Discrete Problem Research Articles

Efficient quantum algorithms for some instances of the semidirect discrete logarithm problem

The semidirect discrete logarithm problem (SDLP) is the following analogue of the standard discrete logarithm problem in the semidirect product semigroup G⋊End(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G\\rtimes {{\\,\ extrm{End}\\,}}(G)$$\\end{document} for a finite semigroup G. Given g∈G,σ∈End(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g\\in G, \\sigma \\in {{\\,\ extrm{End}\\,}}(G)$$\\end{document}, and h=∏i=0t-1σi(g)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h=\\prod _{i=0}^{t-1}\\sigma ^i(g)$$\\end{document} for some integer t, the SDLP(G,σ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(G,\\sigma )$$\\end{document}, for g and h, asks to determine t. As Shor’s algorithm crucially depends on commutativity, it is believed not to be applicable to the SDLP. For generic semigroups, the best known algorithm for the SDLP is based on Kuperberg’s subexponential time quantum algorithm. Still, the problem plays a central role in the security of certain proposed cryptosystems in the family of semidirect product key exchange. This includes a recently proposed signature protocol called SPDH-Sign. In this paper, we show that the SDLP is even easier in some important special cases. Specifically, for a finite group G, we describe quantum algorithms for the SDLP in G⋊Aut(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G\\rtimes {\ extrm{Aut}}(G)$$\\end{document} for the following two classes of instances: the first one is when G is solvable and the second is when G is a matrix group and a power of σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document} with a polynomially small exponent is an inner automorphism of G. We further extend the results to groups composed of factors from these classes. A consequence is that SPDH-Sign and similar cryptosystems whose security assumption is based on the presumed hardness of the SDLP in the cases described above are insecure against quantum attacks. The quantum ingredients we rely on are not new: these are Shor’s factoring and discrete logarithm algorithms and well-known generalizations.

Finite element method for fractional order parabolic obstacle problem with nonlinear source term

This paper considers the finite element method for a fractional order parabolic obstacle problem in time with a nonlinear source term. An error estimation of the spatially semi-discrete problem is discussed. A fully discrete problem is proposed by introducing the finite difference approximation for the time variable. The L2-unconditional stability and convergence of the approximate solution have been proven.

Efficient function approximation in enriched approximation spaces

Abstract An enriched approximation space is the span of a conventional basis with a few extra functions included, for example to capture known features of the solution to a computational problem. Adding functions to a basis makes it overcomplete and, consequently, the corresponding discretized approximation problem may require solving an ill-conditioned system. Recent research indicates that these systems can still provide highly accurate numerical approximations under reasonable conditions. In this paper we propose an efficient algorithm to compute such approximations. It is based on the AZ algorithm for overcomplete sets and frames, which simplifies in the case of an enriched basis. In addition, analysis of the original AZ algorithm and of the proposed variant gives constructive insights on how to achieve optimal and stable discretizations using enriched bases. We apply the algorithm to examples of enriched approximation spaces in literature, including a few nonstandard approximation problems and an enriched spectral method for a 2D boundary value problem, and show that the simplified AZ algorithm is indeed stable, accurate and efficient.

Variational data assimilation with finite-element discretization for second-order parabolic interface equation

Abstract In this paper, we propose and analyze a finite-element method of variational data assimilation for a second-order parabolic interface equation on a two-dimensional bounded domain. The Tikhonov regularization plays a key role in translating the data assimilation problem into an optimization problem. Then the existence, uniqueness and stability are analyzed for the solution of the optimization problem. We utilize the finite-element method for spatial discretization and backward Euler method for the temporal discretization. Then based on the Lagrange multiplier idea, we derive the optimality systems for both the continuous and the discrete data assimilation problems for the second-order parabolic interface equation. The convergence and the optimal error estimate are proved with the recovery of Galerkin orthogonality. Moreover, three iterative methods, which decouple the optimality system and significantly save computational cost, are developed to solve the discrete time evolution optimality system. Finally, numerical results are provided to validate the proposed method.

