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In this study, we examine numerical approximations for 2nd-order linear-nonlinear differential equations with diverse boundary conditions, followed by the residual corrections of the first approximations. We first obtain numerical results using the Galerkin weighted residual approach with Bernstein polynomials. The generation of residuals is brought on by the fact that our first approximation is computed using numerical methods. To minimize these residuals, we use the compact finite difference scheme of 4th-order convergence to solve the error differential equations in accordance with the error boundary conditions. We also introduce the formulation of the compact finite difference method of fourth-order convergence for the nonlinear BVPs. The improved approximations are produced by adding the error values derived from the approximations of the error differential equation to the weighted residual values. Numerical results are compared to the exact solutions and to the solutions available in the published literature to validate the proposed scheme, and high accuracy is achieved in all cases

The problem Power Dominating Set (PDS) is motivated by the placement of phasor measurement units to monitor electrical networks. It asks for a minimum set of vertices in a graph that observes all remaining vertices by exhaustively applying two observation rules. Our contribution is twofold. First, we determine the parameterized complexity of PDS by proving it is $W[P]$-complete when parameterized with respect to the solution size. We note that it was only known to be $W[2]$-hard before. Our second and main contribution is a new algorithm for PDS that efficiently solves practical instances. Our algorithm consists of two complementary parts. The first is a set of reduction rules for PDS that can also be used in conjunction with previously existing algorithms. The second is an algorithm for solving the remaining kernel based on the implicit hitting set approach. Our evaluation on a set of power grid instances from the literature shows that our solver outperforms previous state-of-the-art solvers for PDS by more than one order of magnitude on average. Furthermore, our algorithm can solve previously unsolved instances of continental scale within a few minutes.

A novel positive dependence property is introduced, called positive measure inducing (PMI for short), being fulfilled by numerous copula classes, including Gaussian, Fr\'echet, Farlie-Gumbel-Morgenstern and Frank copulas; it is conjectured that even all positive quadrant dependent Archimedean copulas meet this property. From a geometric viewpoint, a PMI copula concentrates more mass near the main diagonal than in the opposite diagonal. A striking feature of PMI copulas is that they impose an ordering on a certain class of copula-induced measures of concordance, the latter originating in Edwards et al. (2004) and including Spearman's rho $\rho$ and Gini's gamma $\gamma$, leading to numerous new inequalities such as $3 \gamma \geq 2 \rho$. The measures of concordance within this class are estimated using (classical) empirical copulas and the intrinsic construction via empirical checkerboard copulas, and the estimators' asymptotic behaviour is determined. Building upon the presented inequalities, asymptotic tests are constructed having the potential of being used for detecting whether the underlying dependence structure of a given sample is PMI, which in turn can be used for excluding certain copula families from model building. The excellent performance of the tests is demonstrated in a simulation study and by means of a real-data example.

There has a major problem in the current theory of hypothesis testing in which no unified indicator to evaluate the goodness of various test methods since the cost function or utility function usually relies on the specific application scenario, resulting in no optimal hypothesis testing method. In this paper, the problem of optimal hypothesis testing is investigated based on information theory. We propose an information-theoretic framework of hypothesis testing consisting of five parts: test information (TI) is proposed to evaluate the hypothesis testing, which depends on the a posteriori probability distribution function of hypotheses and independent of specific test methods; accuracy with the unit of bit is proposed to evaluate the degree of validity of specific test methods; the sampling a posteriori (SAP) probability test method is presented, which makes stochastic selections on the hypotheses according to the a posteriori probability distribution of the hypotheses; the probability of test failure is defined to reflect the probability of the failed decision is made; test theorem is proved that all accuracy lower than the TI is achievable. Specifically, for every accuracy lower than TI, there exists a test method with the probability of test failure tending to zero. Conversely, there is no test method whose accuracy is more than TI. Numerical simulations are performed to demonstrate that the SAP test is asymptotically optimal. In addition, the results show that the accuracy of the SAP test and the existing test methods, such as the maximum a posteriori probability, expected a posteriori probability, and median a posteriori probability tests, are not more than TI.

In this paper, we address the problem of distributed power allocation in a $K$ user fading multiple access wiretap channel, where global channel state information is limited, i.e., each user has knowledge of their own channel state with respect to Bob and Eve but only knows the distribution of other users' channel states. We model this problem as a Bayesian game, where each user is assumed to selfishly maximize his average \emph{secrecy capacity} with partial channel state information. In this work, we first prove that there is a unique Bayesian equilibrium in the proposed game. Additionally, the price of anarchy is calculated to measure the efficiency of the equilibrium solution. We also propose a fast convergent iterative algorithm for power allocation. Finally, the results are validated using simulation results.

