亚洲男人的天堂2018av,欧美草比,久久久久久免费视频精选,国色天香在线看免费,久久久久亚洲av成人片仓井空

This study investigates the misclassification excess risk bound in the context of 1-bit matrix completion, a significant problem in machine learning involving the recovery of an unknown matrix from a limited subset of its entries. Matrix completion has garnered considerable attention in the last two decades due to its diverse applications across various fields. Unlike conventional approaches that deal with real-valued samples, 1-bit matrix completion is concerned with binary observations. While prior research has predominantly focused on the estimation error of proposed estimators, our study shifts attention to the prediction error. This paper offers theoretical analysis regarding the prediction errors of two previous works utilizing the logistic regression model: one employing a max-norm constrained minimization and the other employing nuclear-norm penalization. Significantly, our findings demonstrate that the latter achieves the minimax-optimal rate without the need for an additional logarithmic term. These novel results contribute to a deeper understanding of 1-bit matrix completion by shedding light on the predictive performance of specific methodologies.

This paper proposes well-conditioned boundary integral equations based on the Burton-Miller method for solving transmission problems. The Burton-Miller method is widely accepted as a highly accurate numerical method based on the boundary integral equation for solving exterior wave problems. While this method can also be applied to solve the transmission problems, a straightforward formulation may yield ill-conditioned integral equations. Consequently, the resulting linear algebraic equations may involve a coefficient matrix with a huge condition number, leading to poor convergence of Krylov-based linear solvers. To address this challenge, our study enhances Burton-Miller-type boundary integral equations tailored for transmission problems by exploiting the Calderon formula. In cases where a single material exists in an unbounded host medium, we demonstrate the formulation of the boundary integral equation such that the underlying integral operator ${\cal A}$ is spectrally well-conditioned. Specifically, ${\cal A}$ can be designed in a simple manner that ensures ${\cal A}^2$ is bounded and has only a single eigenvalue accumulation point. Furthermore, we extend our analysis to the multi-material case, proving that the square of the proposed operator has only a few eigenvalues except for a compact perturbation. Through numerical examples of several benchmark problems, we illustrate that our formulation reduces the iteration number required by iterative linear solvers, even in the presence of material junction points; locations where three or more sub-domains meet on the boundary.

This paper studies the infinite-time stability of the numerical scheme for stochastic McKean-Vlasov equations (SMVEs) via stochastic particle method. The long-time propagation of chaos in mean-square sense is obtained, with which the almost sure propagation in infinite horizon is proved by exploiting the Chebyshev inequality and the Borel-Cantelli lemma. Then the mean-square and almost sure exponential stabilities of the Euler-Maruyama scheme associated with the corresponding interacting particle system are shown through an ingenious manipulation of empirical measure. Combining the assertions enables the numerical solutions to reproduce the stabilities of the original SMVEs. The examples are demonstrated to reveal the importance of this study.

We derive and analyze a symmetric interior penalty discontinuous Galerkin scheme for the approximation of the second-order form of the radiative transfer equation in slab geometry. Using appropriate trace lemmas, the analysis can be carried out as for more standard elliptic problems. Supporting examples show the accuracy and stability of the method also numerically, for different polynomial degrees. For discretization, we employ quad-tree grids, which allow for local refinement in phase-space, and we show exemplary that adaptive methods can efficiently approximate discontinuous solutions. We investigate the behavior of hierarchical error estimators and error estimators based on local averaging.

The Neumann--Neumann method is a commonly employed domain decomposition method for linear elliptic equations. However, the method exhibits slow convergence when applied to semilinear equations and does not seem to converge at all for certain quasilinear equations. We therefore propose two modified Neumann--Neumann methods that have better convergence properties and require less computations. We provide numerical results that show the advantages of these methods when applied to both semilinear and quasilinear equations. We also prove linear convergence with mesh-independent error reduction under certain assumptions on the equation. The analysis is carried out on general Lipschitz domains and relies on the theory of nonlinear Steklov--Poincar\'e operators.

