In this work we make us of Livens principle (sometimes also referred to as Hamilton-Pontryagin principle) in order to obtain a novel structure-preserving integrator for mechanical systems. In contrast to the canonical Hamiltonian equations of motion, the Euler-Lagrange equations pertaining to Livens principle circumvent the need to invert the mass matrix. This is an essential advantage with respect to singular mass matrices, which can yield severe difficulties for the modelling and simulation of multibody systems. Moreover, Livens principle unifies both Lagrangian and Hamiltonian viewpoints on mechanics. Additionally, the present framework avoids the need to set up the system's Hamiltonian. The novel scheme algorithmically conserves a general energy function and aims at the preservation of momentum maps corresponding to symmetries of the system. We present an extension to mechanical systems subject to holonomic constraints. The performance of the newly devised method is studied in representative examples.
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In this work, a Generalized Finite Difference (GFD) scheme is presented for effectively computing the numerical solution of a parabolic-elliptic system modelling a bacterial strain with density-suppressed motility. The GFD method is a meshless method known for its simplicity for solving non-linear boundary value problems over irregular geometries. The paper first introduces the basic elements of the GFD method, and then an explicit-implicit scheme is derived. The convergence of the method is proven under a bound for the time step, and an algorithm is provided for its computational implementation. Finally, some examples are considered comparing the results obtained with a regular mesh and an irregular cloud of points.
In this article we consider an aggregate loss model with dependent losses. The losses occurrence process is governed by a two-state Markovian arrival process (MAP2), a Markov renewal process process that allows for (1) correlated inter-losses times, (2) non-exponentially distributed inter-losses times and, (3) overdisperse losses counts. Some quantities of interest to measure persistence in the loss occurrence process are obtained. Given a real operational risk database, the aggregate loss model is estimated by fitting separately the inter-losses times and severities. The MAP2 is estimated via direct maximization of the likelihood function, and severities are modeled by the heavy-tailed, double-Pareto Lognormal distribution. In comparison with the fit provided by the Poisson process, the results point out that taking into account the dependence and overdispersion in the inter-losses times distribution leads to higher capital charges.
We propose a novel algorithm for the support estimation of partially known Gaussian graphical models that incorporates prior information about the underlying graph. In contrast to classical approaches that provide a point estimate based on a maximum likelihood or a maximum a posteriori criterion using (simple) priors on the precision matrix, we consider a prior on the graph and rely on annealed Langevin diffusion to generate samples from the posterior distribution. Since the Langevin sampler requires access to the score function of the underlying graph prior, we use graph neural networks to effectively estimate the score from a graph dataset (either available beforehand or generated from a known distribution). Numerical experiments demonstrate the benefits of our approach.
In this article, we decrease the degree of the polynomials on the boundary of the weak functions and modify the definition of the weak laplacian which are introduced in \cite{BiharmonicSFWG} to use the SFWG method for the biharmonic equation. Then we propose the relevant numerical format and obtain the optimal order of error estimates in $H^2$ and $L^2$ norms. Finally, we confirm the estimates using numerical experiments.
Inequality measures are quantitative measures that take values in the unit interval, with a zero value characterizing perfect equality. Although originally proposed to measure economic inequalities, they can be applied to several other situations, in which one is interested in the mutual variability between a set of observations, rather than in their deviations from the mean. While unidimensional measures of inequality, such as the Gini index, are widely known and employed, multidimensional measures, such as Lorenz Zonoids, are difficult to interpret and computationally expensive and, for these reasons, are not much well known. To overcome the problem, in this paper we propose a new scaling invariant multidimensional inequality index, based on the Fourier transform, which exhibits a number of interesting properties, and whose application to the multidimensional case is rather straightforward to calculate and interpret.
We propose a two-step Newton's method for refining an approximation of a singular zero whose deflation process terminates after one step, also known as a deflation-one singularity. Given an isolated singular zero of a square analytic system, our algorithm exploits an invertible linear operator obtained by combining the Jacobian and a projection of the Hessian in the direction of the kernel of the Jacobian. We prove the quadratic convergence of the two-step Newton method when it is applied to an approximation of a deflation-one singular zero. Also, the algorithm requires a smaller size of matrices than the existing methods, making it more efficient. We demonstrate examples and experiments to show the efficiency of the method.
In this work, we aim to establish a Bayesian adaptive learning framework by focusing on estimating latent variables in deep neural network (DNN) models. Latent variables indeed encode both transferable distributional information and structural relationships. Thus the distributions of the source latent variables (prior) can be combined with the knowledge learned from the target data (likelihood) to yield the distributions of the target latent variables (posterior) with the goal of addressing acoustic mismatches between training and testing conditions. The prior knowledge transfer is accomplished through Variational Bayes (VB). In addition, we also investigate Maximum a Posteriori (MAP) based Bayesian adaptation. Experimental results on device adaptation in acoustic scene classification show that our proposed approaches can obtain good improvements on target devices, and consistently outperforms other cut-edging algorithms.
In this paper, we propose to consider various models of pattern recognition. At the same time, it is proposed to consider models in the form of two operators: a recognizing operator and a decision rule. Algebraic operations are introduced on recognizing operators, and based on the application of these operators, a family of recognizing algorithms is created. An upper estimate is constructed for the model, which guarantees the completeness of the extension.
In this paper, we provide conditions that hulls of generalized Reed-Solomon (GRS) codes are also GRS codes from algebraic geometry codes. If the conditions are not satisfied, we provide a method of linear algebra to find the bases of hulls of GRS codes and give formulas to compute their dimensions. Besides, we explain that the conditions are too good to be improved by some examples. Moreover, we show self-orthogonal and self-dual GRS codes.
In this note we consider the approximation of the Greeks Delta and Gamma of American-style options through the numerical solution of time-dependent partial differential complementarity problems (PDCPs). This approach is very attractive as it can yield accurate approximations to these Greeks at essentially no additional computational cost during the numerical solution of the PDCP for the pertinent option value function. For the temporal discretization, the Crank-Nicolson method is arguably the most popular method in computational finance. It is well-known, however, that this method can have an undesirable convergence behaviour in the approximation of the Greeks Delta and Gamma for American-style options, even when backward Euler damping (Rannacher smoothing) is employed. In this note we study for the temporal discretization an interesting family of diagonally implicit Runge-Kutta (DIRK) methods together with the two-stage Lobatto IIIC method. Through ample numerical experiments for one- and two-asset American-style options, it is shown that these methods can yield a regular second-order convergence behaviour for the option value as well as for the Greeks Delta and Gamma. A mutual comparison reveals that the DIRK method with suitably chosen parameter $\theta$ is preferable.
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