Efficient attribute-based strong designated verifier signature scheme based on elliptic curve cryptography.

In an attribute-based strong designated verifier signature, a signer who satisfies the access structure signs the message and assigns it to a verifier who satisfies the access structure to verify it, which enables fine-grained access control for signers and verifiers. Such signatures are used in scenarios where the identity of the signer needs to be protected, or where the public verifiability of the signature is avoided and only the designated recipient can verify the validity of the signature. To address the problem that the overall overhead of the traditional attribute-based strong designated verifier signature scheme is relatively large, an efficient attribute-based strong designated verifier signature scheme based on elliptic curve cryptography is proposed, as well as a security analysis of the new scheme given in the standard model under the difficulty of the elliptic curve discrete logarithm problem (ECDLP). On the one hand, the proposed scheme is based on elliptic curve cryptography and uses scalar multiplication on elliptic curves, which is computationally lighter, instead of bilinear pairing, which has a higher computational overhead in traditional attribute-based signature schemes. This reduces the computational overhead of signing and verification in the system, improves the efficiency of the system, and makes the scheme more suitable for resource-constrained cloud end-user scenarios. On the other hand, the proposed scheme uses LSSS (Linear Secret Sharing Schemes) access structure with stronger access policy expression, which is more efficient than the "And" gate or access tree access structure, making the computational efficiency of the proposed scheme meet the needs of resource-constrained cloud end-users.

Stokes problem with slip boundary conditions using stabilized finite elements combined with Nitsche

We discuss how slip conditions for the Stokes equation can be handled using Nitsche method, for a stabilized finite element discretization. Emphasis is made on the interplay between stabilization and Nitsche terms. Well-posedness of the discrete problem and optimal convergence rates, in natural norm for the velocity and the pressure, are established, and illustrated with various numerical experiments. The proposed method fits naturally in the context of a finite element implementation while being accurate, and allows an increased flexibility in the choice of the finite element pairs.

An efficient multi-field dynamic model for 3D wave propagation in saturated anisotropic porous media

An efficient multi–field coupled dynamic model for saturated anisotropic porous materials is proposed here. The mathematical formulation of the dynamic deformation–diffusion problem is developed starting from the mixture theory and the definition of the effective stress for anisotropic poro–elasticity, taking into account also the fluid phase compressibility and anisotropic permeability. The effective stress principle is properly extended through the Biot tensor to predict the coupling between the shear stress of the solid skeleton and the pore fluid pressure. Numerical solution of the coupled problem is obtained by inf-sup stable Finite Element spaces. A fundamental issue for the computational efficiency of the coupled model is the numerical solution of the resulting large–size and non–symmetric discrete problem. In this work, we develop a fully implicit monolithic solver based on the Bi–Conjugate Gradient Stabilized (Bi-CGStab) algorithm accelerated via an ad–hoc Multi–Physics Reduction (MPR) preconditioning technique. The proposed advances are implemented in the 3D GeoMatFem research code and several numerical analyses are performed to test the potential and computational efficiency of the proposed tool. In particular, we focus on 3D wave propagation applications in fully saturated single and multi-layered anisotropic media. Numerical results show that our implementation is able to identify the shear wave splitting phenomenon due to the different degree of material symmetry between the solid phase and the whole porous material.

On the use of dynamical systems in cryptography

Ever since the link between nonlinear science and cryptography became apparent, the problem of applying chaotic dynamics to the construction of cryptographic systems has gained a broad audience and has been the subject of thousands of papers. Yet, the field has not found its place in mainstream cryptography, largely due to persistent weaknesses in the presented systems. The goal of this paper is to help remedy this problem in two ways. The first is by providing a new algorithm that can be used to attack – and hence test the security of – stream ciphers based on the iteration of a chaotic map of the interval. The second is to cast discrete dynamical systems problems in a modern cryptographic and complexity theoretic language, so that researchers working in chaos-based cryptography can begin designing cryptographic protocols that have a better chance of meeting the extreme standards of modern cryptography.

