相关论文: Fast Adaptive Algorithms in the Non-Standard Form …

We present a fast, direct and adaptive Poisson solver for complex two-dimensional geometries based on potential theory and fast multipole acceleration. More precisely, the solver relies on the standard decomposition of the solution as the…

Existing fast algorithms for bilateral and nonlocal means filtering mostly work with grayscale images. They cannot easily be extended to high-dimensional data such as color and hyperspectral images, patch-based data, flow-fields, etc. In…

This paper introduces a directional multiscale algorithm for the two dimensional $N$-body problem of the Helmholtz kernel with applications to high frequency scattering. The algorithm follows the approach in [Engquist and Ying, SIAM Journal…

Developments of nonlocal operators for modeling processes that traditionally have been described by local differential operators have been increasingly active during the last few years. One example is peridynamics for brittle materials and…

It is well-known that the high computational complexity and the insufficient samples in large-scale array signal processing restrict the real-world applications of the conventional full-dimensional adaptive beamforming (sample matrix…

We introduce a new class of multilevel, adaptive, dual-space methods for computing fast convolutional transforms. These methods can be applied to a broad class of kernels, from the Green's functions for classical partial differential…

We introduce a Fourier-based fast algorithm for Gaussian process regression in low dimensions. It approximates a translationally-invariant covariance kernel by complex exponentials on an equispaced Cartesian frequency grid of $M$ nodes.…

This paper introduces a parallel directional fast multipole method (FMM) for solving N-body problems with highly oscillatory kernels, with a focus on the Helmholtz kernel in three dimensions. This class of oscillatory kernels requires a…

The convolution potential arises in a wide variety of application areas, and its efficient and accurate evaluation encounters three challenges: singularity, nonlocality and anisotropy. We introduce a fast algorithm based on a far-field…

Quantum kernel methods are a promising branch of quantum machine learning, yet their effectiveness on diverse, high-dimensional, real-world data remains unverified. Current research has largely been limited to low-dimensional or synthetic…

To accelerate kernel methods, we propose a near input sparsity time algorithm for sampling the high-dimensional feature space implicitly defined by a kernel transformation. Our main contribution is an importance sampling method for…

This paper introduces significant advancements in fractional neural operators (FNOs) through the integration of adaptive hybrid kernels and stochastic multiscale analysis. We address several open problems in the existing literature by…

In the last decade, a considerable research effort has been devoted to developing adaptive algorithms based on kernel functions. One of the main features of these algorithms is that they form a family of universal approximation techniques,…

This paper presents an accelerated quadrature scheme for the evaluation of layer potentials in three dimensions. Our scheme combines a generic, high order quadrature method for singular kernels called Quadrature by Expansion (QBX) with a…

In the era of big data, it is desired to develop efficient machine learning algorithms to tackle massive data challenges such as storage bottleneck, algorithmic scalability, and interpretability. In this paper, we develop a novel efficient…

We present a fast direct solver for boundary integral equations on complex surfaces in three dimensions using an extension of the recently introduced recursive strong skeletonization scheme. For problems that are not highly oscillatory, our…

We present a detailed comparison between two adaptive numerical approaches to solve partial differential equations (PDEs), adaptive multiresolution (MR) and adaptive mesh refinement (AMR). Both discretizations are based on finite volumes in…

The present article is concerned scattered data approximation for higher dimensional data sets which exhibit an anisotropic behavior in the different dimensions. Tailoring sparse polynomial interpolation to this specific situation, we…

Kernel methods approximate nonlinear maps in a data-driven manner by projecting the target map onto a finite-dimensional Hilbert space called the solution space. Traditionally, this space is a subspace of a fixed ambient reproducing kernel…

In this work, the fast-convolving reproducing kernel particle method (FC-RKPM) is introduced. This method is hundreds to millions of times faster than the traditional RKPM for 3D meshfree simulations. In this approach, the meshfree…