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This paper presents a point-wise divergence-free projection method for numerical approximations of photonic quasicrystals problems. The original three-dimensional quasiperiodic Maxwell's system is transformed into a periodic one in higher dimensions through a variable substitution involving the projection matrix, such that periodic boundary condition can be readily applied. To deal with the intrinsic divergence-free constraint of the Maxwell's equations, we present a quasiperiodic de Rham complex and its associated commuting diagram, based on which a point-wise divergence-free quasiperiodic Fourier spectral basis is proposed. With the help of this basis, we then propose an efficient solution algorithm for the quasiperiodic source problem and conduct its rigorous error estimate. Moreover, by analyzing the decay rate of the Fourier coefficients of the eigenfunctions, we further propose a divergence-free reduced projection method for the quasiperiodic Maxwell eigenvalue problem, which significantly alleviates the computational cost. Several numerical experiments are presented to validate the efficiency and accuracy of the proposed method., Comment: 34 pages, 7 figures

Dual function radar and communication (DFRC) is a promising research direction within integrated sensing and communication (ISAC), improving hardware and spectrum efficiency by merging sensing and communication (S&C) functionalities into a shared platform. However, the DFRC receiver (DFRC-R) is tasked with both uplink communication signal detection and simultaneously target-related parameter estimation from the echoes, leading to issues with mutual interference. In this paper, a projection-based scheme is proposed to equivalently transform the joint signal detection and target estimation problem into a joint signal detection process across multiple snapshots. Compared with conventional successive interference cancellation (SIC) schemes, our proposed approach achieves a higher signal-to-noise ratio (SNR), and a higher ergodic rate when the radar signal is non-negligible. Nonetheless, it introduces an ill-conditioned signal detection problem, which is addressed using a non-linear detector. By jointly processing an increased number of snapshots, the proposed scheme can achieve high S&C performance simultaneously., Comment: 5 pages, 3 figures, accepted by IEEE WCL

We investigate a fast-reaction--diffusion system modelling the effect of autotoxicity on plant-growth dynamics, in which the fast-reaction terms are based on the dichotomy between healthy and exposed roots depending on the toxicity. The model was proposed in [Giannino, Iuorio, Soresina, forthcoming] to account for stable stationary spacial patterns considering only biomass and toxicity, and its fast-reaction (cross-diffusion) limit was formally derived and numerically investigated. In this paper, the cross-diffusion limiting system is rigorously obtained as the fast-reaction limit of the reaction-diffusion system with fast-reaction terms by performing a bootstrap argument involving energies. Then, a thorough well-posedness analysis of the cross-diffusion system is presented, including a $L^\infty_{x,t}$-bound, uniqueness, stability, and regularity of weak solutions. This analysis, in turn, becomes crucial to establish the convergence rate for the fast reaction limit, thanks to the key idea of using an inverse Neumann Laplacian operator. Finally, numerical experiments illustrate the analytical findings on the convergence rate.

Many applications using large datasets require efficient methods for minimizing a proximable convex function subject to satisfying a set of linear constraints within a specified tolerance. For this task, we present a proximal projection (PP) algorithm, which is an instance of Douglas-Rachford splitting that directly uses projections onto the set of constraints. Formal guarantees are presented to prove convergence of PP estimates to optimizers. Unlike many methods that obtain feasibility asymptotically, each PP iterate is feasible. Numerically, we show PP either matches or outperforms alternatives (e.g. linearized Bregman, primal dual hybrid gradient, proximal augmented Lagrangian, proximal gradient) on problems in basis pursuit, stable matrix completion, stable principal component pursuit, and the computation of earth mover's distances.

