Narrow Results

This paper establishes a theory framework of a class of nonlinear Lagrangians for solving nonlinear programming problems with inequality constraints. A set of conditions are proposed to guarantee the convergence of nonlinear Lagrangian algorithms, to analyze condition numbers of nonlinear Lagrangian Hessians as well as to develop the dual approaches. These conditions are satisfied by well-known nonlinear Lagrangians appearing in literature. The convergence theorem shows that the dual algorithm based on any nonlinear Lagrangian in the class is locally convergent when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions and the error bound solution, depending on the penalty parameter, is also established. The paper also develops the dual problems based on the proposed nonlinear Lagrangians, and the related duality theorem and saddle point theorem are demonstrated. Furthermore, it is shown that the condition numbers of Lagrangian Hessians at optimal solutions are proportional to the controlling penalty parameters. We report some numerical results obtained by using nonlinear Lagrangians.

The quantitative analysis of seismic performance under small and moderate earthquakes is of great significance. However, only using the stiffness distribution index is no longer applicable to seismic performance evaluation, and the influence of mass distribution must be introduced. In this study, under the action of small and medium earthquakes, the equation of motion can be simplified into the equilibrium equation of the dynamic matrix by the quasi-static method. Furthermore, a condition number index of dynamic matrix is proposed to evaluate the seismic performance of truss structures, which can consider both stiffness distribution and mass distribution. Then, the correctness of the proposed index is verified by comparing with the displacement and strain indexes. In addition, the performance of truss structure is studied by the condition number of stiffness matrix, mass matrix and dynamic matrix. It is found that since the condition number of stiffness matrix and the condition number of mass matrix cannot consider both stiffness and mass, it is not comprehensive to use these two indexes to evaluate the seismic performance of truss structures. The evaluation result based on the condition number of dynamic matrix is more reasonable under small and medium earthquakes.

In this paper, we propose a construction of a new cubic spline-wavelet basis on the hypercube satisfying homogeneous Dirichlet boundary conditions. Wavelets have two vanishing moments. Stiffness matrices arising from discretization of elliptic problems using a constructed wavelet basis have uniformly bounded condition numbers and we show that these condition numbers are small. We present quantitative properties of the constructed basis and we provide a numerical example to show the efficiency of the Galerkin method using the constructed basis.

A new method has been introduced which allows us to determine the stability of protein complexes with point changes of amino acid residues that also take into account the three-dimensional structure of the complex. This formulated and proven theorem is aimed at determining the criterion for the stability of protein molecules. The algorithm and software package were developed for analyzing protein interactions, taking into account their three-dimensional structure from the PDB database.

Linear systems of equations lie at the heart of numerous scientific and engineering challenges. In cutting-edge arena like artificial intelligence, machine learning and neuro-computation, these systems serve as a fundamental tool for mathematical modeling. Classical algorithms for solving linear systems have been extensively developed and forms the backbone of diverse applications across various scientific disciplines. While classical algorithms exist for solving linear systems, they often encounter limitations termed “NP-completeness” as data complexity increases. The emerging field of quantum computing offers a revolutionary approach to deal with these kinds of problems. The Harrow–Hassidim–Lloyd (HHL) algorithm tackles these challenges and opens new avenues for research. This study delves into the contemporary effectiveness of the HHL algorithm to address systems of linear equations. By examining recent research in quantum machine learning, we aim to assess the HHL algorithm’s potential to revolutionize the process of optimizing hyperparameters for machine learning models, resulting in increased efficiency and cost savings. This paper meticulously analyzes the HHL algorithm and explores its evolution from conception to the latest advancements. A comprehensive examination of the HHL algorithm, including its evolution over time, is thoroughly explored. The investigation delves into the potential challenges and limitations that might hinder the practical deployment of the HHL algorithm. Identifying these roadblocks will pave the way for future research and development efforts.

Following a recent attempt by Waziri et al. [2019] to find an appropriate choice for the nonnegative parameter of the Hager–Zhang conjugate gradient method, we have proposed two adaptive options for the Hager–Zhang nonnegative parameter by analyzing the search direction matrix. We also used the proposed parameters with the projection technique to solve convex constraint monotone equations. Furthermore, the global convergence of the methods is proved using some proper assumptions. Finally, the efficacy of the proposed methods is demonstrated using a number of numerical examples.

