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

In this work we analyze the inverse problem of recovering the space-dependent potential coefficient in an elliptic / parabolic problem from distributed observation. We establish novel (weighted) conditional stability estimates under very mild conditions on the problem data. Then we provide an error analysis of a standard reconstruction scheme based on the standard output least-squares formulation with Tikhonov regularization (by an $H^1$-seminorm penalty), which is then discretized by the Galerkin finite element method with continuous piecewise linear finite elements in space (and also backward Euler method in time for parabolic problems). We present a detailed analysis of the discrete scheme, and provide convergence rates in a weighted $L^2(\Omega)$ for discrete approximations with respect to the exact potential. The error bounds are explicitly dependent on the noise level, regularization parameter and discretization parameter(s). Under suitable conditions, we also derive error estimates in the standard $L^2(\Omega)$ and interior $L^2$ norms. The analysis employs sharp a priori error estimates and nonstandard test functions. Several numerical experiments are given to complement the theoretical analysis.

Image reconstruction using deep learning algorithms offers improved reconstruction quality and lower reconstruction time than classical compressed sensing and model-based algorithms. Unfortunately, clean and fully sampled ground-truth data to train the deep networks is often unavailable in several applications, restricting the applicability of the above methods. We introduce a novel metric termed the ENsemble Stein's Unbiased Risk Estimate (ENSURE) framework, which can be used to train deep image reconstruction algorithms without fully sampled and noise-free images. The proposed framework is the generalization of the classical SURE and GSURE formulation to the setting where the images are sampled by different measurement operators, chosen randomly from a set. We evaluate the expectation of the GSURE loss functions over the sampling patterns to obtain the ENSURE loss function. We show that this loss is an unbiased estimate for the true mean-square error, which offers a better alternative to GSURE, which only offers an unbiased estimate for the projected error. Our experiments show that the networks trained with this loss function can offer reconstructions comparable to the supervised setting. While we demonstrate this framework in the context of MR image recovery, the ENSURE framework is generally applicable to arbitrary inverse problems.

Many applications, such as system identification, classification of time series, direct and inverse problems in partial differential equations, and uncertainty quantification lead to the question of approximation of a non-linear operator between metric spaces $\mathfrak{X}$ and $\mathfrak{Y}$. We study the problem of determining the degree of approximation of such operators on a compact subset $K_\mathfrak{X}\subset \mathfrak{X}$ using a finite amount of information. If $\mathcal{F}: K_\mathfrak{X}\to K_\mathfrak{Y}$, a well established strategy to approximate $\mathcal{F}(F)$ for some $F\in K_\mathfrak{X}$ is to encode $F$ (respectively, $\mathcal{F}(F)$) in terms of a finite number $d$ (repectively $m$) of real numbers. Together with appropriate reconstruction algorithms (decoders), the problem reduces to the approximation of $m$ functions on a compact subset of a high dimensional Euclidean space $\mathbb{R}^d$, equivalently, the unit sphere $\mathbb{S}^d$ embedded in $\mathbb{R}^{d+1}$. The problem is challenging because $d$, $m$, as well as the complexity of the approximation on $\mathbb{S}^d$ are all large, and it is necessary to estimate the accuracy keeping track of the inter-dependence of all the approximations involved. In this paper, we establish constructive methods to do this efficiently; i.e., with the constants involved in the estimates on the approximation on $\mathbb{S}^d$ being $\mathcal{O}(d^{1/6})$. We study different smoothness classes for the operators, and also propose a method for approximation of $\mathcal{F}(F)$ using only information in a small neighborhood of $F$, resulting in an effective reduction in the number of parameters involved.

We provide a concise review of the exponentially convergent multiscale finite element method (ExpMsFEM) for efficient model reduction of PDEs in heterogeneous media without scale separation and in high-frequency wave propagation. ExpMsFEM is built on the non-overlapped domain decomposition in the classical MsFEM while enriching the approximation space systematically to achieve a nearly exponential convergence rate regarding the number of basis functions. Unlike most generalizations of MsFEM in the literature, ExpMsFEM does not rely on any partition of unity functions. In general, it is necessary to use function representations dependent on the right-hand side to break the algebraic Kolmogorov $n$-width barrier to achieve exponential convergence. Indeed, there are online and offline parts in the function representation provided by ExpMsFEM. The online part depends on the right-hand side locally and can be computed in parallel efficiently. The offline part contains basis functions that are used in the Galerkin method to assemble the stiffness matrix; they are all independent of the right-hand side, so the stiffness matrix can be used repeatedly in multi-query scenarios.

