Abstract. 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, Green's 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.Key words. Monte Carlo method, random walk, diffusion, reaction rate, electrostatic energy, molecule AMS subject classifications. 65C05, 65N99, 78M25, 92C451. Introduction. Elliptic partial differential equations such as the Laplace, Poisson, Poisson-Boltzmann, etc. are effectively used as mathematical models in different branches of computational biophysics and chemistry. Calculation of the diffusionlimited reaction rate, the electrostatic potential and field, the internal energy -are all problems that can be reduced to the solution of a diffusion equation with some conditions on the boundary and at the infinity. The intrinsic analogy between diffusion and electrostatics makes it possible to apply the same computational techniques to solve problems coming from these different fields. It is worth noting that the solution domain in this class of problems is usually infinite. So, it is natural that since the days of Maxwell, the boundary-element method has been used as an effective tool for solving such problems. In particular, one can calculate the capacitance and the diffusionlimited reaction rate as a surface integral [18]. The boundary-element method, as well as finite-difference and finite-element methods, are still commonly used for solving electrostatics and diffusion problems arising in biophysics. Review of these and other techniques used in the computation of molecular electrostatic properties is given in [2].Another possible way of computationally treating these problems comes from the probabilistic representation of solutions to elliptic and parabolic partial differential equations as functionals of diffusion process trajectories [15,7,8]. Direct computational simulation of physical diffusion in this case coincides with the approximation of Brownian motion as the solution to a stochastic differential equation via a firstorder Euler scheme [19,16]. This approach was applied [4] to simulation studies of diffusion-limited reactions. Though computationally far from being optimal, it allows one to include different physical phenomena (hydrodynamic, electrostatic, etc.) into the computational scheme, and, what is essential, it is efficient enough to be competitive with deterministic methods. Later, this algorithm was modified [30] by incorporating different boundary conditions [28,26]. In addition, other variants of the algorithm were suggested. In [31], in particular, the Brownian dynamics simulation method was compared to the algorithms based on survival probab...

The monograph presents new probabilistic representations for classical boundary value problems of mathematical physics. When comparing with the well known probabilistic representations in the form of Wiener and diffusion path integrals, the trajectories of random walks in our representations are simulated on the boundary of the domain as Markov chains generated by the kernels of the boundary integral equations equivalent to the original boundary value problem. The Monte Carlo methods based on the walk on boundary processes have a series of advantages: (1) high-dimensional problems can be solved, (2) the method is grid-free and gives the solution simultaneously in arbitrary points, (3) external and internal boundary value problems are solved using one and the same random walk on boundary process, (4) when comparing with the classical probabilistic representations, there is no ε-error generated by the approximations in the ε-boundary, and (5) parallel implementation of the walk on boundary algorithms is straightforward and much easier. This is the first book devoted to the walk on boundary algorithms. First introduced by K. Sabelfeld for solving the interior and exterior boundary value problems for the Laplace and heat equations, the method was then extended to all the main boundary value problems of the potential and elasticity theories. For specialists in applied and computational mathematics, applied probabilists, for students and postgraduates studying new numerical methods for solving PDEs.

This paper describes a fast-forward electromagnetic solver (FFS) for the image reconstruction algorithm of our microwave tomography system. Our apparatus is a preclinical prototype of a biomedical imaging system, designed for the purpose of early breast cancer detection. It operates in the 3-6-GHz frequency band using a circular array of probe antennas immersed in a matching liquid; it produces image reconstructions of the permittivity and conductivity profiles of the breast under examination. Our reconstruction algorithm solves the electromagnetic (EM) inverse problem and takes into account the real EM properties of the probe antenna array as well as the influence of the patient's body and that of the upper metal screen sheet. This FFS algorithm is much faster than conventional EM simulation solvers. In comparison, in the same PC, the CST solver takes ~45 min, while the FFS takes ~1 s of effective simulation time for the same EM model of a numerical breast phantom.

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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.

The mixing rules for the permittivity and permeability of composites are known to depend greatly on the microscopic structure of the composite. This dependence can be quantified in terms of Bergman’s spectral function. In this paper, the spectral function of actual magnetic composites is reconstructed from their measured microwave constitutive parameters. The samples under study are composed of carbonyl iron or Fe–Cr–Al alloy powders embedded in a paraffin wax matrix. The permittivity and permeability of the samples is measured in the 0.1–10 GHz frequency band. The proposed approach to process the measured data allows the spectral function of the composite and frequency dependence of intrinsic permeability of inclusions to be derived. The obtained results are in agreement with available tabulated data.