Abstract This article proposes a method for constructing computer experiment designs, specifically for Strauss processes marked by three types of marks. This method is based on the theory of marked point processes. The designs obtained using the Monte Carlo Markov Chain (MCMC) method and the Metropolis–Hastings algorithm are easily adaptable and can meet various objectives. A detailed study on the convergence of the Markov chain was conducted. In addition, a comparison was made between our approach and other existing computer experiment designs.

Abstract We show that Lasso and Bayesian Lasso are very close when the sparsity is large and the noise is small. We propose to solve Bayesian Lasso using multivalued stochastic differential equation. We derive four discretization algorithms, and present highly efficient multilevel Monte Carlo (MLMC) simulations. Additionally, we perform a numerical comparison of the Monte Carlo (MC), MLMC and proximal Markov chain Monte Carlo algorithm (PMALA).

Abstract In this paper, we study the class of first order Periodic Generalized Autoregressive Conditional heteroscedasticity processes ( PGARCH ⁢ ( 1 , 1 ) {\mathrm{PGARCH}(1,1)} for short) in which the parameters in the volatility process are allowed to switch between different regimes. First, we establish necessary and sufficient conditions for a PGARCH ⁢ ( 1 , 1 ) {\mathrm{PGARCH}(1,1)} process to have a unique stationary solution (in periodic sense) and for the existence of moments of any order. Next, we are able to estimate the unknown parameters involved in model via the so-called generalized method of moments (GMM). We construct an estimator and establish its asymptotic properties. Specifically, we demonstrate its consistency and asymptotic normality. The GMM estimator in some cases can be more efficient than the least squares estimator (LSE) and the quasi maximum likelihood estimator (QMLE). Some simulation studies are also performed to highlight the impact of our theoretical results.

Abstract This study deals with randomized algorithms and random projection methods for solving systems of linear algebraic equations with Toeplitz matrices. A preconditioning of such systems with circulant matrices is used that improves the convergence of the stochastic projection method. The developed stochastic algorithms are applied to first kind boundary integral equations for the Laplace, screened Poisson, and Helmholtz equations. Another application concerns the inverse problem for a wave equation where the task is to recover the unknown coefficient of this equation. A series of computer simulations are carried out to analyze the efficiency of the developed algorithm.

Abstract Analyzing COVID-19 data presents a challenge in Bayesian computations of the Poisson process because the experimental conditions are not under control. This lack of homogeneity can lead to inconsistent model parameters, which violates the assumptions of Bayesian inference. In this paper, we study the multiple change-point detection problem from this viewpoint for a non-homogeneous sample path of the Poisson process as the response variable. The rate parameters are linked to some explanatory using a generalized linear model. The number of change-points is considered to be unknown as well as their locations. We introduce a Bayesian paradigm to estimate the number and location of change-points. We also present an adaptive RJMCMC algorithm to generate pseudo-random samples from the posterior distributions. We apply the proposed model to analyze the COVID-19 infection curves from different countries and identify patterns of cases. We also assess the efficacy of interventions, such as vaccination and public health emergency responses, implemented by different countries. The results of the analysis provide valuable insights into the spread of COVID-19 and the effectiveness of interventions. The proposed model can be used to inform public health decision-making and help to improve the management of the pandemic.

Abstract The dependence of the binding energies of adsorbed In and Al atoms on the (001) surfaces of InAs and AlAs with β 2 {\beta_{2}} (2 × {\times} 4) reconstruction on the biaxial compression and tension strains has been investigated using the density functional theory. The maps of the potential energy surface of the adsorbed In atom on the (001) surface of InAs and AlAs with β 2 {\beta_{2}} (2 × {\times} 4) reconstruction have been calculated. According to the calculation results, a general trend towards an increase in the absolute values of the binding energy of the adsorbed In and Al atoms on the (001) surface of InAs and AlAs with an increase in the biaxial tension strain has been observed. For certain positions of the adsorbed atoms, a change in the sign of the derivative of the binding energy dependence on the biaxial strain has been observed with a change in the symmetry of the adsorbed state of the atom.

Abstract In this paper, we suggest joint application of computational probabilistic analysis and the Monte Carlo method for numerical stochastic modeling problems. We use all the capabilities of computational probabilistic analysis while maintaining all the advantages of the Monte Carlo method. Our approach allows us to efficiently implement a computational hybrid scheme. In this way, we reduce the computation time and present the results in the form of distributions. The crucial new points of our method are arithmetic operations on probability density functions and procedures for constructing on the probabilistic extensions. Relying on specific numerical examples of solving systems of linear algebraic equations with random coefficients, we present the advantages of our approach.

Abstract This paper considers three approaches to choosing the constant H in the expression H ⁢ 𝐃 ⁢ ζ / n {H\sqrt{{\mathbf{D}}\zeta}/\sqrt{n}} for the error of the Monte Carlo method for numerical calculation of mathematical expectation 𝐄 ⁢ ζ {{\mathbf{E}}\zeta} of a random variable ζ: in probability, in mean square and in mean.In practical studies using the Monte Carlo method, when estimating the calculation error, it is recommended to use the “in mean” approach with the constant H = 2 π = 0.79788456079 ⁢ … {H=\sqrt{\frac{2}{\pi}}=0.79788456079\dots} .

Abstract We develop a Monte Carlo method to solve backward stochastic differential equations (BSDEs) in high dimensions. The proposed algorithm is based on the regression-later approach using multivariate Hermite polynomials and their gradients. We propose numerical experiments to illustrate its performance.

Monte Carlo Methods and appliCations is a quarterly published journal presenting original articles on the theory and applications of Monte Carlo and quasi-Monte Carlo methods. Launched in 1995 the journal covers all stochastic numerics topics with emphasis on the theory of Monte Carlo methods and new applications in all branches of science and technology. The following topics will be covered: theory of Monte Carlo methods, quasi-Monte Carlo methods, integration, boundary value problems for PDE's, numerics of stochastic differential equations, simulation of random variables, stochastic processes and fields, and stochastic models in all fields of applied sciences.