In this paper, we propose multi-scale deep neural networks (MscaleDNNs) using the idea of radial scaling in frequency domain and activation functions with compact support. The radial scaling converts the problem of approximation of high frequency contents of PDEs' solutions to a problem of learning about lower frequency functions, and the compact support activation functions facilitate the separation of frequency contents of the target function to be approximated by corresponding DNNs. As a result, the MscaleDNNs achieve fast uniform convergence over multiple scales. The proposed MscaleDNNs are shown to be superior to traditional fully connected DNNs and be an effective mesh-less numerical method for Poisson-Boltzmann equations with ample frequency contents over complex and singular domains.
Accurate calculation of electrostatic potential and gradient on the molecular surface is highly desirable for the continuum and hybrid modeling of large scale deformation of biomolecules in solvent. In this article a new numerical method is proposed to calculate these quantities on the dielectric interface from the numerical solutions of the Poisson-Boltzmann equation. Our method reconstructs a potential field locally in the least square sense on the polynomial basis enriched with Green's functions, the latter characterize the Coulomb potential induced by charges near the position of reconstru…
We are examining a new kind of lattice gas that closely resembles modern lattice Boltzmann methods. This new kind of lattice gas, that we call a Monte Carlo Lattice Gas, has interesting properties that shed light on the origin of the multi-relaxation time collision operator and it derives the equilibrium distribution for entropic lattice Boltzmann. Furthermore these lattice gas methods have Galilean invariant fluctuations given by a Poisson statistics, giving further insight into the properties that we should expect for fluctuating lattice Boltzmann methods.
The computational study of chemical reactions in complex, wet environments is critical for applications in many fields. It is often essential to study chemical reactions in the presence of applied electrochemical potentials, taking into account the non-trivial electrostatic screening coming from the solvent and the electrolytes. As a consequence the electrostatic potential has to be found by solving the generalized Poisson and the Poisson-Boltzmann equation for neutral and ionic solutions, respectively. In the present work solvers for both problems have been developed. A preconditioned conjuga…
The Poisson-Boltzmann equation is often presented via a variational formulation based on the electrostatic potential. However, the functional has the defect of being non-convex. It can not be used as a local minimization principle while coupled to other dynamic degrees of freedom. We formulate a convex dual functional which is numerically equivalent at its minimum and which is more suited to local optimization.
In this paper, we present a GPU-accelerated direct-sum boundary integral method to solve the linear Poisson-Boltzmann (PB) equation. In our method, a well-posed boundary integral formulation is used to ensure the fast convergence of Krylov subspace based linear algebraic solver such as the GMRES. The molecular surfaces are discretized with flat triangles and centroid collocation. To speed up our method, we take advantage of the parallel nature of the boundary integral formulation and parallelize the schemes within CUDA shared memory architecture on GPU. The schemes use only $11N+6N_c$ size-of-…
Electroconvective flow between two infinitely long parallel electrodes is investigated via a multiphysics computational model. The model solves for spatiotemporal flow properties using two-relaxation-time Lattice Boltzmann Method for fluid and charge transport coupled to Fast Fourier Transport Poisson solver for the electric potential. The segregated model agrees with the previous analytical and numerical results providing a robust approach for modeling electrohydrodynamic flows.
The Poisson-Boltzmann model is an effective and popular approach for modeling solvated biomolecules in continuum solvent with dissolved electrolytes. In this paper, we report our recent work in developing a Galerkin boundary integral method for solving the Poisson-Boltzmann (PB) equation. The solver has combined advantages in accuracy, efficiency, and memory usage as it applies a well-posed boundary integral formulation to circumvent many numerical difficulties associated with the PB equation and uses an O(N) Cartesian Fast Multipole Method (FMM) to accelerate the GMRES iteration. In addition,…
The Poisson-Boltzmann equation is a widely used model to study the electrostatics in molecular solvation. Its numerical solution using a boundary integral formulation requires a mesh on the molecular surface only, yielding accurate representations of the solute, which is usually a complicated geometry. Here, we utilize adjoint-based analyses to form two goal-oriented error estimates that allows us to determine the contribution of each discretization element (panel) to the numerical error in the solvation free energy. This information is useful to identify high-error panels to then refine them …
A Domain Decomposition Method for the Poisson-Boltzmann Solvation Models
Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, and Institut des Sciences du Calcul et des Données（索邦大学，巴黎第六大学，UMR 7598，雅克-路易·利翁斯实验室，计算与数据科学研究所）；Center for Computational Engineering Science, RWTH Aachen University（亚琛工业大学计算工程科学中心）；Computational Biomedicine, Institute for Advanced Simulation IAS-5 and Institute of Neuroscience and Medicine INM-9, Forschungszentrum Jülich（朱利希研究中心计算生物医学，先进模拟研究所IAS-5，神经科学与医学研究所INM-9）；Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, and Institut Universitaire de France（索邦大学，巴黎第六大学，UMR 7598，雅克-路易·利翁斯实验室，法国高等研究院）；Division of Applied Mathematics, Brown University（布朗大学应用数学系）
