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The scope of this paper is the analysis and approximation of an optimal control problem related to the Allen-Cahn equation. A tracking functional is minimized subject to the Allen-Cahn equation using distributed controls that satisfy point-wise control constraints. First and second order necessary and sufficient conditions are proved. The lowest order discontinuous Galerkin - in time - scheme is considered for the approximation of the control to state and adjoint state mappings. Under a suitable restriction on maximum size of the temporal and spatial discretization parameters $k$, $h$ respectively in terms of the parameter $\epsilon$ that describes the thickness of the interface layer, a-priori estimates are proved with constants depending polynomially upon $1/ \epsilon$. Unlike to previous works for the uncontrolled Allen-Cahn problem our approach does not rely on a construction of an approximation of the spectral estimate, and as a consequence our estimates are valid under low regularity assumptions imposed by the optimal control setting. These estimates are also valid in cases where the solution and its discrete approximation do not satisfy uniform space-time bounds independent of $\epsilon$. These estimates and a suitable localization technique, via the second order condition (see \cite{Arada-Casas-Troltzsch_2002,Casas-Mateos-Troltzsch_2005,Casas-Raymond_2006,Casas-Mateos-Raymond_2007}), allows to prove error estimates for the difference between local optimal controls and their discrete approximation as well as between the associated state and adjoint state variables and their discrete approximations

We study the computational complexity of two hard problems on determinantal point processes (DPPs). One is maximum a posteriori (MAP) inference, i.e., to find a principal submatrix having the maximum determinant. The other is probabilistic inference on exponentiated DPPs (E-DPPs), which can sharpen or weaken the diversity preference of DPPs with an exponent parameter $p$. We present several complexity-theoretic hardness results that explain the difficulty in approximating MAP inference and the normalizing constant for E-DPPs. We first prove that unconstrained MAP inference for an $n \times n$ matrix is $\textsf{NP}$-hard to approximate within a factor of $2^{\beta n}$, where $\beta = 10^{-10^{13}} $. This result improves upon the best-known inapproximability factor of $(\frac{9}{8}-\epsilon)$, and rules out the existence of any polynomial-factor approximation algorithm assuming $\textsf{P} \neq \textsf{NP}$. We then show that log-determinant maximization is $\textsf{NP}$-hard to approximate within a factor of $\frac{5}{4}$ for the unconstrained case and within a factor of $1+10^{-10^{13}}$ for the size-constrained monotone case. In particular, log-determinant maximization does not admit a polynomial-time approximation scheme unless $\textsf{P} = \textsf{NP}$. As a corollary of the first result, we demonstrate that the normalizing constant for E-DPPs of any (fixed) constant exponent $p \geq \beta^{-1} = 10^{10^{13}}$ is $\textsf{NP}$-hard to approximate within a factor of $2^{\beta pn}$, which is in contrast to the case of $p \leq 1$ admitting a fully polynomial-time randomized approximation scheme.

Despite the vast empirical success of neural networks, theoretical understanding of the training procedures remains limited, especially in providing performance guarantees of testing performance due to the non-convex nature of the optimization problem. Inspired by a recent work of (Juditsky & Nemirovsky, 2019), instead of using the traditional loss function minimization approach, we reduce the training of the network parameters to another problem with convex structure -- to solve a monotone variational inequality (MVI). The solution to MVI can be found by computationally efficient procedures, and importantly, this leads to performance guarantee of $\ell_2$ and $\ell_{\infty}$ bounds on model recovery accuracy and prediction accuracy under the theoretical setting of training one-layer linear neural network. In addition, we study the use of MVI for training multi-layer neural networks and propose a practical algorithm called \textit{stochastic variational inequality} (SVI), and demonstrates its applicability in training fully-connected neural networks and graph neural networks (GNN) (SVI is completely general and can be used to train other types of neural networks). We demonstrate the competitive or better performance of SVI compared to the stochastic gradient descent (SGD) on both synthetic and real network data prediction tasks regarding various performance metrics.

A nonlinear partial differential equation (PDE) that models the possible shapes that a periodic Miura tessellation can take in the homogenization limit has been established recently and solved only in specific cases. In this paper, the existence and uniqueness of a solution to the PDE is proved for general Dirichlet boundary conditions. Then a H^2-conforming discretization is introduced to approximate the solution of the PDE and a fixed point algorithm is proposed to solve the associated discrete problem. A convergence proof for the method is given as well as a convergence rate. Finally, numerical experiments show the robustness of the method and that non trivial shapes can be achieved using periodic Miura tessellations.

The asymptotic distribution of a wide class of V- and U-statistics with estimated parameters is derived in the case when the kernel is not necessarily differentiable along the parameter. The results have their application in goodness-of-fit problems.

