Search

In the least-squares fitting framework, the Vandermonde with Arnoldi (V+A) method presented in [Brubeck, Nakatsukasa, and Trefethen, SIAM Review, 63 (2021), pp. 405-415] is an effective approach to compute a polynomial that approximates an underlying univariate function f. Extensions of V+A include its multivariate version and the univariate confluent V+A; the latter enables us to use the information of the derivative of f in obtaining the approximation polynomial. In this paper, we shall extend V+A further to the multivariate confluent V+A. Besides the technical generalization of the univariate confluent V+A, we also introduce a general and application-dependent G-orthogonalization in the Arnoldi process. We shall demonstrate with several applications that, by specifying an application-related G-inner product, the desired approximate multivariate polynomial as well as its certain partial derivatives can be computed accurately from a well-conditioned least-squares problem whose coefficient matrix is orthonormal. The desired multivariate polynomial is represented in a discrete G-orthogonal polynomials basis which admits an explicit recurrence, and therefore, facilitates evaluating function values and certain partial derivatives at new nodes efficiently. We demonstrate its flexibility by applying it to solve the multivariate Hermite least-squares problem and PDEs with various boundary conditions in irregular domains., Comment: 7 figures

Low-light image enhancement (LLIE) aims to improve low-illumination images. However, existing methods face two challenges: (1) uncertainty in restoration from diverse brightness degradations; (2) loss of texture and color information caused by noise suppression and light enhancement. In this paper, we propose a novel enhancement approach, CodeEnhance, by leveraging quantized priors and image refinement to address these challenges. In particular, we reframe LLIE as learning an image-to-code mapping from low-light images to discrete codebook, which has been learned from high-quality images. To enhance this process, a Semantic Embedding Module (SEM) is introduced to integrate semantic information with low-level features, and a Codebook Shift (CS) mechanism, designed to adapt the pre-learned codebook to better suit the distinct characteristics of our low-light dataset. Additionally, we present an Interactive Feature Transformation (IFT) module to refine texture and color information during image reconstruction, allowing for interactive enhancement based on user preferences. Extensive experiments on both real-world and synthetic benchmarks demonstrate that the incorporation of prior knowledge and controllable information transfer significantly enhances LLIE performance in terms of quality and fidelity. The proposed CodeEnhance exhibits superior robustness to various degradations, including uneven illumination, noise, and color distortion., Comment: 10 pages, 13 figures

Given samples of a real or complex-valued function on a set of distinct nodes, the traditional linear Chebyshev approximation is to compute the best minimax approximation on a prescribed linear functional space. Lawson's iteration is a classical and well-known method for that task. However, Lawson's iteration converges linearly and in many cases, the convergence is very slow. In this paper, by the duality theory of linear programming, we first provide an elementary and self-contained proof for the well-known Alternation Theorem in the real case. Also, relying upon the Lagrange duality, we further establish an $L_q$-weighted dual programming for the linear Chebyshev approximation. In this framework, we revisit the convergence of Lawson's iteration, and moreover, propose a Newton type iteration, the interior-point method, to solve the $L_2$-weighted dual programming. Numerical experiments are reported to demonstrate its fast convergence and its capability in finding the reference points that characterize the unique minimax approximation., Comment: 29 pages, 8 figures

Computing the discrete rational minimax approximation in the complex plane is challenging. Apart from Ruttan's sufficient condition, there are few other sufficient conditions for global optimality. The state-of-the-art rational approximation algorithms, such as the adaptive Antoulas-Anderson (AAA), AAA-Lawson, and the rational Krylov fitting (RKFIT) method, perform highly efficiently, but the computed rational approximations may not be minimax solutions. In this paper, we propose a convex programming approach, the solution of which is guaranteed to be the rational minimax approximation under Ruttan's sufficient condition. Furthermore, we present a new version of Lawson's iteration for solving this convex programming problem. The computed solution can be easily verified as the rational minimax approximation. Our numerical experiments demonstrate that this updated version of Lawson's iteration generally converges monotonically with respect to the objective function of the convex optimization. It is an effective competitive approach for computing the rational minimax approximation, compared to the highly efficient AAA, AAA-Lawson, and the stabilized Sanathanan-Koerner iteration., Comment: 40 pages, 11 figures