Related papers: The Random Feature Method for Solving Interface Pr…
Random media introduce large degrees of freedom in device design and can thus address challenges in manipulating optical waves. Wave shaping with metasurfaces has mainly utilized periodic or quasi-periodic grids, and, the potential of…
In this paper, a shape optimization problem constrained by a random elliptic partial differential equation with a pure Neumann boundary is presented. The model is motivated by applications in interface identification, where we assume…
We consider the reliable implementation of an adaptive high-order unfitted finite element method on Cartesian meshes for solving elliptic interface problems with geometrically curved singularities. We extend our previous work on the…
Self-affine rough interfaces are ubiquitous in experimental systems, and display characteristic scaling properties as a signature of the nature of disorder in their supporting medium, i.e. of the statistical features of its heterogeneities.…
In this work, we investigate data fitting problems with random noises. A randomized progressive iterative regularization method is proposed. It works well for large-scale matrix computations and converges in expectation to the least-squares…
We consider the random deposition of objects of variable width and height over a line. The successive additions of these structures create a random interface. We focus on the regime of heavy tailed distributions of the structure width. When…
This article presents new immersed finite element (IFE) methods for solving the popular second order elliptic interface problems on structured Cartesian meshes even if the involved interfaces have nontrivial geometries. These IFE methods…
Localized features such as singularities, sharp gradients, discontinuities, and moving sources require adaptive finite element discretizations. Conventional refinement strategies introduce significant computational overhead through…
Immersed finite element (IFE) methods are a group of long-existing numerical methods for solving interface problems on unfitted meshes. A core argument of the methods is to avoid mesh regeneration procedure when solving moving interface…
For uncertainty propagation of highly complex and/or nonlinear problems, one must resort to sample-based non-intrusive approaches [1]. In such cases, minimizing the number of function evaluations required to evaluate the response surface is…
Non-overlapping domain decomposition methods are natural for solving interface problems arising from various disciplines, however, the numerical simulation requires technical analysis and is often available only with the use of high-quality…
Random features is one of the most popular techniques to speed up kernel methods in large-scale problems. Related works have been recognized by the NeurIPS Test-of-Time award in 2017 and the ICML Best Paper Finalist in 2019. The body of…
Kernel methods are powerful and flexible approach to solve many problems in machine learning. Due to the pairwise evaluations in kernel methods, the complexity of kernel computation grows as the data size increases; thus the applicability…
Kernel method has been developed as one of the standard approaches for nonlinear learning, which however, does not scale to large data set due to its quadratic complexity in the number of samples. A number of kernel approximation methods…
In this paper, we present a new immersed finite element scheme for solving elliptic interface problems on unfitted meshes by combining the skeletal finite element method (FEM) with the standard FEM. The skeletal FEM is used for the…
We propose a stochastic approximation method for approximating the efficient frontier of chance-constrained nonlinear programs. Our approach is based on a bi-objective viewpoint of chance-constrained programs that seeks solutions on the…
In this paper, we define new unfitted finite element methods for numerically approximating the solution of surface partial differential equations using bulk finite elements. The key idea is that the $n$-dimensional hypersurface, $\Gamma…
Frequently, the design of physicochemical processes requires screening of large numbers of alternative designs with complex geometries. These geometries may result in conformal meshes which introduce stability issues, significant…
The random feature method (RFM), a mesh-free machine learning-based framework, has emerged as a promising alternative for solving PDEs on complex domains. However, for large three-dimensional nonlinear problems, attaining high accuracy…
Feature selection is playing an increasingly significant role with respect to many computer vision applications spanning from object recognition to visual object tracking. However, most of the recent solutions in feature selection are not…