mathematics 2018-07-04 Lingling Ma , Qiuqi Li , Lijian Jiang

This work aims to construct and analyze a discontinuous Galerkin method on polytopal grids (PolydG) to solve the pseudo-stress formulation of the unsteady Stokes problem. The pseudo-stress variable is introduced due to the growing interest…

Numerical Analysis · Mathematics 2025-12-03 Paola F. Antonietti , Michele Botti , Alessandra Cancrini , Ilario Mazzieri

Meshless methods inherently do not require mesh topologies and are practically used for solving continuum equations. However, these methods generally tend to have a higher computational load than conventional mesh-based methods because…

Fluid Dynamics · Physics 2024-04-29 Takeharu Matsuda , Kohsuke Tsukui , Satoshi Ii

A probabilistic representation for initial value semilinear parabolic problems based on generalized random trees has been derived. Two different strategies have been proposed, both requiring generating suitable random trees combined with a…

Numerical Analysis · Mathematics 2024-02-13 Juan A. Acebron , Angel Rodriguez-Rozas

The position-based dynamics (PBD) algorithm is a popular and versatile technique for real-time simulation of deformable bodies, but is only applicable to forces that can be expressed as linearly compliant constraints. In this work, we…

Graphics · Computer Science 2025-12-01 Manas Chaudhary , Chandradeep Pokhariya , Rahul Narain

Stochastic convex optimization algorithms are the most popular way to train machine learning models on large-scale data. Scaling up the training process of these models is crucial, but the most popular algorithm, Stochastic Gradient Descent…

Machine Learning · Statistics 2018-10-30 Ashok Cutkosky , Robert Busa-Fekete

This work considers a weighted POD-greedy method to estimate statistical outputs parabolic PDE problems with parametrized random data. The key idea of weighted reduced basis methods is to weight the parameter-dependent error estimate…

Numerical Analysis · Mathematics 2017-12-21 Christopher Spannring , Sebastian Ullmann , Jens Lang

In this paper we present an arbitrary-order fully discrete Stokes complex on general polyhedral meshes. We enriche the fully discrete de Rham complex with the addition of a full gradient operator defined on vector fields and fitting into…

Numerical Analysis · Mathematics 2024-01-18 Marien-Lorenzo Hanot

A parallel implementation of a compatible discretization scheme for steady-state Stokes problems is presented in this work. The scheme uses generalized moving least squares to generate differential operators and apply boundary conditions.…

Numerical Analysis · Mathematics 2021-04-30 Quang-Thinh Ha , Paul A. Kuberry , Nathaniel A. Trask , Emily M. Ryan

In this work, we propose a new stochastic domain decomposition method for solving steady-state partial differential equations (PDEs) with random inputs. Based on the efficiency of the Variable-separation (VS) method in simulating stochastic…

Numerical Analysis · Mathematics 2025-02-06 Liang Chen , Yaru Chen , Qiuqi Li , Zhiwen Zhang

The integral equation approach to partial differential equations (PDEs) provides significant advantages in the numerical solution of the incompressible Navier-Stokes equations. In particular, the divergence-free condition and boundary…

Numerical Analysis · Mathematics 2020-02-26 Ludvig af Klinteberg , Travis Askham , Mary Catherine Kropinski

The gradient discretisation method (GDM) is a generic framework designed recently, as a discretise in spatial space, to partial differential equations. This paper aims to use the GDM to establish a first general error estimate for numerical…

Numerical Analysis · Mathematics 2020-09-22 Yahya Alnashri

The present work focuses on the geometric parametrization and the reduced order modeling of the Stokes equation. We discuss the concept of a parametrized geometry and its application within a reduced order modeling technique. The full order…

Numerical Analysis · Mathematics 2021-06-01 Nirav Vasant Shah , Martin Hess , Gianluigi Rozza

In this paper we propose a new class of preconditioners for the isogeometric discretization of the Stokes system. Their application involves the solution of a Sylvester-like equation, which can be done efficiently thanks to the Fast…

Numerical Analysis · Mathematics 2018-05-23 Monica Montardini , Giancarlo Sangalli , Mattia Tani

Stiff ordinary differential equations (ODEs) play an important role in many scientific and engineering applications. Often, the dependence of the solution of the ODE on additional parameters is of interest, e.g.\ when dealing with…

Numerical Analysis · Mathematics 2025-11-11 Idoia Cortes Garcia , P. Förster , W. Schilders , S. Schöps

This work recasts time-dependent optimal control problems governed by partial differential equations in a Dynamic Mode Decomposition with control framework. Indeed, since the numerical solution of such problems requires a lot of…

Optimization and Control · Mathematics 2022-03-25 Eleonora Donadini , Maria Strazzullo , Marco Tezzele , Gianluigi Rozza

Optimization problem, which is aimed at finding the global minimal value of a given cost function, is one of the central problem in science and engineering. Various numerical methods have been proposed to solve this problem, among which the…

Optimization and Control · Mathematics 2022-10-07 Shaojun Dong , Fengyu Le , Meng Zhang , Si-Jing Tao , Chao Wang , Yong-Jian Han , Guo-Ping Guo

Huge scale machine learning problems are nowadays tackled by distributed optimization algorithms, i.e. algorithms that leverage the compute power of many devices for training. The communication overhead is a key bottleneck that hinders…

Machine Learning · Computer Science 2018-11-30 Sebastian U. Stich , Jean-Baptiste Cordonnier , Martin Jaggi
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