In this paper, we study the almost sure boundedness and the convergence of the stochastic approximation (SA) algorithm. At present, most available convergence proofs are based on the ODE method, and the almost sure boundedness of the iterations is an assumption and not a conclusion. In Borkar-Meyn (2000), it is shown that if the ODE has only one globally attractive equilibrium, then under additional assumptions, the iterations are bounded almost surely, and the SA algorithm converges to the desired solution. Our objective in the present paper is to provide an alternate proof of the above, based on martingale methods, which are simpler and less technical than those based on the ODE method. As a prelude, we prove a new sufficient condition for the global asymptotic stability of an ODE. Next we prove a ``converse'' Lyapunov theorem on the existence of a suitable Lyapunov function with a globally bounded Hessian, for a globally exponentially stable system. Both theorems are of independent interest to researchers in stability theory. Then, using these results, we provide sufficient conditions for the almost sure boundedness and the convergence of the SA algorithm. We show through examples that our theory covers some situations that are not covered by currently known results, specifically Borkar-Meyn (2000).

We propose a conservative energy method based on neural networks with subdomains for solving variational problems (CENN), where the admissible function satisfying the essential boundary condition without boundary penalty is constructed by the radial basis function (RBF), particular solution neural network, and general neural network. The loss term is the potential energy, optimized based on the principle of minimum potential energy. The loss term at the interfaces has the lower order derivative compared to the strong form PINN with subdomains. The advantage of the proposed method is higher efficiency, more accurate, and less hyperparameters than the strong form PINN with subdomains. Another advantage of the proposed method is that it can apply to complex geometries based on the special construction of the admissible function. To analyze its performance, the proposed method CENN is used to model representative PDEs, the examples include strong discontinuity, singularity, complex boundary, non-linear, and heterogeneous problems. Furthermore, it outperforms other methods when dealing with heterogeneous problems.

We obtain Marcinkiewicz-Zygmund strong laws of large numbers for weighted sums of pairwise positively quadrant dependent random variables stochastically dominated by a random variable $X \in \mathscr{L}_{p}$, $1 \leqslant p < 2$. We use our results to establish the strong consistency of estimators which emerge from regression models having pairwise positively quadrant dependent errors.

The time-varying quadratic miniaturization (TVQM) problem, as a hotspot currently, urgently demands a more reliable and faster--solving model. To this end, a novel adaptive coefficient constructs framework is presented and realized to improve the performance of the solution model, leading to the adaptive zeroing-type neural dynamics (AZTND) model. Then the AZTND model is applied to solve the TVQM problem. The adaptive coefficients can adjust the step size of the model online so that the solution model converges faster. At the same time, the integration term develops to enhance the robustness of the model in a perturbed environment. Experiments demonstrate that the proposed model shows faster convergence and more reliable robustness than existing approaches. Finally, the AZTND model is applied in a target tracking scheme, proving the practicality of our proposed model.

Two combined numerical methods for solving time-varying semilinear differential-algebraic equations (DAEs) are obtained. These equations are also called degenerate DEs, descriptor systems, operator-differential equations and DEs on manifolds. The convergence and correctness of the methods are proved. When constructing methods we use, in particular, time-varying spectral projectors which can be numerically found. This enables to numerically solve and analyze the considered DAE in the original form without additional analytical transformations. To improve the accuracy of the second method, recalculation (a ``predictor-corrector'' scheme) is used. Note that the developed methods are applicable to the DAEs with the continuous nonlinear part which may not be continuously differentiable in $t$, and that the restrictions of the type of the global Lipschitz condition, including the global condition of contractivity, are not used in the theorems on the global solvability of the DAEs and on the convergence of the numerical methods. This enables to use the developed methods for the numerical solution of more general classes of mathematical models. For example, the functions of currents and voltages in electric circuits may not be differentiable or may be approximated by nondifferentiable functions. Presented conditions for the global solvability of the DAEs ensure the existence of an unique exact global solution for the corresponding initial value problem, which enables to compute approximate solutions on any given time interval (provided that the conditions of theorems or remarks on the convergence of the methods are fulfilled). In the paper, the numerical analysis of the mathematical model for a certain electrical circuit, which demonstrates the application of the presented theorems and numerical methods, is carried out.

There are existing standard solvers for tackling discrete optimization problems. However, in practice, it is uncommon to apply them directly to the large input space typical of this class of problems. Rather, the input is preprocessed to look for simplifications and to extract the core subset of the problem space, which is called the Kernel. This pre-processing procedure is known in the context of parameterized complexity theory as Kernelization. In this thesis, I implement parallel versions of some Kernelization algorithms and evaluate their performance. The performance of Kernelization algorithms is measured either by the size of the output Kernel or by the time it takes to compute the kernel. Sometimes the Kernel is the same as the original input, so it is desirable to know this, as soon as possible. The problem scope is limited to a particular type of discrete optimisation problem which is a version of the K-clique problem in which nodes of the given graph are pre-coloured legally using k colours. The final evaluation shows that my parallel implementations achieve over 50% improvement in efficiency for at least one of these algorithms. This is attained not just in terms of speed, but it is also able to produce a smaller kernel.