This paper presents a novel approach to Bayesian nonparametric spectral analysis of stationary multivariate time series. Starting with a parametric vector-autoregressive model, the parametric likelihood is nonparametrically adjusted in the frequency domain to account for potential deviations from parametric assumptions. We show mutual contiguity of the nonparametrically corrected likelihood, the multivariate Whittle likelihood approximation and the exact likelihood for Gaussian time series. A multivariate extension of the nonparametric Bernstein-Dirichlet process prior for univariate spectral densities to the space of Hermitian positive definite spectral density matrices is specified directly on the correction matrices. An infinite series representation of this prior is then used to develop a Markov chain Monte Carlo algorithm to sample from the posterior distribution. The code is made publicly available for ease of use and reproducibility. With this novel approach we provide a generalization of the multivariate Whittle-likelihood-based method of Meier et al. (2020) as well as an extension of the nonparametrically corrected likelihood for univariate stationary time series of Kirch et al. (2019) to the multivariate case. We demonstrate that the nonparametrically corrected likelihood combines the efficiencies of a parametric with the robustness of a nonparametric model. Its numerical accuracy is illustrated in a comprehensive simulation study. We illustrate its practical advantages by a spectral analysis of two environmental time series data sets: a bivariate time series of the Southern Oscillation Index and fish recruitment and time series of windspeed data at six locations in California.

We analyze Newton's method with lazy Hessian updates for solving general possibly non-convex optimization problems. We propose to reuse a previously seen Hessian for several iterations while computing new gradients at each step of the method. This significantly reduces the overall arithmetical complexity of second-order optimization schemes. By using the cubic regularization technique, we establish fast global convergence of our method to a second-order stationary point, while the Hessian does not need to be updated each iteration. For convex problems, we justify global and local superlinear rates for lazy Newton steps with quadratic regularization, which is easier to compute. The optimal frequency for updating the Hessian is once every $d$ iterations, where $d$ is the dimension of the problem. This provably improves the total arithmetical complexity of second-order algorithms by a factor $\sqrt{d}$.

We consider finding flat, local minimizers by adding average weight perturbations. Given a nonconvex function $f: \mathbb{R}^d \rightarrow \mathbb{R}$ and a $d$-dimensional distribution $\mathcal{P}$ which is symmetric at zero, we perturb the weight of $f$ and define $F(W) = \mathbb{E}[f({W + U})]$, where $U$ is a random sample from $\mathcal{P}$. This injection induces regularization through the Hessian trace of $f$ for small, isotropic Gaussian perturbations. Thus, the weight-perturbed function biases to minimizers with low Hessian trace. Several prior works have studied settings related to this weight-perturbed function by designing algorithms to improve generalization. Still, convergence rates are not known for finding minima under the average perturbations of the function $F$. This paper considers an SGD-like algorithm that injects random noise before computing gradients while leveraging the symmetry of $\mathcal{P}$ to reduce variance. We then provide a rigorous analysis, showing matching upper and lower bounds of our algorithm for finding an approximate first-order stationary point of $F$ when the gradient of $f$ is Lipschitz-continuous. We empirically validate our algorithm for several image classification tasks with various architectures. Compared to sharpness-aware minimization, we note a 12.6% and 7.8% drop in the Hessian trace and top eigenvalue of the found minima, respectively, averaged over eight datasets. Ablation studies validate the benefit of the design of our algorithm.

In this work we connect two notions: That of the nonparametric mode of a probability measure, defined by asymptotic small ball probabilities, and that of the Onsager-Machlup functional, a generalized density also defined via asymptotic small ball probabilities. We show that in a separable Hilbert space setting and under mild conditions on the likelihood, modes of a Bayesian posterior distribution based upon a Gaussian prior exist and agree with the minimizers of its Onsager-Machlup functional and thus also with weak posterior modes. We apply this result to inverse problems and derive conditions on the forward mapping under which this variational characterization of posterior modes holds. Our results show rigorously that in the limit case of infinite-dimensional data corrupted by additive Gaussian or Laplacian noise, nonparametric maximum a posteriori estimation is equivalent to Tikhonov-Phillips regularization. In comparison with the work of Dashti, Law, Stuart, and Voss (2013), the assumptions on the likelihood are relaxed so that they cover in particular the important case of white Gaussian process noise. We illustrate our results by applying them to a severely ill-posed linear problem with Laplacian noise, where we express the maximum a posteriori estimator analytically and study its rate of convergence in the small noise limit.

Quantum subspace diagonalization methods are an exciting new class of algorithms for solving large\rev{-}scale eigenvalue problems using quantum computers. Unfortunately, these methods require the solution of an ill-conditioned generalized eigenvalue problem, with a matrix pair corrupted by a non-negligible amount of noise that is far above the machine precision. Despite pessimistic predictions from classical \rev{worst-case} perturbation theories, these methods can perform reliably well if the generalized eigenvalue problem is solved using a standard truncation strategy. By leveraging and advancing classical results in matrix perturbation theory, we provide a theoretical analysis of this surprising phenomenon, proving that under certain natural conditions, a quantum subspace diagonalization algorithm can accurately compute the smallest eigenvalue of a large Hermitian matrix. We give numerical experiments demonstrating the effectiveness of the theory and providing practical guidance for the choice of truncation level. Our new results can also be of independent interest to solving eigenvalue problems outside the context of quantum computation.