The paper aims to study the performance of the amplitude-based model \newline $\widehat{\mathbf x} \in argmin{{\mathbf x}\in \mathbb{C}^d}\sum_{j=1}^m\left(|\langle {\mathbf a}_j,{\mathbf x}\rangle|-b_j\right)^2$, where $b_j:=|\langle {\mathbf a}_j,{\mathbf x}_0\rangle|+\eta_j$ and ${\mathbf x}_0\in \mathbb{C}^d$ is a target signal. The model is raised in phase retrieval as well as in absolute value rectification neural networks. Many efficient algorithms have been developed to solve it in the past decades. {However, there are very few results available regarding the estimation performance in the complex case under noisy conditions.} In this paper, {we present a theoretical guarantee on the amplitude-based model for the noisy complex phase retrieval problem}. Specifically, we show that $\min_{\theta\in[0,2\pi)}\|\widehat{\mathbf x}-\exp(\mathrm{i}\theta)\cdot{\mathbf x}_0\|_2 \lesssim \frac{\|{\mathbf \eta}\|_2}{\sqrt{m}}$ holds with high probability provided the measurement vectors ${\mathbf a}_j\in \mathbb{C}^d,$ $j=1,\ldots,m,$ are {i.i.d.} complex sub-Gaussian random vectors and $m\gtrsim d$. Here ${\mathbf \eta}=(\eta_1,\ldots,\eta_m)\in \mathbb{R}^m$ is the noise vector without any assumption on the distribution. Furthermore, we prove that the reconstruction error is sharp. For the case where the target signal ${\mathbf x}_0\in \mathbb{C}^{d}$ is sparse, we establish a similar result for the nonlinear constrained $\ell_1$ minimization model. { To accomplish this, we leverage a strong version of restricted isometry property for an operator on the space of simultaneous low-rank and sparse matrices.}
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The numerical solution of continuum damage mechanics (CDM) problems suffers from convergence-related challenges during the material softening stage, and consequently existing iterative solvers are subject to a trade-off between computational expense and solution accuracy. In this work, we present a novel unified arc-length (UAL) method, and we derive the formulation of the analytical tangent matrix and governing system of equations for both local and non-local gradient damage problems. Unlike existing versions of arc-length solvers that monolithically scale the external force vector, the proposed method treats the latter as an independent variable and determines the position of the system on the equilibrium path based on all the nodal variations of the external force vector. This approach renders the proposed solver substantially more efficient and robust than existing solvers used in CDM problems. We demonstrate the considerable advantages of the proposed algorithm through several benchmark 1D problems with sharp snap-backs and 2D examples under various boundary conditions and loading scenarios. The proposed UAL approach exhibits a superior ability of overcoming critical increments along the equilibrium path. Moreover, the proposed UAL method is 1-2 orders of magnitude faster than force-controlled arc-length and monolithic Newton-Raphson solvers.
Over the last two decades, the field of geometric curve evolutions has attracted significant attention from scientific computing. One of the most popular numerical methods for solving geometric flows is the so-called BGN scheme, which was proposed by Barrett, Garcke, and Nurnberg (J. Comput. Phys., 222 (2007), pp. 441{467), due to its favorable properties (e.g., its computational efficiency and the good mesh property). However, the BGN scheme is limited to first-order accuracy in time, and how to develop a higher-order numerical scheme is challenging. In this paper, we propose a fully discrete, temporal second-order parametric finite element method, which incorporates a mesh regularization technique when necessary, for solving geometric flows of curves. The scheme is constructed based on the BGN formulation and a semi-implicit Crank-Nicolson leap-frog time stepping discretization as well as a linear finite element approximation in space. More importantly, we point out that the shape metrics, such as manifold distance and Hausdorff distance, instead of function norms, should be employed to measure numerical errors. Extensive numerical experiments demonstrate that the proposed BGN-based scheme is second-order accurate in time in terms of shape metrics. Moreover, by employing the classical BGN scheme as a mesh regularization technique when necessary, our proposed second-order scheme exhibits good properties with respect to the mesh distribution.