Electrostatic correlations and fluctuations in ionic systems can be described within an extended Poisson-Boltzmann theory using a Gaussian variational form. The resulting equations are challenging to solve because they require the solution of a non-linear partial differential equation for the pair correlation function. This has limited existing studies to simple approximations or to one-dimensional geometries. In this paper we show that the numerical solution of the equations is greatly simplified by the use of selective inversion of a finite difference operator which occurs in the theory. Thi…
In this paper, a nonuniform size modified Poisson-Boltzmann ion channel (nuSMPBIC) model is presented as a nonlinear system of an electrostatic potential and multiple ionic concentrations. It mixes nonlinear algebraic equations with a Poisson boundary value problem involving Dirichlet-Neumann mixed boundary value conditions and a membrane surface charge density to reflect the effects of ion sizes and membrane charges on electrostatics and ionic concentrations. To overcome the difficulties of strong singularities and exponential nonlinearities, it is split into three submodels with a solution o…
We present the theory and implementation of a Poisson-Boltzmann implicit solvation model for electrolyte solutions. This model can be combined with arbitrary electronic structure methods that provide an accurate charge density of the solute. A hierarchy of approximations for this model includes a linear approximation for weak electrostatic potentials, finite size of the mobile electrolyte ions and a Stern-layer correction. Recasting the Poisson-Boltzmann equations into Euler-Lagrange equations then significantly simplifies the derivation of the free energy of solvation for these approximate mo…
A Self-Adjusting FEM-BEM Coupling Scheme for the Nonlinear Poisson-Boltzmann Equation
Universidad Técnica Federico Santa María（弗雷德里科·圣塔玛丽亚技术大学）；Kingston University London（金斯顿大学伦敦分校）；Centro Científico Tecnológico de Valparaíso (CCTVal)（瓦尔帕莱索科技中心）
Wereportonanewmultiscalemethodapproachforthestudyofsystemswith wide separation of short-range forces acting on short time scales and long-range forces acting on much slower scales. We consider the case of the Poisson-Boltzmann equation that describes the long-range forces using the Boltzmann formula (i.e. we assume the medium to be in quasi local thermal equilibrium). We developed a new approach where fields and particle information (mediated by the equations for their moments) are solved self-consistently. The new approach is implicit and numerically stable, providing exact energy conservatio…
The Poisson-Boltzmann (PB) implicit solvent model is a popular framework for studying the electrostatics of biomolecules immersed in water with dissolved salt. In this model the dielectric interface between the biomolecule and solvent is often taken to be the molecular surface or solvent-excluded surface (SES), and the quality of the SES triangulation is critical in boundary element simulations of the PB model. In this work we compare the MSMS and NanoShaper surface triangulation codes for a set of 38 biomolecules. While MSMS produces triangles of exceedingly small area and large aspect ratio,…
This paper studies an efficient numerical method for solving modified Poisson-Boltzmann (MPB) equations with the self Green's function as a state equation to describe electrostatic correlations in ionic systems. Previously, the most expensive point of the MPB solver is the evaluation of Green's function. The evaluation of Green's function requires solving high-dimensional partial differential equations, which is the computational bottleneck for solving MPB equations. Numerically, the MPB solver only requires the evaluation of Green's function as the diagonal part of the inverse of the discrete…
Implicit-solvent models are widely used to study the electrostatics in dissolved biomolecules, which are parameterized using force fields. Standard force fields treat the charge distribution with point charges, however, other force fields have emerged which offer a more realistic description by considering polarizability. In this work, we present the implementation of the polarizable and multipolar force field AMOEBA, in the boundary integral Poisson-Boltzmann solver \texttt{PyGBe}. Previous work from other researchers coupled AMOEBA with the finite-difference solver APBS, and found difficulti…
We present a new approach to find accurate solutions to the Poisson equation, as obtained from the steady-state limit of a diffusion equation with strong source terms. For this purpose, we start from Boltzmann's kinetic theory and investigate the influence of higher order terms on the resulting macroscopic equations. By performing an appropriate expansion of the equilibrium distribution, we provide a method to remove the unnecessary terms up to a desired order and show that it is possible to find, with high level of accuracy, the steady-state solution of the diffusion equation for sizeable Knu…
Interactions between surfaces and proteins occur in many vital processes and are crucial in biotechnology: the ability to control specific interactions is essential in fields like biomaterials, biomedical implants and biosensors. In the latter case, biosensor sensitivity hinges on ligand proteins adsorbing on bioactive surfaces with a favorable orientation, exposing reaction sites to target molecules. Protein adsorption, being a free-energy-driven process, is difficult to study experimentally. This paper develops and evaluates a computational model to study electrostatic interactions of protei…
Electrostatic interactions play crucial roles in biophysical processes such as protein folding and molecular recognition. Poisson-Boltzmann equation (PBE)-based models have emerged as widely used in modeling these important processes. Though great efforts have been put into developing efficient PBE numerical models, challenges still remain due to the high dimensionality of typical biomolecular systems. In this study, we implemented and analyzed commonly used linear PBE solvers for the ever-improving graphics processing units (GPU) for biomolecular simulations, including both standard and preco…