In this paper, we propose a new algorithm, the irrational-window-filter projection method (IWFPM), for solving arbitrary dimensional global quasiperiodic systems. Based on the projection method (PM), IWFPM further utilizes the concentrated distribution of Fourier coefficients to filter out relevant spectral points using an irrational window. Moreover, a corresponding index-shift transform is designed to make the Fast Fourier Transform available. The corresponding error analysis on the function approximation level is also given. We apply IWFPM to 1D, 2D, and 3D quasiperiodic Schr\"odinger eigenproblems to demonstrate its accuracy and efficiency. IWFPM exhibits a significant computational advantage over PM for both extended and localized quantum states. Furthermore, the widespread existence of such spectral point distribution feature can endow IWFPM with significant potential for broader applications in quasiperiodic systems.

The projection method is the standard approach for numerically integrating the incompressible Navier-Stokes equation initial-boundary-value problem. Typical boundary conditions specify either the velocity or the gradient velocity on the boundary. Here, we consider the pressure-tangential-velocity boundary condition in which the tangential components of the velocity and the pressure at the boundary are specified., Comment: 12 pages, 1 figure

In this study, we address the challenge of solving elliptic equations with quasiperiodic coefficients. To achieve accurate and efficient computation, we introduce the projection method, which enables the embedding of quasiperiodic systems into higher-dimensional periodic systems. To enhance the computational efficiency, we propose a compressed storage strategy for the stiffness matrix by its multi-level block circulant structure, significantly reducing memory requirements. Furthermore, we design a diagonal preconditioner to efficiently solve the resulting high-dimensional linear system by reducing the condition number of the stiffness matrix. These techniques collectively contribute to the computational effectiveness of our proposed approach. Convergence analysis shows the spectral accuracy of the proposed method. We demonstrate the effectiveness and accuracy of our approach through a series of numerical examples. Moreover, we apply our method to achieve a highly accurate computation of the homogenized coefficients for a quasiperiodic multiscale elliptic equation.

We propose and analyze a novel approach to construct structure preserving approximations for the Poisson-Nernst-Planck equations, focusing on the positivity preserving and mass conservation properties. The strategy consists of a standard time marching step with a projection (or correction) step to satisfy the desired physical constraints (positivity and mass conservation). Based on the $L^2$ projection, we construct a second order Crank-Nicolson type finite difference scheme, which is linear (exclude the very efficient $L^2$ projection part), positivity preserving and mass conserving. Rigorous error estimates in $L^2$ norm are established, which are both second order accurate in space and time. The other choice of projection, e.g. $H^1$ projection, is discussed. Numerical examples are presented to verify the theoretical results and demonstrate the efficiency of the proposed method.

We present the class of projection methods for community detection that generalizes many popular community detection methods. In this framework, we represent each clustering (partition) by a vector on a high-dimensional hypersphere. A community detection method is a projection method if it can be described by the following two-step approach: 1) the graph is mapped to a query vector on the hypersphere; and 2) the query vector is projected on the set of clustering vectors. This last projection step is performed by minimizing the distance between the query vector and the clustering vector, over the set of clusterings. We prove that optimizing Markov stability, modularity, the likelihood of planted partition models and correlation clustering fit this framework. A consequence of this equivalence is that algorithms for each of these methods can be modified to perform the projection step in our framework. In addition, we show that these different methods suffer from the same granularity problem: they have parameters that control the granularity of the resulting clustering, but choosing these to obtain clusterings of the desired granularity is nontrivial. We provide a general heuristic to address this granularity problem, which can be applied to any projection method. Finally, we show how, given a generator of graphs with community structure, we can optimize a projection method for this generator in order to obtain a community detection method that performs well on this generator.

Variable projection methods prove highly efficient in solving separable nonlinear least squares problems by transforming them into a reduced nonlinear least squares problem, typically solvable via the Gauss-Newton method. When solving large-scale separable nonlinear inverse problems with general-form Tikhonov regularization, the computational demand for computing Jacobians in the Gauss-Newton method becomes very challenging. To mitigate this, iterative methods, specifically LSQR, can be used as inner solvers to compute approximate Jacobians. This article analyzes the impact of these approximate Jacobians within the variable projection method and introduces stopping criteria to ensure convergence. We also present numerical experiments where we apply the proposed method to solve a blind deconvolution problem to illustrate and confirm our theoretical results., Comment: 18 pages, 5 figures