This paper studies the extreme singular values of non-harmonic Fourier matrices. Such a matrix of size $m\times s$ can be written as $\Phi=[ e^{-2\pi i j x_k}]_{j=0,1,\dots,m-1, k=1,2,\dots,s}$ for some set $\mathcal{X}=\{x_k\}_{k=1}^s$. The main results provide explicit lower bounds for the smallest singular value of $\Phi$ under the assumption $m\geq 6s$ and without any restrictions on $\mathcal{X}$. They show that for an appropriate scale $\tau$ determined by a density criteria, interactions between elements in $\mathcal{X}$ at scales smaller than $\tau$ are most significant and depends on the multiscale structure of $\mathcal{X}$ at fine scales, while distances larger than $\tau$ are less important and only depend on the local sparsity of the far away points. Theoretical and numerical comparisons show that the main results significantly improve upon classical bounds and achieve the same rate that was previously discovered for more restrictive settings.
We prove tight bounds on the site percolation threshold for $k$-uniform hypergraphs of maximum degree $\Delta$ and for $k$-uniform hypergraphs of maximum degree $\Delta$ in which any pair of edges overlaps in at most $r$ vertices. The hypergraphs that achieve these bounds are hypertrees, but unlike in the case of graphs, there are many different $k$-uniform, $\Delta$-regular hypertrees. Determining the extremal tree for a given $k, \Delta, r$ involves an optimization problem, and our bounds arise from a convex relaxation of this problem. By combining our percolation bounds with the method of disagreement percolation we obtain improved bounds on the uniqueness threshold for the hard-core model on hypergraphs satisfying the same constraints. Our uniqueness conditions imply exponential weak spatial mixing, and go beyond the known bounds for rapid mixing of local Markov chains and existence of efficient approximate counting and sampling algorithms. Our results lead to natural conjectures regarding the aforementioned algorithmic tasks, based on the intuition that uniqueness thresholds for the extremal hypertrees for percolation determine computational thresholds.
We study the maximum-average submatrix problem, in which given an $N \times N$ matrix $J$ one needs to find the $k \times k$ submatrix with the largest average of entries. We study the problem for random matrices $J$ whose entries are i.i.d. random variables by mapping it to a variant of the Sherrington-Kirkpatrick spin-glass model at fixed magnetization. We characterize analytically the phase diagram of the model as a function of the submatrix average and the size of the submatrix $k$ in the limit $N\to\infty$. We consider submatrices of size $k = m N$ with $0 < m < 1$. We find a rich phase diagram, including dynamical, static one-step replica symmetry breaking and full-step replica symmetry breaking. In the limit of $m \to 0$, we find a simpler phase diagram featuring a frozen 1-RSB phase, where the Gibbs measure is composed of exponentially many pure states each with zero entropy. We discover an interesting phenomenon, reminiscent of the phenomenology of the binary perceptron: there exist efficient algorithms that provably work in the frozen 1-RSB phase.
We present a novel stabilized isogeometric formulation for the Stokes problem, where the geometry of interest is obtained via overlapping NURBS (non-uniform rational B-spline) patches, i.e., one patch on top of another in an arbitrary but predefined hierarchical order. All the visible regions constitute the computational domain, whereas independent patches are coupled through visible interfaces using Nitsche's formulation. Such a geometric representation inevitably involves trimming, which may yield trimmed elements of extremely small measures (referred to as bad elements) and thus lead to the instability issue. Motivated by the minimal stabilization method that rigorously guarantees stability for trimmed geometries [1], in this work we generalize it to the Stokes problem on overlapping patches. Central to our method is the distinct treatments for the pressure and velocity spaces: Stabilization for velocity is carried out for the flux terms on interfaces, whereas pressure is stabilized in all the bad elements. We provide a priori error estimates with a comprehensive theoretical study. Through a suite of numerical tests, we first show that optimal convergence rates are achieved, which consistently agrees with our theoretical findings. Second, we show that the accuracy of pressure is significantly improved by several orders using the proposed stabilization method, compared to the results without stabilization. Finally, we also demonstrate the flexibility and efficiency of the proposed method in capturing local features in the solution field.
