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This paper aims to develop an efficient adaptive finite element method for the second-order elliptic problem. Although the theory for adaptive finite element methods based on residual-type a posteriori error estimator and bisection…

In this work, we consider ill-posed inverse problems in which the forward operator is continuous and weakly closed, and the sought solution belongs to a weakly closed constraint set. We propose a regularization method based on minimizing…

Subspace learning and matrix factorization problems have great many applications in science and engineering, and efficient algorithms are critical as dataset sizes continue to grow. Many relevant problem formulations are non-convex, and in…

Discrete transforms such as the discrete Fourier transform (DFT) and the discrete Hartley transform (DHT) are important tools in numerical analysis. The successful application of transform techniques relies on the existence of efficient…

We derive the asymptotic distribution of the domination number of a new family of random digraph called proximity catch digraph (PCD), which has application to statistical testing of spatial point patterns and to pattern recognition. The…

Classical simulation is essential in quantum algorithm development and quantum device verification. With the increasing complexity and diversity of quantum circuit structures, existing classical simulation algorithms need to be improved and…

Machine learning (ML) is becoming increasingly popular for predicting material properties to accelerate materials discovery. Because material properties are strongly affected by its crystal structure, a key issue is converting the crystal…

Simulation-based verification algorithms can provide formal safety guarantees for nonlinear and hybrid systems. The previous algorithms rely on user provided model annotations called discrepancy function, which are crucial for computing…

Optimization models with non-convex constraints arise in many tasks in machine learning, e.g., learning with fairness constraints or Neyman-Pearson classification with non-convex loss. Although many efficient methods have been developed…

The proximal point algorithm is a widely used tool for solving a variety of convex optimization problems such as finding zeros of maximally monotone operators, fixed points of nonexpansive mappings, as well as minimizing convex functions.…

Diffusion tensor imaging (DTI) holds significant importance in clinical diagnosis and neuroscience research. However, conventional model-based fitting methods often suffer from sensitivity to noise, leading to decreased accuracy in…

The discrete cosine transform (DCT) is a central tool for image and video coding because it can be related to the Karhunen-Lo\`eve transform (KLT), which is the optimal transform in terms of retained transform coefficients and data…

Delaunay triangulations of a point set in the Euclidean plane are ubiquitous in a number of computational sciences, including computational geometry. Delaunay triangulations are not well defined as soon as 4 or more points are concyclic but…

A fast Discrete Cosine Transform (DCT) algorithm is introduced that can be of particular interest in image processing. The main features of the algorithm are regularity of the graph and very low arithmetic complexity. The 16-point version…

We study the problem of minimizing a $m$-weakly convex and possibly nonsmooth function. Weak convexity provides a broad framework that subsumes convex, smooth, and many composite nonconvex functions. In this work, we propose a…

Randomized numerical linear algebra is proved to bridge theoretical advancements to offer scalable solutions for approximating tensor decomposition. This paper introduces fast randomized algorithms for solving the fixed Tucker-rank problem…

Dynamic programming is an important optimization technique, but designing efficient dynamic programming algorithms can be difficult for even professional programmers. Thinning, a technique developed for systematically deriving efficient…

A new computational algorithm, the discrete singular convolution (DSC), is introduced for computational electromagnetics. The basic philosophy behind the DSC algorithm for the approximation of functions and their derivatives is studied.…

This paper aims to accelerate the test-time computation of deep convolutional neural networks (CNNs). Unlike existing methods that are designed for approximating linear filters or linear responses, our method takes the nonlinear units into…

When modeling scientific and industrial problems, geometries are typically modeled by explicit boundary representations obtained from computer-aided design software. Unfitted (also known as embedded or immersed) finite element methods offer…