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

We study least-squares trace regression when the parameter is the sum of a $r$-low-rank and a $s$-sparse matrices and a fraction $\epsilon$ of the labels is corrupted. For subgaussian distributions, we highlight three design properties. The first, termed $\PP$, handles additive decomposition and follows from a product process inequality. The second, termed $\IP$, handles both label contamination and additive decomposition. It follows from Chevet's inequality. The third, termed $\MP$, handles the interaction between the design and featured-dependent noise. It follows from a multiplier process inequality. Jointly, these properties entail the near-optimality of a tractable estimator with respect to the effective dimensions $d_{\eff,r}$ and $d_{\eff,s}$ for the low-rank and sparse components, $\epsilon$ and the failure probability $\delta$. This rate has the form $$ \mathsf{r}(n,d_{\eff,r}) + \mathsf{r}(n,d_{\eff,s}) + \sqrt{(1+\log(1/\delta))/n} + \epsilon\log(1/\epsilon). $$ Here, $\mathsf{r}(n,d_{\eff,r})+\mathsf{r}(n,d_{\eff,s})$ is the optimal uncontaminated rate, independent of $\delta$. Our estimator is adaptive to $(s,r,\epsilon,\delta)$ and, for fixed absolute constant $c>0$, it attains the mentioned rate with probability $1-\delta$ uniformly over all $\delta\ge\exp(-cn)$. Disconsidering matrix decomposition, our analysis also entails optimal bounds for a robust estimator adapted to the noise variance. Finally, we consider robust matrix completion. We highlight a new property for this problem: one can robustly and optimally estimate the incomplete matrix regardless of the \emph{magnitude of the corruption}. Our estimators are based on ``sorted'' versions of Huber's loss. We present simulations matching the theory. In particular, it reveals the superiority of ``sorted'' Huber loss over the classical Huber's loss.

Computer model calibration involves using partial and imperfect observations of the real world to learn which values of a model's input parameters lead to outputs that are consistent with real-world observations. When calibrating models with high-dimensional output (e.g. a spatial field), it is common to represent the output as a linear combination of a small set of basis vectors. Often, when trying to calibrate to such output, what is important to the credibility of the model is that key emergent physical phenomena are represented, even if not faithfully or in the right place. In these cases, comparison of model output and data in a linear subspace is inappropriate and will usually lead to poor model calibration. To overcome this, we present kernel-based history matching (KHM), generalising the meaning of the technique sufficiently to be able to project model outputs and observations into a higher-dimensional feature space, where patterns can be compared without their location necessarily being fixed. We develop the technical methodology, present an expert-driven kernel selection algorithm, and then apply the techniques to the calibration of boundary layer clouds for the French climate model IPSL-CM.

Given a gamma population with known shape parameter $\alpha$, we develop a general theory for estimating a function $g(\cdot)$ of the scale parameter $\beta$ with bounded variance. We begin by defining a sequential sampling procedure with $g(\cdot)$ satisfying some desired condition in proposing the stopping rule, and show the procedure enjoys appealing asymptotic properties. After these general conditions, we substitute $g(\cdot)$ with specific functions including the gamma mean, the gamma variance, the gamma rate parameter, and a gamma survival probability as four possible illustrations. For each illustration, Monte Carlo simulations are carried out to justify the remarkable performance of our proposed sequential procedure. This is further substantiated with a real data study on weights of newly born babies.

We construct and analyze finite element approximations of the Einstein tensor in dimension $N \ge 3$. We focus on the setting where a smooth Riemannian metric tensor $g$ on a polyhedral domain $\Omega \subset \mathbb{R}^N$ has been approximated by a piecewise polynomial metric $g_h$ on a simplicial triangulation $\mathcal{T}$ of $\Omega$ having maximum element diameter $h$. We assume that $g_h$ possesses single-valued tangential-tangential components on every codimension-1 simplex in $\mathcal{T}$. Such a metric is not classically differentiable in general, but it turns out that one can still attribute meaning to its Einstein curvature in a distributional sense. We study the convergence of the distributional Einstein curvature of $g_h$ to the Einstein curvature of $g$ under refinement of the triangulation. We show that in the $H^{-2}(\Omega)$-norm, this convergence takes place at a rate of $O(h^{r+1})$ when $g_h$ is an optimal-order interpolant of $g$ that is piecewise polynomial of degree $r \ge 1$. We provide numerical evidence to support this claim.

Contraction in Wasserstein 1-distance with explicit rates is established for generalized Hamiltonian Monte Carlo with stochastic gradients under possibly nonconvex conditions. The algorithms considered include splitting schemes of kinetic Langevin diffusion. As consequence, quantitative Gaussian concentration bounds are provided for empirical averages. Convergence in Wasserstein 2-distance, total variation and relative entropy are also given, together with numerical bias estimates.