When discrete fronts and pulses form a single family: FPU chain with hardening-softening springs

We consider a version of the classical Hamiltonian FPU (Fermi–Pasta–Ulam) problem with nonlinear force-strain relation in which a hardening response is taken over by a softening regime above a critical strain value. We show that in addition to pulses (solitary waves) this discrete system also supports non-topological and dissipation-free fronts (kinks). Moreover, we demonstrate that these two types of supersonic traveling wave solutions belong to the same family. Within this family, solitary waves exist for continuous ranges of velocity that extend up to a limiting speed corresponding to kinks. As the kink velocity limit is approached from above or below, the solitary waves become progressively more broad and acquire the structure of a kink–antikink bundle. Direct numerical simulations and Floquet analysis of linear stability suggest that all of the obtained solutions are effectively stable. To motivate and support our study of the discrete problem we also analyze a quasicontinuum approximation with temporal dispersion. We show that this model captures the main effects observed in the discrete problem both qualitatively and quantitatively.

Multilevel Monte Carlo Methods for Stochastic Convection–Diffusion Eigenvalue Problems

We develop new multilevel Monte Carlo (MLMC) methods to estimate the expectation of the smallest eigenvalue of a stochastic convection–diffusion operator with random coefficients. The MLMC method is based on a sequence of finite element (FE) discretizations of the eigenvalue problem on a hierarchy of increasingly finer meshes. For the discretized, algebraic eigenproblems we use both the Rayleigh quotient (RQ) iteration and implicitly restarted Arnoldi (IRA), providing an analysis of the cost in each case. By studying the variance on each level and adapting classical FE error bounds to the stochastic setting, we are able to bound the total error of our MLMC estimator and provide a complexity analysis. As expected, the complexity bound for our MLMC estimator is superior to plain Monte Carlo. To improve the efficiency of the MLMC further, we exploit the hierarchy of meshes and use coarser approximations as starting values for the eigensolvers on finer ones. To improve the stability of the MLMC method for convection-dominated problems, we employ two additional strategies. First, we consider the streamline upwind Petrov–Galerkin formulation of the discrete eigenvalue problem, which allows us to start the MLMC method on coarser meshes than is possible with standard FEs. Second, we apply a homotopy method to add stability to the eigensolver for each sample. Finally, we present a multilevel quasi-Monte Carlo method that replaces Monte Carlo with a quasi-Monte Carlo (QMC) rule on each level. Due to the faster convergence of QMC, this improves the overall complexity. We provide detailed numerical results comparing our different strategies to demonstrate the practical feasibility of the MLMC method in different use cases. The results support our complexity analysis and further demonstrate the superiority over plain Monte Carlo in all cases.

Twisted Hessian curve over a local ring

In this paper, we study the twisted Hessian curve denoted $H^{n}_{a,d}$ over the ring $\mathbb{F}_{q} [X] / (X^{n})$, where $\mathbb{F}_{q}$ is a finite field of q elements, with q is a power of a prime number $p \geq 5$ and $n \geq 5$. In a first time, we describe these curves over this ring. In addition, we prove that when $p$ doesn't divide $\# ( H_{\pi(a), \ \pi(d)})$, then $H^{n}_{a,d}$ is a direct sum of $ H_{\pi(a), \ \pi(d)}$ and $\mathbb{F}_{q}^{n-1}$, where $ H_{\pi(a), \ \pi(d)}$ is the twisted Hessian curve over $\mathbb{F}_{q}$. Other results are deduced from, we cite the equivalence of the discrete logarithm problem on the twisted Hessian curves $H^{n}_{a,d}$ and $ H_{\pi (a), \ \pi (d)}$, which is beneficial for cryptography and cryptanalysis as well.