We consider Gaussian measures $\mu, \tilde{\mu}$ on a separable Hilbert space, with fractional-order covariance operators $A^{-2\beta}$ resp. $\tilde{A}^{-2\tilde{\beta}}$, and derive necessary and sufficient conditions on $A, \tilde{A}$ and $\beta, \tilde{\beta} > 0$ for I. equivalence of the measures $\mu$ and $\tilde{\mu}$, and II. uniform asymptotic optimality of linear predictions for $\mu$ based on the misspecified measure $\tilde{\mu}$. These results hold, e.g., for Gaussian processes on compact metric spaces. As an important special case, we consider the class of generalized Whittle-Mat\'ern Gaussian random fields, where $A$ and $\tilde{A}$ are elliptic second-order differential operators, formulated on a bounded Euclidean domain $\mathcal{D}\subset\mathbb{R}^d$ and augmented with homogeneous Dirichlet boundary conditions. Our outcomes explain why the predictive performances of stationary and non-stationary models in spatial statistics often are comparable, and provide a crucial first step in deriving consistency results for parameter estimation of generalized Whittle-Mat\'ern fields.

We propose a new simulation-based estimation method, adversarial estimation, for structural models. The estimator is formulated as the solution to a minimax problem between a generator (which generates synthetic observations using the structural model) and a discriminator (which classifies if an observation is synthetic). The discriminator maximizes the accuracy of its classification while the generator minimizes it. We show that, with a sufficiently rich discriminator, the adversarial estimator attains parametric efficiency under correct specification and the parametric rate under misspecification. We advocate the use of a neural network as a discriminator that can exploit adaptivity properties and attain fast rates of convergence. We apply our method to the elderly's saving decision model and show that our estimator uncovers the bequest motive as an important source of saving across the wealth distribution, not only for the rich.

We analyze the dynamics of a random sequential message passing algorithm for approximate inference with large Gaussian latent variable models in a student-teacher scenario. To model nontrivial dependencies between the latent variables, we assume random covariance matrices drawn from rotation invariant ensembles. Moreover, we consider a model mismatching setting, where the teacher model and the one used by the student may be different. By means of dynamical functional approach, we obtain exact dynamical mean-field equations characterizing the dynamics of the inference algorithm. We also derive a range of model parameters for which the sequential algorithm does not converge. The boundary of this parameter range coincides with the de Almeida Thouless (AT) stability condition of the replica symmetric ansatz for the static probabilistic model.

This paper is concerned with two improved variants of the Hutch++ algorithm for estimating the trace of a square matrix, implicitly given through matrix-vector products. Hutch++ combines randomized low-rank approximation in a first phase with stochastic trace estimation in a second phase. In turn, Hutch++ only requires $O\left(\varepsilon^{-1}\right)$ matrix-vector products to approximate the trace within a relative error $\varepsilon$ with high probability. This compares favorably with the $O\left(\varepsilon^{-2}\right)$ matrix-vector products needed when using stochastic trace estimation alone. In Hutch++, the number of matrix-vector products is fixed a priori and distributed in a prescribed fashion among the two phases. In this work, we derive an adaptive variant of Hutch++, which outputs an estimate of the trace that is within some prescribed error tolerance with a controllable failure probability, while splitting the matrix-vector products in a near-optimal way among the two phases. For the special case of symmetric positive semi-definite matrix, we present another variant of Hutch++, called Nystr\"om++, which utilizes the so called Nystr\"om approximation and requires only one pass over the matrix, as compared to two passes with Hutch++. We extend the analysis of Hutch++ to Nystr\"om++. Numerical experiments demonstrate the effectiveness of our two new algorithms.

Sampling methods (e.g., node-wise, layer-wise, or subgraph) has become an indispensable strategy to speed up training large-scale Graph Neural Networks (GNNs). However, existing sampling methods are mostly based on the graph structural information and ignore the dynamicity of optimization, which leads to high variance in estimating the stochastic gradients. The high variance issue can be very pronounced in extremely large graphs, where it results in slow convergence and poor generalization. In this paper, we theoretically analyze the variance of sampling methods and show that, due to the composite structure of empirical risk, the variance of any sampling method can be decomposed into \textit{embedding approximation variance} in the forward stage and \textit{stochastic gradient variance} in the backward stage that necessities mitigating both types of variance to obtain faster convergence rate. We propose a decoupled variance reduction strategy that employs (approximate) gradient information to adaptively sample nodes with minimal variance, and explicitly reduces the variance introduced by embedding approximation. We show theoretically and empirically that the proposed method, even with smaller mini-batch sizes, enjoys a faster convergence rate and entails a better generalization compared to the existing methods.