The filtered Lie splitting scheme is an established method for the numerical integration of the periodic nonlinear Schr\"{o}dinger equation at low regularity. Its temporal convergence was recently analyzed in a framework of discrete Bourgain spaces in one and two space dimensions for initial data in $H^s$ with $0<s\leq 2$. Here, this analysis is extended to dimensions $d=3, 4, 5$ for data satisfying $d/2-1 < s \leq 2$. In this setting, convergence of order $s/2$ in $L^2$ is proven. Numerical examples illustrate these convergence results.

By incorporating a new matrix splitting and the momentum acceleration into the relaxed-based matrix splitting (RMS) method \cite{soso2023}, a generalization of the RMS (GRMS) iterative method for solving the generalized absolute value equations (GAVEs) is proposed. Unlike some existing methods, by using the Cauchy's convergence principle, we give some sufficient conditions for the existence and uniqueness of the solution to the GAVEs and prove that our method can converge to the unique solution of the GAVEs. Moreover, we can obtain a few new and weaker convergence conditions for some existing methods. Preliminary numerical experiments show that the proposed method is efficient.

In this paper, a two-sided variable-coefficient space-fractional diffusion equation with fractional Neumann boundary condition is considered. To conquer the weak singularity caused by the nonlocal space-fractional differential operators, by introducing an auxiliary fractional flux variable and using piecewise linear interpolations, a fractional block-centered finite difference (BCFD) method on general nonuniform grids is proposed. However, like other numerical methods, the proposed method still produces linear algebraic systems with unstructured dense coefficient matrices under the general nonuniform grids.Consequently, traditional direct solvers such as Gaussian elimination method shall require $\mathcal{O}(M^2)$ memory and $\mathcal{O}(M^3)$ computational work per time level, where $M$ is the number of spatial unknowns in the numerical discretization. To address this issue, we combine the well-known sum-of-exponentials (SOE) approximation technique with the fractional BCFD method to propose a fast version fractional BCFD algorithm. Based upon the Krylov subspace iterative methods, fast matrix-vector multiplications of the resulting coefficient matrices with any vector are developed, in which they can be implemented in only $\mathcal{O}(MN_{exp})$ operations per iteration, where $N_{exp}\ll M$ is the number of exponentials in the SOE approximation. Moreover, the coefficient matrices do not necessarily need to be generated explicitly, while they can be stored in $\mathcal{O}(MN_{exp})$ memory by only storing some coefficient vectors. Numerical experiments are provided to demonstrate the efficiency and accuracy of the method.

In this paper, a new two-relaxation-time regularized (TRT-R) lattice Boltzmann (LB) model for convection-diffusion equation (CDE) with variable coefficients is proposed. Within this framework, we first derive a TRT-R collision operator by constructing a new regularized procedure through the high-order Hermite expansion of non-equilibrium. Then a first-order discrete-velocity form of discrete source term is introduced to improve the accuracy of the source term. Finally and most importantly, a new first-order space-derivative auxiliary term is proposed to recover the correct CDE with variable coefficients. To evaluate this model, we simulate a classic benchmark problem of the rotating Gaussian pulse. The results show that our model has better accuracy, stability and convergence than other popular LB models, especially in the case of a large time step.

We investigate the performance of two approximation algorithms for the Hafnian of a nonnegative square matrix, namely the Barvinok and Godsil-Gutman estimators. We observe that, while there are examples of matrices for which these algorithms fail to provide a good approximation, the algorithms perform surprisingly well for adjacency matrices of random graphs. In most cases, the Godsil-Gutman estimator provides a far superior accuracy. For dense graphs, however, both estimators demonstrate a slow growth of the variance. For complete graphs, we show analytically that the relative variance $\sigma / \mu$ grows as a square root of the size of the graph. Finally, we simulate a Gaussian Boson Sampling experiment using the Godsil-Gutman estimator and show that the technique used can successfully reproduce low-order correlation functions.