The Regularized Global GMERR Method for Solving Large-Scale Linear Discrete Ill-Posed Problems

The Regularized Global GMERR Method for Solving Large-Scale Linear Discrete Ill-Posed Problems

On the convergence of discrete dynamic unbalanced transport models

A generalized unbalanced optimal transport distance WBΛ on matrix-valued measures M(Ω, 𝕊n+) was defined in Li and Zou (arXiv:2011.05845) à la Benamou–Brenier, which extends the Kantorovich–Bures and the Wasserstein–Fisher–Rao distances. In this work, we investigate the convergence properties of the discrete transport problems associated with WBΛ. We first present a convergence framework for abstract discretization. Then, we propose a specific discretization scheme that aligns with this framework, whose convergence relies on the assumption that the initial and final distributions are absolutely continuous with respect to the Lebesgue measure. Further, in the case of the Wasserstein–Fisher–Rao distance, thanks to the static formulation, we show that such an assumption can be removed.

Advancing feature ranking with hybrid feature ranking weighted majority model: a weighted majority voting strategy enhanced by the Harris hawks optimizer

Abstract Feature selection (FS) is vital in improving the performance of machine learning (ML) algorithms. Despite its importance, identifying the most important features remains challenging, highlighting the need for advanced optimization techniques. In this study, we propose a novel hybrid feature ranking technique called the Hybrid Feature Ranking Weighted Majority Model (HFRWM2). HFRWM2 combines ML models with the Harris Hawks Optimizer (HHO) metaheuristic. HHO is known for its versatility in addressing various optimization challenges, thanks to its ability to handle continuous, discrete, and combinatorial optimization problems. It achieves a balance between exploration and exploitation by mimicking the cooperative hunting behavior of Harris’s hawks, thus thoroughly exploring the search space and converging toward optimal solutions. Our approach operates in two phases. First, an odd number of ML models, in conjunction with HHO, generate feature encodings along with performance metrics. These encodings are then weighted based on their metrics and vertically aggregated. This process produces feature rankings, facilitating the extraction of the top-K features. The motivation behind our research is 2-fold: to enhance the precision of ML algorithms through optimized FS and to improve the overall efficiency of predictive models. To evaluate the effectiveness of HFRWM2, we conducted rigorous tests on two datasets: “Australian” and “Fertility.” Our findings demonstrate the effectiveness of HFRWM2 in navigating the search space and identifying optimal solutions. We compared HFRWM2 with 12 other feature ranking techniques and found it to outperform them. This superiority was particularly evident in the graphical comparison of the “Australian” dataset, where HFRWM2 showed significant advancements in feature ranking.

Augmented flexible Krylov subspace methods with applications to Bayesian inverse problems

This paper presents two new augmented flexible (AF)-Krylov subspace methods, AF-GMRES and AF-LSQR, to compute solutions of large-scale linear discrete ill-posed problems that can be modeled as the sum of two independent random variables, exhibiting smooth and sparse stochastic characteristics respectively. Following a Bayesian modeling approach, this corresponds to adding a covariance-weighted quadratic term and a sparsity enforcing ℓ1 term in the original least-squares minimization scheme. To handle the ℓ1 regularization term, the proposed approach constructs a sequence approximating quadratic problems that are partially solved using augmented flexible Krylov–Tikhonov methods.Compared to other traditional methods used to solve this minimization problem, such as those based on iteratively reweighted norm schemes, the new algorithms build a single (augmented, flexible) approximation (Krylov) subspace that encodes information about the different regularization terms through adaptable “preconditioning”. The solution space is then expanded as soon as a new problem within the sequence is defined. This also allows for the regularization parameters to be chosen on-the-fly at each iteration. Compared to most recent work on generalized flexible Krylov methods [6], our methods offer theoretical assurance of convergence and a more stable numerical performance. The efficiency of the new methods is shown through a variety of experiments, including a synthetic image deblurring problem, a synthetic atmospheric transport problem, and fluorescence molecular tomography reconstructions using both synthetic and real-world experimental data.

