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Register a new userEfficient mesh refinement for the Poisson-Boltzmann equation with boundary elements
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Added by
Christopher Cooper
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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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 adaptively to find optimal surface meshes. We present results for spheres and real molecular geometries, and see that elements with large error tend to be in regions where there is a high electrostatic potential. We also find that even though both estimates predict different total errors, they have similar performance as part of an adaptive mesh refinement scheme. Our test cases suggest that the adaptive mesh refinement scheme is very effective, as we are able to reduce the error one order of magnitude by increasing the mesh size less than 20%. This result sets the basis towards efficient automatic mesh refinement schemes that produce optimal meshes for solvation energy calculations.
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The Poisson-Boltzmann equation (PBE) models the electrostatic interactions of charged bodies such as molecules and proteins in an electrolyte solvent. The PBE is a challenging equation to solve numerically due to the presence of singularities, discontinuous coefficients and boundary conditions. Hence, there is often large error in the numerical solution of the PBE that needs to be quantified. In this work, we use adjoint based a posteriori analysis to accurately quantify the error in an important quantity of interest, the solvation free energy, for the finite element solution of the PBE. We identify various sources of error and propose novel refinement strategies based on a posteriori error estimates.
This work further improves the pseudo-transient approach for the Poisson Boltzmann equation (PBE) in the electrostatic analysis of solvated biomolecules. The numerical solution of the nonlinear PBE is known to involve many difficulties, such as exponential nonlinear term, strong singularity by the source terms, and complex dielectric interface. Recently, a pseudo-time ghost-fluid method (GFM) has been developed in [S. Ahmed Ullah and S. Zhao, Applied Mathematics and Computation, 380, 125267, (2020)], by analytically handling both nonlinearity and singular sources. The GFM interface treatment not only captures the discontinuity in the regularized potential and its flux across the molecular surface, but also guarantees the stability and efficiency of the time integration. However, the molecular surface definition based on the MSMS package is known to induce instability in some cases, and a nontrivial Lagrangian-to-Eulerian conversion is indispensable for the GFM finite difference discretization. In this paper, an Eulerian Solvent Excluded Surface (ESES) is implemented to replace the MSMS for defining the dielectric interface. The electrostatic analysis shows that the ESES free energy is more accurate than that of the MSMS, while being free of instability issues. Moreover, this work explores, for the first time in the PBE literature, adaptive time integration techniques for the pseudo-transient simulations. A major finding is that the time increment $Delta t$ should become smaller as the time increases, in order to maintain the temporal accuracy. This is opposite to the common practice for the steady state convergence, and is believed to be due to the PBE nonlinearity and its time splitting treatment. Effective adaptive schemes have been constructed so that the pseudo-time GFM methods become more efficient than the constant $Delta t$ ones.
It has recently been demonstrated that dynamical low-rank algorithms can provide robust and efficient approximation to a range of kinetic equations. This is true especially if the solution is close to some asymptotic limit where it is known that the solution is low-rank. A particularly interesting case is the fluid dynamic limit that is commonly obtained in the limit of small Knudsen number. However, in this case the Maxwellian which describes the corresponding equilibrium distribution is not necessarily low-rank; because of this, the methods known in the literature are only applicable to the weakly compressible case. In this paper, we propose an efficient dynamical low-rank integrator that can capture the fluid limit -- the Navier-Stokes equations -- of the Boltzmann-BGK model even in the compressible regime. This is accomplished by writing the solution as $f=Mg$, where $M$ is the Maxwellian and the low-rank approximation is only applied to $g$. To efficiently implement this decomposition within a low-rank framework requires, in the isothermal case, that certain coefficients are evaluated using convolutions, for which fast algorithms are known. Using the proposed decomposition also has the advantage that the rank required to obtain accurate results is significantly reduced compared to the previous state of the art. We demonstrate this by performing a number of numerical experiments and also show that our method is able to capture sharp gradients/shock waves.
This work develops entropy-stable positivity-preserving DG methods as a computational scheme for Boltzmann-Poisson systems modeling the pdf of electronic transport along energy bands in semiconductor crystal lattices. We pose, using spherical or energy-angular variables as momentum coordinates, the corresponding Vlasov Boltzmann eq. with a linear collision operator with a singular measure modeling the scattering as functions of the energy band. We show stability results of semi-discrete DG schemes under an entropy norm for 1D-position 2D-momentum, and 2D-position 3D-momentum, using the dissipative properties of the collisional operator given its entropy inequality, which depends on the whole Hamiltonian rather than only the kinetic energy. For the 1D problem, knowledge of the analytic solution to Poisson and of the convergence to a constant current is crucial to obtain full stability. For the 2D problem, specular reflection BC are considered in addition to periodicity in the estimate for stability under an entropy norm. Regarding positivity preservation (1D position), we treat the collision operator as a source term and find convex combinations of the transport and collision terms which guarantee the positivity of the cell average of our numerical pdf at the next time step. The positivity of the numerical pdf in the whole domain is guaranteed by applying the natural limiters that preserve the cell average but modify the slope of the piecewise linear solutions in order to make the function non-negative. The use of a spherical coordinate system $vec{p}(|vec{p}|,mu=costheta,varphi)$ is slightly different to the choice in previous DG solvers for BP, since the proposed DG formulation gives simpler integrals involving just piecewise polynomial functions for both transport and collision terms, which is more adequate for Gaussian quadrature than previous approaches.
In numerical simulations of many charged systems at the micro/nano scale, a common theme is the repeated solution of the Poisson-Boltzmann equation. This task proves challenging, if not entirely infeasible, largely due to the nonlinearity of the equation and the high dimensionality of the physical and parametric domains with the latter emulating the system configuration. In this paper, we for the first time adapt a mathematically rigorous and computationally efficient model order reduction paradigm, the so-called reduced basis method (RBM), to mitigate this challenge. We adopt a finite difference method as the mandatory underlying scheme to produce the {em truth approximations} of the RBM upon which the fast algorithm is built and its performance is measured against. Numerical tests presented in this paper demonstrate the high efficiency and accuracy of the fast algorithm, the reliability of its error estimation, as well as its capability in effectively capturing the boundary layer.
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