We study the convergence rate of the alternating projection method (APM) applied to the intersection of an affine subspace and the second-order cone. We show that when they intersect non-transversally, the convergence rate is $O(k^{-1/2})$, where $k$ is the number of iterations of the APM. In particular, when the intersection is not at the origin or forms a half-line with the origin as the endpoint, the obtained convergence rate can be exact because a lower bound of the convergence rate is evaluated. These results coincide with the worst-case convergence rate obtained from the error bound discussed in [Borwein et al., SIOPT, 2014] and [Drusvyatskiy et al., Math. Prog., 2017]. Moreover, we consider the convergence rate of the APM for the intersection of an affine subspace and the product of two second-order cones. We provide an example that the worst-case convergence rate of the APM is better than the rate expected from the error bound for the example., Comment: All comments and suggestions are welcome

In this paper, we present a new method to solve a certain type of Semidefinite Programming (SDP) problems. These types of SDPs naturally arise in the Quadratic Convex Reformulation (QCR) method and can be used to obtain dual bounds of Quadratic Unconstrained Binary Optimization (QUBO) problems. QUBO problems have recently become the focus of attention in the quantum computing and optimization communities as they are well suited to both gate-based and annealing-based quantum computers on one side, and can encompass an exceptional variety of combinatorial optimization problems on the other. Our new method can be easily warm-started, making it very effective when embedded into a branch-and-bound scheme and used to solve the QUBO problem to global optimality. We test our method in this setting and present computational results showing the effectiveness of our approach.

We prove that the Douglas--Rachford method applied to two closed convex cones in the Euclidean plane converges in finitely many steps if and only if the set of fixed points of the Douglas--Rachford operator is nontrivial. We analyze this special case using circle dynamics. We also construct explicit examples for a broad family of projection methods for which the set of fixed points of the relevant projection method operator is nontrivial, but the convergence is not finite. This three-parametric family is well known in the projection method literature and includes both the Douglas--Rachford method and the classic method of alternating projections. Even though our setting is fairly elementary, this work contributes in a new way to the body of theoretical research justifying the superior performance of the Douglas--Rachford method compared to other techniques. Moreover, our result leads to a neat sufficient condition for finite convergence of the Douglas--Rachford method in the locally polyhedral case on the plane, unifying and expanding several special cases available in the literature., Comment: 18 pages, 7 figures

The numerical solution of the Stokes equations on an evolving domain with a moving boundary is studied based on the arbitrary Lagrangian-Eulerian finite element method and a second-order projection method along the trajectories of the evolving mesh for decoupling the unknown solutions of velocity and pressure. The error of the semidiscrete arbitrary Lagrangian-Eulerian method is shown to be $O(h^{r+1})$ for the Taylor--Hood finite elements of degree $r\ge 2$, using Nitsche's duality argument adapted to an evolving mesh, by proving that the material derivative and the Stokes--Ritz projection commute up to terms which have optimal-order convergence in the $L^2$ norm. Additionally, the error of the fully discrete finite element method, with a second-order projection method along the trajectories of the evolving mesh, is shown to be $O(\ln(1/\tau+1)\tau^{2}+\ln(1/h+1)h^{r+1})$ in the discrete $L^\infty(0,T; L^2)$ norm using newly developed energy techniques and backward parabolic duality arguments that are applicable to the Stokes equations with an evolving mesh. To maintain consistency between the notations of the numerical scheme in a moving domain and those in a fixed domain, we introduce the equivalence class of finite element spaces across time levels. Numerical examples are provided to support the theoretical analysis and to illustrate the performance of the method in simulating Navier--Stokes flow in a domain with a rotating propeller.