Generalized multiscale finite element method for language competition modeling II: Online approach

In this paper, we consider language competition in a heterogeneous medium. The mathematical model considers bilingualism, population growth, and social pressure to use specific languages (diglossia). The transition from one language group to another is determined by probabilities that depend on status, group size, and spatial location. The model considers not only diffusive spread but also convective spread, which models seasonal migrations of speaker populations. We develop multiscale approaches for coarse grid modeling while maintaining reasonable accuracy. We present offline and online approaches based on the Generalized Multiscale Finite Element Method. We consider a model problem in a heterogeneous environment to test these approaches. The results show that both approaches can significantly reduce the discrete problem’s size. However, the online approach achieves better accuracy.

Mathematical Models for the Single-Channel and Multi-Channel PMU Allocation Problem and Their Solution Algorithms

Phasor measurement units (PMUs) are deployed at power grid nodes around the transmission grid, determining precise power system monitoring conditions. In real life, it is not realistic to place a PMU at every power grid node; thus, the lowest PMU number is optimally selected for the full observation of the entire network. In this study, the PMU placement model is reconsidered, taking into account single- and multi-capacity placement models rather than the well-studied PMU placement model with an unrestricted number of channels. A restricted number of channels per monitoring device is used, instead of supposing that a PMU is able to observe all incident buses through the transmission connectivity lines. The optimization models are declared closely to the power dominating set and minimum edge cover problem in graph theory. These discrete optimization problems are directly related with the minimum set covering problem. Initially, the allocation model is declared as a constrained mixed-integer linear program implemented by mathematical and stochastic algorithms. Then, the 0/1 integer linear problem is reformulated into a non-convex constraint program to find optimality. The mathematical models are solved either in binary form or in the continuous domain using specialized optimization libraries, and are all implemented in YALMIP software in conjunction with MATLAB. Mixed-integer linear solvers, nonlinear programming solvers, and heuristic algorithms are utilized in the aforementioned software packages to locate the global solution for each instance solved in this application, which considers the transformation of the existing power grids to smart grids.

Benders decomposition for the discrete ordered median problem

Ordered median optimization has been proven to be a powerful tool to generalize many well-known problems from the literature. In Location Theory, the Discrete Ordered Median Problem (DOMP) is a facility location problem where clients are first ranked according to their allocation cost to the nearest open facility, and then these costs are multiplied by a suitable weight vector λ. That way, DOMP generalizes many well-known discrete location problems including p-median, p-center or centdian. In this article, we also allow negative entries of λ, allowing us to derive models for better addressing equity and fairness in facility location, for modeling obnoxious facility location problems or for including other client preference models. We present new mixed integer programming models for DOMP along with algorithmic enhancements for solving the DOMP to optimality using mixed integer programming techniques. Specifically, starting from state-of-the-art formulations from the literature, we present several Benders decomposition reformulations applied to them. Using these approaches, new state-of-the-art results have been obtained for different ordered weighting vectors.

Learning to sample initial solution for solving 0–1 discrete optimization problem by local search

Learning to sample initial solution for solving 0–1 discrete optimization problem by local search

Normalized homoclinic solutions of discrete nonlocal double phase problems

The aim of this paper is to discuss the existence of normalized solutions to the following nonlocal double phase problems driving by the discrete fractional Laplacian: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] if [Formula: see text], [Formula: see text] if [Formula: see text], and [Formula: see text]([Formula: see text] or [Formula: see text], [Formula: see text] or [Formula: see text]) is the discrete fractional [Formula: see text]-Laplacian. By variational methods, we discuss the existence of non-negative normalized homoclinic solutions under the conditions that the nonlinear term satisfies sublinear growth or superlinear growth conditions. In particular, we establish the compactness of the associated energy functional of the problem without weights. Our paper is the first time to deal with the existence of normalized solutions for discrete double phase problems.