Some properties that nominally involve the eigenvalues of the Gaussian Unitary Ensemble (GUE) can instead be phrased in terms of singular values. By discarding the signs of the eigenvalues, we gain access to a surprising decomposition: the singular values of the GUE are distributed as the union of the singular values of two independent ensembles of Laguerre type. This independence is remarkable given the well-known phenomenon of eigenvalue repulsion. The structure of this decomposition reveals that several existing observations about large $n$ limits of the GUE are in fact manifestations of phenomena that are already present for finite random matrices. We relate the semicircle law to the quarter-circle law by connecting Hermite polynomials to generalized Laguerre polynomials with parameter $\pm 1/2$. Similarly, we write the absolute value of the determinant of the $n\times n$ GUE as a product $n$ independent random variables to gain new insight into its asymptotic log-normality. The decomposition also provides a description of the distribution of the smallest singular value of the GUE, which in turn permits the study of the leading order behavior of the condition number of GUE matrices. The study is motivated by questions involving the enumeration of orientable maps, and is related to questions involving powers of complex Ginibre matrices. The inescapable conclusion of this work is that the singular values of the GUE play an unpredictably important role that had gone unnoticed for decades even though, in hindsight, so many clues had been around.

In this paper, we study the condition number of a random Toeplitz matrix. As a Toeplitz matrix is a diagonal constant matrix, its rows or columns cannot be stochastically independent. This situation does not permit us to use the classic strategies to analyze its minimum singular value when all the entries of a random matrix are stochastically independent. Using a circulant embedding as a decoupling technique, we break the stochastic dependence of the structure of the Toeplitz matrix and reduce the problem to analyze the extreme singular values of a random circulant matrix. A circulant matrix is, in fact, a particular case of a Toeplitz matrix, but with a more specific structure, where it is possible to obtain explicit formulas for its eigenvalues and also for its singular values. Among our results, we show the condition number of a non-symmetric random circulant matrix ${{𝒞}}_n$ of dimension $n$ under the existence of the moment generating function of the random entries is $\kappa ({{𝒞}}_n)=\text{O}(\frac{1}{𝜀}{n}^{\rho +1/2}{(logn)}^{1/2})$ with probability $1-\text{O}(({𝜀}^2+𝜀)n^{-2\rho }+n^{-1/2+\text{o}(1)})$ for any $𝜀>0$, $\rho \in (0,1/4)$. Moreover, if the random entries only have the second moment, the condition number satisfies $\kappa ({{𝒞}}_n)=\text{O}(\frac{1}{𝜀}{n}^{\rho +1/2}logn)$ with probability $1-\text{O}(({𝜀}^2+𝜀)n^{-2\rho }+{(logn)}^{-1/2})$. Also, we analyze the condition number of a random symmetric circulant matrix ${{𝒞}}_n^{\mathrm{\text{sym}}}$. For the condition number of a random (non-symmetric or symmetric) Toeplitz matrix ${{𝒯}}_n$ we establish $\kappa ({{𝒯}}_n)\le \kappa ({{𝒞}}_{2n}){({\sigma }_{min}(C_{2n}){\sigma }_{min}({{𝒮}}_n))}^{-1}$, where ${\sigma }_{min}(A)$ is the minimum singular value of the matrix $A$. The matrix $C_{2n}$ is a random circulant matrix and ${{𝒮}}_n:=F_{2,n}^{\ast }{D}_{1,n}^{-1}{F}_{2,n}+F_{4,n}^{\ast }{D}_2^{-1}{F}_{4,n}$, where $F_{2,n},F_{4,n}$ are deterministic matrices, $F^{\ast }$ indicates the conjugate transpose of $F$ and $D_{1,n},D_{2,n}$ are random diagonal matrices. From random experiments, we conjecture that ${{𝒮}}_n$ is well-conditioned if the moment generating function of the random entries of ${{𝒞}}_{2n}$ exists.

No abstract received.

We present some recent results on the probabilistic behaviour of interior point methods for the convex conic feasibility problem and for homotopy methods solving complex polynomial equations. As suggested by Spielman and Teng, the goal is to prove that for all inputs (even ill-posed ones), and all slight random perturbations of that input, it is unlikely that the running time will be large. These results are obtained through a probabilistic analysis of the condition of the corresponding computational problems.

This paper discusses the numerical uncertainty related to reference data pairs used for software validation. Various methods for assigning a numerical accuracy bound to reference data pairs are compared. Several performance metrics are discussed which can be used to summarize the results of software testing. They require the calculation of the condition number of the problem solved by the software, or a related quantity called numerical sensitivity. These performance metrics may be used to demonstrate traceability of metrology software.