The formulation of Bayesian inverse problems involves choosing prior distributions; choices that seem equally reasonable may lead to significantly different conclusions. We develop a computational approach to better understand the impact of the hyperparameters defining the prior on the posterior statistics of the quantities of interest. Our approach relies on global sensitivity analysis (GSA) of Bayesian inverse problems with respect to the hyperparameters defining the prior. This, however, is a challenging problem--a naive double loop sampling approach would require running a prohibitive number of Markov chain Monte Carlo (MCMC) sampling procedures. The present work takes a foundational step in making such a sensitivity analysis practical through (i) a judicious combination of efficient surrogate models and (ii) a tailored importance sampling method. In particular, we can perform accurate GSA of posterior prediction statistics with respect to prior hyperparameters without having to repeat MCMC runs. We demonstrate the effectiveness of the approach on a simple Bayesian linear inverse problem and a nonlinear inverse problem governed by an epidemiological model

This work presents a comparative study to numerically compute impulse approximate controls for parabolic equations with various boundary conditions. Theoretical controllability results have been recently investigated using a logarithmic convexity estimate at a single time based on a Carleman commutator approach. We propose a numerical algorithm for computing the impulse controls with minimal $L^2$-norms by adapting a penalized Hilbert Uniqueness Method (HUM) combined with a Conjugate Gradient (CG) method. We consider static boundary conditions (Dirichlet and Neumann) and dynamic boundary conditions. Some numerical experiments based on our developed algorithm are given to validate and compare the theoretical impulse controllability results.

We introduce a convergent hierarchy of lower bounds on the minimum value of a real homogeneous polynomial over the sphere. The main practical advantage of our hierarchy over the sum-of-squares (SOS) hierarchy is that the lower bound at each level of our hierarchy is obtained by a minimum eigenvalue computation, as opposed to the full semidefinite program (SDP) required at each level of SOS. In practice, this allows us to go to much higher levels than are computationally feasible for the SOS hierarchy. For both hierarchies, the underlying space at the $k$-th level is the set of homogeneous polynomials of degree $2k$. We prove that our hierarchy converges as $O(1/k)$ in the level $k$, matching the best-known convergence of the SOS hierarchy when the number of variables $n$ is less than the half-degree $d$ (the best-known convergence of SOS when $n \geq d$ is $O(1/k^2)$). More generally, we introduce a convergent hierarchy of minimum eigenvalue computations for minimizing the inner product between a real tensor and an element of the spherical Segre-Veronese variety, with similar convergence guarantees. As examples, we obtain hierarchies for computing the (real) tensor spectral norm, and for minimizing biquadratic forms over the sphere. Hierarchies of eigencomputations for more general constrained polynomial optimization problems are discussed.

For problems of time-harmonic scattering by rational polygonal obstacles, embedding formulae express the far-field pattern induced by any incident plane wave in terms of the far-field patterns for a relatively small (frequency-independent) set of canonical incident angles. Although these remarkable formulae are exact in theory, here we demonstrate that: (i) they are highly sensitive to numerical errors in practice, and; (ii) direct calculation of the coefficients in these formulae may be impossible for particular sets of canonical incident angles, even in exact arithmetic. Only by overcoming these practical issues can embedding formulae provide a highly efficient approach to computing the far-field pattern induced by a large number of incident angles. Here we propose solutions for problems (i) and (ii), backed up by theory and numerical experiments. Problem (i) is solved using techniques from computational complex analysis: we reformulate the embedding formula as a complex contour integral and prove that this is much less sensitive to numerical errors. In practice, this contour integral can be efficiently evaluated by residue calculus. Problem (ii) is addressed using techniques from numerical linear algebra: we oversample, considering more canonical incident angles than are necessary, thus expanding the space of valid coefficients vectors. The coefficients vectors can then be selected using either a least squares approach or column subset selection.

Despite the growing interest in parallel-in-time methods as an approach to accelerate numerical simulations in atmospheric modelling, improving their stability and convergence remains a substantial challenge for their application to operational models. In this work, we study the temporal parallelization of the shallow water equations on the rotating sphere combined with time-stepping schemes commonly used in atmospheric modelling due to their stability properties, namely an Eulerian implicit-explicit (IMEX) method and a semi-Lagrangian semi-implicit method (SL-SI-SETTLS). The main goal is to investigate the performance of parallel-in-time methods, namely Parareal and Multigrid Reduction in Time (MGRIT), when these well-established schemes are used on the coarse discretization levels and provide insights on how they can be improved for better performance. We begin by performing an analytical stability study of Parareal and MGRIT applied to a linearized ordinary differential equation depending on the choice of a coarse scheme. Next, we perform numerical simulations of two standard tests to evaluate the stability, convergence and speedup provided by the parallel-in-time methods compared to a fine reference solution computed serially. We also conduct a detailed investigation on the influence of artificial viscosity and hyperviscosity approaches, applied on the coarse discretization levels, on the performance of the temporal parallelization. Both the analytical stability study and the numerical simulations indicate a poorer stability behaviour when SL-SI-SETTLS is used on the coarse levels, compared to the IMEX scheme. With the IMEX scheme, a better trade-off between convergence, stability and speedup compared to serial simulations can be obtained under proper parameters and artificial viscosity choices, opening the perspective of the potential competitiveness for realistic models.