Wenyuan Wu

Yu Wang, Wenyuan Wu, Bican Xia

A special homotopy continuation method, as a combination of the polyhedral homotopy and the linear product homotopy, is proposed for computing all the isolated solutions to a special class of polynomial systems. The root number bound of this method is between the total degree bound and the mixed volume bound and can be easily computed. The new algorithm has been implemented as a program called LPH using C++. Our experiments show its efficiency compared to the polyhedral or other homotopies on such systems. As an application, the algorithm can be used to find witness points on each connected component of a real variety.

Xiaolin Qin, Wenyuan Wu, Yong Feng et al.

This paper deals with the structural analysis problem of dynamic lumped process high-index DAE models. We consider two methods for index reduction of such models by differentiation: Pryce's method and the symbolic differential elimination algorithm rifsimp. Discussion and comparison of these methods are given via a class of fundamental process simulation examples. In particular, the efficiency of the Pryce method is illustrated as a function of the number of tanks in process design.

Zhenyi Ji, Wenyuan Wu, Yi Li et al.

In this paper, based on the homotopy continuation method and the interval Newton method, an efficient algorithm is introduced to isolate the real roots of semi-algebraic system. Tests on some random examples and a variety of problems including transcendental functions arising in many applications show that the new algorithm reduces the cost substantially compared with the traditional symbolic approaches.

Wenyuan Wu, Wenqiang Yang

Solving Linear Ordinary Differential Equations (ODEs) plays an important role in many applications. There are various numerical methods and solvers to obtain approximate solutions. However, few work about global error estimation can be found in the literature. In this paper, we first give a definition of the residual, based on the piecewise Hermit interpolation, which is a kind of the backward-error of ODE solvers. It indicates the reliability and quality of numerical solution. Secondly, the global error between the exact solution and an approximate solution is the forward error and a bound of it can be given by using the backward-error. The examples in the paper show that our estimate works well for a large class of ODE models.

Yong Feng, Wenyuan Wu, Jingzhong Zhang

Factorization of polynomials is one of the foundations of symbolic computation. Its applications arise in numerous branches of mathematics and other sciences. However, the present advanced programming languages such as C++ and J++, do not support symbolic computation directly. Hence, it leads to difficulties in applying factorization in engineering fields. In this paper, we present an algorithm which use numerical method to obtain exact factors of a bivariate polynomial with rational coefficients. Our method can be directly implemented in efficient programming language such C++ together with the GNU Multiple-Precision Library. In addition, the numerical computation part often only requires double precision and is easily parallelizable.

Likang Peng, Chao Su, Wenyuan Wu et al.

Cross-modal hashing (CMH) facilitates efficient retrieval across different modalities (e.g., image and text) by encoding data into compact binary representations. While recent methods have achieved remarkable performance, they often rely heavily on fully annotated datasets, which are costly and labor-intensive to obtain. In real-world scenarios, particularly in multi-label datasets, label noise is prevalent and severely degrades retrieval performance. Moreover, existing CMH approaches typically overlook the partial semantic overlaps inherent in multi-label data, limiting their robustness and generalization. To tackle these challenges, we propose a novel framework named Semantic-Consistent Bidirectional Contrastive Hashing (SCBCH). The framework comprises two complementary modules: (1) Cross-modal Semantic-Consistent Classification (CSCC), which leverages cross-modal semantic consistency to estimate sample reliability and reduce the impact of noisy labels; (2) Bidirectional Soft Contrastive Hashing (BSCH), which dynamically generates soft contrastive sample pairs based on multi-label semantic overlap, enabling adaptive contrastive learning between semantically similar and dissimilar samples across modalities. Extensive experiments on four widely-used cross-modal retrieval benchmarks validate the effectiveness and robustness of our method, consistently outperforming state-of-the-art approaches under noisy multi-label conditions.

Rui Li, Wenyuan Wu, Weijie Miao

Fully homomorphic encryption (FHE) supports only additions and multiplications, so FHE-only neural-network inference typically replaces ReLU with polynomials fitted over empirical activation intervals. Such interval fitting often requires higher-degree polynomials to control activation error, incurring homomorphic evaluation costs, while classification is determined by the final logit decision. We revisit ReLU replacement from a decision-aware perspective: given a trained single-hidden-layer ReLU MLP and a specified calibration set, can an HE-friendly low-degree polynomial replace ReLU without retraining while preserving calibration-set decisions? We focus on quadratic replacement, the lowest-degree choice that retains a genuine per-unit nonlinearity. For calibration sets positive-margin separable in the lifted space, we formulate quadratic replacement as a linear separation problem, yielding necessary and sufficient conditions for calibration-lossless replacement and a constructive algorithm for the coefficients. When the positive-margin condition fails -- typically due to a few misclassified calibration samples -- we extend the same geometric framework via reduced convex hulls and Lagrangian-dual soft-margin relaxations, which bound the influence of any single sample, converting the problem into smaller convex quadratic programs that yield approximately feasible coefficients with high empirical agreement on calibration-set decisions. In particular, at the maximal weight cap $μ=1$, the reduced-convex-hull relaxation reduces to the convex-hull separation of the strictly separable case; the relaxation thus continuously extends the exact theory. Under CKKS, the quadratic replacement matches plaintext top-1 accuracy on multiple benchmarks, running 3.7--4.1$\times$ faster than Remez-7 in the activation module and 1.18--1.68$\times$ faster end-to-end.

Greg Reid, Fei Wang, Henry Wolkowicz et al.

The real radical ideal of a system of polynomials with finitely many complex roots is generated by a system of real polynomials having only real roots and free of multiplicities. It is a central object in computational real algebraic geometry and important as a preconditioner for numerical solvers. Lasserre and co-workers have shown that the real radical ideal of real polynomial systems with finitely many real solutions can be determined by a combination of semi-definite programming (SDP) and geometric involution techniques. A conjectured extension of such methods to positive dimensional polynomial systems has been given recently by Ma, Wang and Zhi. We show that regularity in the form of the Slater constraint qualification (strict feasibility) fails for the resulting SDP feasibility problems. Facial reduction is then a popular technique whereby SDP problems that fail strict feasibility can be regularized by projecting onto a face of the convex cone of semi-definite problems. In this paper we introduce a framework for combining facial reduction with such SDP methods for analyzing $0$ and positive dimensional real ideals of real polynomial systems. The SDP methods are implemented in MATLAB and our geometric involutive form is implemented in Maple. We use two approaches to find a feasible moment matrix. We use an interior point method within the CVX package for MATLAB and also the Douglas-Rachford (DR) projection-reflection method. Illustrative examples show the advantages of the DR approach for some problems over standard interior point methods. We also see the advantage of facial reduction both in regularizing the problem and also in reducing the dimension of the moment matrices. Problems requiring more than one facial reduction are also presented.

Juan Tang, Wenyuan Wu, Xiaolin Qin et al.

Motivated by Pryce's structural index reduction method for differential algebraic equations (DAEs), we show the complexity of the fixed-point iteration algorithm and propose a fixed-point iteration method with parameters. It leads to a block fixed-point iteration method which can be applied to large-scale DAEs with block upper triangular structure. Moreover, its complexity analysis is also given in this paper.