Nikolai A. Simonov

Monte Carlo Methods for Calculating Some Physical Properties of Large Molecules

In this paper we describe Monte Carlo methods for solving some boundary-value problems for elliptic partial differential equations arising in the computation of physical properties of large molecules. The constructed algorithms are based on walk on spheres, Greens function first passage, walk in subdomains techniques, and finite-difference approximations of the boundary condition. The methods are applied to calculating the diffusion-limited reaction rate, the electrostatic energy of a molecule, and point values of an electrostatic field.

Monte Carlo-based linear Poisson-Boltzmann approach makes accurate salt-dependent solvation free energy predictions possible

The prediction of salt-mediated electrostatic effects with high accuracy is highly desirable since many biological processes where biomolecules such as peptides and proteins are key players can be modulated by adjusting the salt concentration of the cellular milieu. With this goal in mind, we present a novel implicit-solvent based linear Poisson-Boltzmann (PB) solver that provides very accurate nonspecific salt-dependent electrostatic properties of biomolecular systems. To solve the linear PB equation by the Monte Carlo method, we use information from the simulation of random walks in the physical space. Due to inherent properties of the statistical simulation method, we are able to account for subtle geometric features in the biomolecular model, treat continuity and outer boundary conditions and interior point charges exactly, and compute electrostatic properties at different salt concentrations in a single PB calculation. These features of the Monte Carlo-based linear PB formulation make it possible to predict the salt-dependent electrostatic properties of biomolecules with very high accuracy. To illustrate the efficiency of our approach, we compute the salt-dependent electrostatic solvation free energies of arginine-rich RNA-binding peptides and compare these Monte Carlo-based PB predictions with computational results obtained using the more mature deterministic numerical methods.

Monte Carlo method for calculating the electrostatic energy of a molecule

The problem of computing the electrostatic energy of a large molecule is considered. It is reduced to solving the Poisson equation inside and the linear Poisson-Boltzmann equation in the exterior, coupled by boundary conditions. A Monte Carlo estimate for the potential point values, their derivatives, and the energy is constructed. The estimate is based on the walk on spheres and Greens function first passage algorithms; the walk in subdomains technique; and finite-difference approximations of the boundary condition. Results of some illustrative calculations are presented.

Walk-on-Spheres Algorithm for Solving Boundary-Value Problems with Continuity Flux Conditions

We consider boundary-value problem for elliptic equations with constant coefficients related through the continuity conditions on the boundary between the domains. To take into account conditions involving the solution’s normal derivative, we apply a new mean-value relation written down at a boundary point. This integral relation is exact and provides a possibility to get rid of the bias caused by usually used finite-difference approximation. Randomization of this mean-value relation makes it possible to continue simulating walk-on-spheres trajectory after it hits the boundary. We prove the convergence of the algorithm and determine its rate. In conclusion, we present the results of some model computations.

Solving BVPs Using Quasirandom Walks on the Boundary

The ”random walk on the boundary” Monte Carlo method has been successfully used for solving boundary-value problems. This method has significant advantages when compared to random walks on spheres, balls or a grid, when solving exterior problems, or when solving a problem at an arbitrary number of points using a single random walk. In this paper we study the properties of the method when we use quasirandom sequences instead of pseudorandom numbers to construct the walks on the boundary. Theoretical estimates of the convergence rate are given and numerical experiments are presented in an attempt to confirm the convergence results. The numerical results show that for “walk on the boundary” quasirandom sequences provide a slight improvement over ordinary Monte Carlo.

Random Walk Algorithms for Estimating Effective Properties of Digitized Porous Media

In this paper we describe a Monte Carlo method for permeability calculations in complex digitized porous structures. The relation between the permeability and the diffusion penetration depth is established. The corresponding Dirichlet boundary value problem is solved by random walk algorithms. The results of computational experiments for some random models of porous media confirm the log-normality hypothesis for the permeability distribution.

Random walks for solving boundary-value problems with flux conditions

We consider boundary-value problems for elliptic equations with constant coefficients and apply Monte Carlo methods to solving these equations. To take into account boundary conditions involving solutions normal derivative, we apply the new mean-value relation written down at boundary point. This integral relation is exact and provides a possibility to get rid of the bias caused by usually used finite-difference approximation. We consider Neumann and mixed boundary-value problems, and also the problem with continuity boundary conditions, which involve fluxes. Randomization of the mean-value relation makes it possible to continue simulating walk-on-spheres trajectory after it hits the boundary. We prove the convergence of the algorithm and determine its rate. In conclusion, we present the results of some model computations.

Parallel Quasirandom Walks on the Boundary

The Monte Carlo method called “random walks on boundary” has been successfully used for solving boundary-value problems. This method has significant advantages when compared with random walks on spheres, balls or on discrete grids when an exterior Dirichlet or Neumann problem is solved, or when we are interested in computing the solution to a problem at an arbitrary number of points using a single random walk. In this paper we will investigate ways: • to increase the convergence rate of this method by using quasirandom sequences instead of pseudorandom numbers for the construction of the boundary walks, • to find an efficient parallel implementation of this method on a cluster using MPI. In our parallel implementation we use disjoint contiguous blocks of quasirandom numbers extracted from a given quasirandom sequence for each processor. In this case, the increased convergence rate does not come at the cost of less trustworthy answers. We also present some numerical examples confirming both the increased rate of convergence and the good parallel efficiency of the method.

Monte Carlo Algorithms for Problems with Partially Reflecting Boundaries.

We consider diffusion problems with partially reflecting boundaries that can be formulated in terms of an elliptic equation. To solve boundary value problems with the Robin condition, we propose a Monte Carlo method based on a randomization of an integral representation. The algorithm behaviour is analysed in its application for solving a model problem.

Monte Carlo algorithm for the Robin boundary conditions in application to solving a model diffusion-recombination problem

Abstract A new Monte Carlo algorithm for solving the Robin boundary-value problem is described and applied to the calculation of the electron beam induced current in a simplified model of the imaging measurements.