相关论文: Adaptive methods for PDE's: wavelets or mesh refin…

It is well known that the quasi-optimality of the Galerkin finite element method for the Helmholtz equation is dependent on the mesh size and the wave-number. In the literature, different criteria have been proposed to ensure uniform…

The construction of B-spline wavelet bases on nonequispaced knots is extended to wavelets that are piecewise segments from any combination of smooth functions. The extended wavelet family thus provides multiresolution basis functions with…

Mesh generation is essential for accurate and efficient computational fluid dynamics simulations. To resolve critical features in the flow, adaptive mesh refinement (AMR) is routinely employed in certain regions of the computational domain,…

We introduce an adaptive sampling method for the Deep Ritz method aimed at solving partial differential equations (PDEs). Two deep neural networks are used. One network is employed to approximate the solution of PDEs, while the other one is…

We develop all of the components needed to construct an adaptive finite element code that can be used to approximate fractional partial differential equations, on non-trivial domains in $d\geq 1$ dimensions. Our main approach consists of…

This work introduces an adaptive mesh refinement technique for hierarchical hybrid grids with the goal to reach scalability and maintain excellent performance on massively parallel computer systems. On the block structured hierarchical…

The convergence analysis for least-squares finite element methods led to various adaptive mesh-refinement strategies: Collective marking algorithms driven by the built-in a posteriori error estimator or an alternative explicit…

We explain four variants of an adaptive finite element method with cubic splines and compare their performance in simple elliptic model problems. The methods in comparison are Truncated Hierarchical B-splines with two different refinement…

An adaptive moving mesh finite element method is studied for the numerical solution of the porous medium equation with and without variable exponents and absorption. The method is based on the so-called moving mesh partial differential…

In this work, we revisit the marking decisions made in the standard adaptive finite element method (AFEM). Experience shows that a na\"{i}ve marking policy leads to inefficient use of computational resources for adaptive mesh refinement…

For a given symmetric refinable mask obeying the sum rule of order $n$, an explicit method is suggested for the construction of mutually symmetric almost frame-like wavelet system providing approximation order $n$. A transformation based on…

The generalization of neural networks is a central challenge in machine learning, especially concerning the performance under distributions that differ from training ones. Current methods, mainly based on the data-driven paradigm such as…

Domain-specific adaptation is critical to maximizing the performance of pre-trained language models (PLMs) on one or multiple targeted tasks, especially under resource-constrained use cases, such as edge devices. However, existing methods…

Recent works emphasized the interest of numerical solution of PDE's with wavelets. In their works, A.Cohen, W.Dahmen and R.DeVore focussed on the non linear approximation aspect of the wavelet approximation of PDE's to prove the relevance…

This paper is concerned with density estimation of directional data on the sphere. We introduce a procedure based on thresholding on a new type of spherical wavelets called {\it needlets}. We establish a minimax result and prove its…

The accuracy of finite element solutions is closely tied to the mesh quality. In particular, geometrically nonlinear problems involving large and strongly localized deformations often result in prohibitively large element distortions. In…

Masked Image Modeling (MIM) has garnered significant attention in self-supervised learning, thanks to its impressive capacity to learn scalable visual representations tailored for downstream tasks. However, images inherently contain…

This study presents constructions of the space-time Conservation Element and Solution Element (CESE) methods to accommodate adaptive unstructured quadrilateral meshes. Subsequently, a novel algorithm is devised to effectively manage the…

In this paper, we propose a new approach to model reduction of parameterized partial differential equations (PDEs) based on the concept of adaptive reduced bases. The presented approach is particularly suited for large-scale nonlinear…

An adaptive mesh refinement and error estimation method for numerically solving optimal control problems is developed using Legendre-Gauss-Radau direct collocation. In regions of the solution where the desired accuracy tolerance has not…