A sequential pattern with negation, or negative sequential pattern, takes the form of a sequential pattern for which the negation symbol may be used in front of some of the pattern's itemsets. Intuitively, such a pattern occurs in a sequence if negated itemsets are absent in the sequence. Recent work has shown that different semantics can be attributed to these pattern forms, and that state-of-the-art algorithms do not extract the same sets of patterns. This raises the important question of the interpretability of sequential pattern with negation. In this study, our focus is on exploring how potential users perceive negation in sequential patterns. Our aim is to determine whether specific semantics are more "intuitive" than others and whether these align with the semantics employed by one or more state-of-the-art algorithms. To achieve this, we designed a questionnaire to reveal the semantics' intuition of each user. This article presents both the design of the questionnaire and an in-depth analysis of the 124 responses obtained. The outcomes indicate that two of the semantics are predominantly intuitive; however, neither of them aligns with the semantics of the primary state-of-the-art algorithms. As a result, we provide recommendations to account for this disparity in the conclusions drawn.
We propose a method to modify a polygonal mesh in order to fit the zero-isoline of a level set function by extending a standard body-fitted strategy to a tessellation with arbitrarily-shaped elements. The novel level set-fitted approach, in combination with a Discontinuous Galerkin finite element approximation, provides an ideal setting to model physical problems characterized by embedded or evolving complex geometries, since it allows skipping any mesh post-processing in terms of grid quality. The proposed methodology is firstly assessed on the linear elasticity equation, by verifying the approximation capability of the level set-fitted approach when dealing with configurations with heterogeneous material properties. Successively, we combine the level set-fitted methodology with a minimum compliance topology optimization technique, in order to deliver optimized layouts exhibiting crisp boundaries and reliable mechanical performances. An extensive numerical test campaign confirms the effectiveness of the proposed method.
Ordinary state-based peridynamic (OSB-PD) models have an unparalleled capability to simulate crack propagation phenomena in solids with arbitrary Poisson's ratio. However, their non-locality also leads to prohibitively high computational cost. In this paper, a fast solution scheme for OSB-PD models based on matrix operation is introduced, with which, the graphics processing units (GPUs) are used to accelerate the computation. For the purpose of comparison and verification, a commonly used solution scheme based on loop operation is also presented. An in-house software is developed in MATLAB. Firstly, the vibration of a cantilever beam is solved for validating the loop- and matrix-based schemes by comparing the numerical solutions to those produced by a FEM software. Subsequently, two typical dynamic crack propagation problems are simulated to illustrate the effectiveness of the proposed schemes in solving dynamic fracture problems. Finally, the simulation of the Brokenshire torsion experiment is carried out by using the matrix-based scheme, and the similarity in the shapes of the experimental and numerical broken specimens further demonstrates the ability of the proposed approach to deal with 3D non-planar fracture problems. In addition, the speed-up of the matrix-based scheme with respect to the loop-based scheme and the performance of the GPU acceleration are investigated. The results emphasize the high computational efficiency of the matrix-based implementation scheme.
We prove that for any graph $G$ of maximum degree at most $\Delta$, the zeros of its chromatic polynomial $\chi_G(x)$ (in $\mathbb{C}$) lie inside the disc of radius $5.94 \Delta$ centered at $0$. This improves on the previously best known bound of approximately $6.91\Delta$. We also obtain improved bounds for graphs of high girth. We prove that for every $g$ there is a constant $K_g$ such that for any graph $G$ of maximum degree at most $\Delta$ and girth at least $g$, the zeros of its chromatic polynomial $\chi_G(x)$ lie inside the disc of radius $K_g \Delta$ centered at $0$, where $K_g$ is the solution to a certain optimization problem. In particular, $K_g < 5$ when $g \geq 5$ and $K_g < 4$ when $g \geq 25$ and $K_g$ tends to approximately $3.86$ as $g \to \infty$. Key to the proof is a classical theorem of Whitney which allows us to relate the chromatic polynomial of a graph $G$ to the generating function of so-called broken-circuit-free forests in $G$. We also establish a zero-free disc for the generating function of all forests in $G$ (aka the partition function of the arboreal gas) which may be of independent interest.
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