This paper presents a reduced projection method for the solution of quasiperiodic Schr\"{o}dinger eigenvalue problems for photonic moir\'e lattices. Using the properties of the Schr\"{o}dinger operator in higher-dimensional space via a projection matrix, we rigorously prove that the generalized Fourier coefficients of the eigenfunctions exhibit faster decay rate along a fixed direction associated with the projection matrix. An efficient reduction strategy of the basis space is then proposed to reduce the degrees of freedom significantly. Rigorous error estimates of the proposed reduced projection method are provided, indicating that a small portion of the degrees of freedom is sufficient to achieve the same level of accuracy as the classical projection method. We present numerical examples of photonic moir\'e lattices in one and two dimensions to demonstrate the accuracy and efficiency of our proposed method., Comment: 25 pages, 9 figures

We propose a novel collocated projection method for solving the incompressible Navier-Stokes equations with arbitrary boundaries. Our approach employs non-graded octree grids, where all variables are stored at the nodes. To discretize the viscosity and projection steps, we utilize supra-convergent finite difference approximations with sharp boundary treatments. We demonstrate the stability of our projection on uniform grids, identify a sufficient stability condition on adaptive grids, and validate these findings numerically. We further demonstrate the accuracy and capabilities of our solver with several canonical two- and three-dimensional simulations of incompressible fluid flows. Overall, our method is second-order accurate, allows for dynamic grid adaptivity with arbitrary geometries, and reduces the overhead in code development through data collocation.

A new approach, iteration projection method, is proposed to solve the saddle point problem obtained after the full discretization of the unsteady Navier-Stokes equations. The proposed method iterates projections in each time step with a proper convection form. We prove that the projection iterations converge with a certain parameter regime. Optimal iteration convergence can be achieved with the modulation of parameter values. This new method has several significant improvements over the Uzawa method and the projection method both theoretically and practically. First, when the iterative projections are fully convergent in each time step, the numerical velocity is weakly divergence free (pointwise divergence free in divergence free finite element methods), and the stability and error estimate are rigorously proven. With proper parameters, this method converges much faster than the Uzawa method. Second, numerical simulations show that with rather relaxed stopping criteria which require only a few iterations each time step, the numerical solution preserves stability and accuracy for high Reynolds numbers, where the convectional projection method would fail. Furthermore, this method retains the efficiency of the traditional projection method by decoupling the velocity and pressure fields, which splits the saddle point system into small elliptic problems. Three dimensional simulations with Taylor-Hood P2/P1 finite elements are presented to demonstrate the performance and efficiency of this method. More importantly, this method is a generic approach and thus there are many potential improvements and extensions of the iterative projection method, including utilization of various convection forms, association with stabilization techniques for high Reynolds numbers, applications to other saddle point problems.

A low-dissipative solution framework integrating various types of high-order time scheme is proposed and implemented based on the open-source C++ library OpenFOAM. This framework aims to introduce different categories of low-dissipative time integration schemes into a unified solver convenient for comparison of scheme performance in finite volume computational fluid dynamics code, contributing to the development of low dissipation scheme appropriate for scale-resolving turbulence simulation. To demonstrate this general framework's ability of including a wide range of time integration method, in addition to typical Runge--Kutta family schemes of linear single-step method, two more complex linear multi-step method, Adams--Bashforth family and Adams--Bashforth--Moutton family schemes, are implemented with the projection algorithm, which increase the options of time discretization. The unified solver obtained by the solution framework select the specified time scheme from a variety of alternatives in runtime rather than maintaining multiple solvers with each compiled for a single scheme, while new scheme can be easily added according to the basic idea of this universal framework. Further research on the stability of the explicit scheme indicates that a multi-step method with an appropriate order may be an optimal choice when taking both the prediction accuracy and computational efficiency into account in unstable problems.

Semi-supervised clustering problems focus on clustering data with labels. In this paper,we consider the semi-supervised hypergraph problems. We use the hypergraph related tensor to construct an orthogonal constrained optimization model. The optimization problem is solved by a retraction method, which employs the polar decomposition to map the gradient direction in the tangent space to the Stefiel manifold. A nonmonotone curvilinear search is implemented to guarantee reduction in the objective function value. Convergence analysis demonstrates that the first order optimality condition is satisfied at the accumulation point. Experiments on synthetic hypergraph and hypergraph given by real data demonstrate the effectivity of our method.