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We show that the error probability of reconstructing kernel matrices from Random Fourier Features for the Gaussian kernel function is at most $\mathcal{O}(R^{2/3} \exp(-D))$, where $D$ is the number of random features and $R$ is the diameter of the data domain. We also provide an information-theoretic method-independent lower bound of $\Omega((1-\exp(-R^2)) \exp(-D))$. Compared to prior work, we are the first to show that the error probability for random Fourier features is independent of the dimensionality of data points. As applications of our theory, we obtain dimension-independent bounds for kernel ridge regression and support vector machines.

A version of the convexification globally convergent numerical method is constructed for a coefficient inverse problem for a wave-like partial differential equation. The presence of the Carleman Weight Function in the corresponding Tikhonov-like cost functional ensures the global strict convexity of this functional. Numerical results are presented to illustrate the effectiveness and efficiency of the proposed method.

In this paper, we propose an eXtended Virtual Element Method (X-VEM) for two-dimensional linear elastic fracture. This approach, which is an extension of the standard Virtual Element Method (VEM), facilitates mesh-independent modeling of crack discontinuities and elastic crack-tip singularities on general polygonal meshes. For elastic fracture in the X-VEM, the standard virtual element space is augmented by additional basis functions that are constructed by multiplying standard virtual basis functions by suitable enrichment fields, such as asymptotic mixed-mode crack-tip solutions. The design of the X-VEM requires an extended projector that maps functions lying in the extended virtual element space onto a set spanned by linear polynomials and the enrichment fields. An efficient scheme to compute the mixed-mode stress intensity factors using the domain form of the interaction integral is described. The formulation permits integration of weakly singular functions to be performed over the boundary edges of the element. Numerical experiments are conducted on benchmark mixed-mode linear elastic fracture problems that demonstrate the sound accuracy and optimal convergence in energy of the proposed formulation.

We present Coordinated Proximal Policy Optimization (CoPPO), an algorithm that extends the original Proximal Policy Optimization (PPO) to the multi-agent setting. The key idea lies in the coordinated adaptation of step size during the policy update process among multiple agents. We prove the monotonicity of policy improvement when optimizing a theoretically-grounded joint objective, and derive a simplified optimization objective based on a set of approximations. We then interpret that such an objective in CoPPO can achieve dynamic credit assignment among agents, thereby alleviating the high variance issue during the concurrent update of agent policies. Finally, we demonstrate that CoPPO outperforms several strong baselines and is competitive with the latest multi-agent PPO method (i.e. MAPPO) under typical multi-agent settings, including cooperative matrix games and the StarCraft II micromanagement tasks.

This paper is concerned with superconvergence properties of the direct discontinuous Galerkin (DDG) method for two-dimensional nonlinear convection-diffusion equations. By using the idea of correction function, we prove that, for any piecewise tensor-product polynomials of degree $k\geq 2$, the DDG solution is superconvergent at nodes and Lobatto points, with an order of ${\cal O}(h^{2k})$ and ${\cal O}(h^{k+2})$, respectively. Moreover, superconvergence properties for the derivative approximation are also studied and the superconvergence points are identified at Gauss points, with an order of ${\cal O}(h^{k+1})$. Numerical experiments are presented to confirm the sharpness of all the theoretical findings.

In this paper, we revisit the $L_2$-norm error estimate for $C^0$-interior penalty analysis of Dirichlet boundary control problem governed by biharmonic operator. In this work, we have relaxed the interior angle condition of the domain from $120$ degrees to $180$ degrees, therefore this analysis can be carried out for any convex domain. The theoretical findings are illustrated by numerical experiments. Moreover, we propose a new analysis to derive the error estimates for the biharmonic equation with Cahn-Hilliard type boundary condition under minimal regularity assumption.

We consider the query complexity of finding a local minimum of a function defined on a graph, where at most $k$ rounds of interaction with the oracle are allowed. Rounds model parallel settings, where each query takes resources to complete and is executed on a separate processor. Thus the query complexity in $k$ rounds informs how many processors are needed to achieve a parallel time of $k$. We focus on the d-dimensional grid $[n]^d$, where the dimension $d$ is a constant, and consider two regimes for the number of rounds: constant and polynomial in n. We give algorithms and lower bounds that characterize the trade-off between the number of rounds of adaptivity and the query complexity of local search. When the number of rounds $k$ is constant, we show that the query complexity of local search in $k$ rounds is $\Theta\bigl(n^{\frac{d^{k+1} - d^k}{d^k - 1}}\bigl)$, for both deterministic and randomized algorithms. When the number of rounds is polynomial, i.e. $k = n^{\alpha}$ for $0 < \alpha < d/2$, the randomized query complexity is $\Theta\left(n^{d-1 - \frac{d-2}{d}\alpha}\right)$ for all $d \geq 5$. For $d=3$ and $d=4$, we show the same upper bound expression holds and give almost matching lower bounds. The local search analysis also enables us to characterize the query complexity of computing a Brouwer fixed point in rounds. Our proof technique for lower bounding the query complexity in rounds may be of independent interest as an alternative to the classical relational adversary method of Aaronson from the fully adaptive setting.

I consider the problem of integrating a function $f$ over the $d$-dimensional unit cube. I describe a multilevel Monte Carlo method that estimates the integral with variance at most $\epsilon^{2}$ in $O(d+\ln(d)d_{t}\epsilon^{-2})$ time, for $\epsilon>0$, where $d_{t}$ is the truncation dimension of $f$. In contrast, the standard Monte Carlo method typically achieves such variance in $O(d\epsilon^{-2})$ time. A lower bound of order $d+d_{t}\epsilon^{-2}$ is described for a class of multilevel Monte Carlo methods.

In this note, we consider the problem of differentially privately (DP) computing an anonymized histogram, which is defined as the multiset of counts of the input dataset (without bucket labels). In the low-privacy regime $\epsilon \geq 1$, we give an $\epsilon$-DP algorithm with an expected $\ell_1$-error bound of $O(\sqrt{n} / e^\epsilon)$. In the high-privacy regime $\epsilon < 1$, we give an $\Omega(\sqrt{n \log(1/\epsilon) / \epsilon})$ lower bound on the expected $\ell_1$ error. In both cases, our bounds asymptotically match the previously known lower/upper bounds due to [Suresh, NeurIPS 2019].

In this paper we discuss the numerical solution on a simple 2D domain of the Helmoltz equation with mixed boundary conditions. The so called radiation problem depends on the wavenumber constant parameter k and it is inspired here by medical applications, where a transducer emits a pulse at a given frequency. This problem has been successfully solved in the past with the classical Finite Element Method (FEM) for relative small values of k. But in modern applications the values of k can be of order of thousands and FEM faces up several numerical difficulties. To overcome these difficulties we solve the radiation problem using the Isogeometric Analysis (IgA), a kind of generalization of FEM. Starting with the variational formulation of the radiation problem, we show with details how to apply the isogeometric approach in order to compute the coefficients of the approximated solution of radiation problem in terms of the B-spline basis functions. Our implementation of IgA using GeoPDEs software shows that isogeometric approach is superior than FEM, since it is able to reduce substantially the pollution error, especially for high values of k, producing additionally smoother solutions which depend on less degrees of freedom.