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Register a new userA reduced basis method for the nonlinear Poisson-Boltzmann equation
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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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Linear kinetic transport equations play a critical role in optical tomography, radiative transfer and neutron transport. The fundamental difficulty hampering their efficient and accurate numerical resolution lies in the high dimensionality of the physical and velocity/angular variables and the fact that the problem is multiscale in nature. Leveraging the existence of a hidden low-rank structure hinted by the diffusive limit, in this work, we design and test the angular-space reduced order model for the linear radiative transfer equation, the first such effort based on the celebrated reduced basis method (RBM). Our method is built upon a high-fidelity solver employing the discrete ordinates method in the angular space, an asymptotic preserving upwind discontinuous Galerkin method for the physical space, and an efficient synthetic accelerated source iteration for the resulting linear system. Addressing the challenge of the parameter values (or angular directions) being coupled through an integration operator, the first novel ingredient of our method is an iterative procedure where the macroscopic density is constructed from the RBM snapshots, treated explicitly and allowing a transport sweep, and then updated afterwards. A greedy algorithm can then proceed to adaptively select the representative samples in the angular space and form a surrogate solution space. The second novelty is a least-squares density reconstruction strategy, at each of the relevant physical locations, enabling the robust and accurate integration over an arbitrarily unstructured set of angular samples toward the macroscopic density. Numerical experiments indicate that our method is highly effective for computational cost reduction in a variety of regimes.
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.
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.
We propose and test the first Reduced Radial Basis Function Method (R$^2$BFM) for solving parametric partial differential equations on irregular domains. The two major ingredients are a stable Radial Basis Function (RBF) solver that has an optimized set of centers chosen through a reduced-basis-type greedy algorithm, and a collocation-based model reduction approach that systematically generates a reduced-order approximation whose dimension is orders of magnitude smaller than the total number of RBF centers. The resulting algorithm is efficient and accurate as demonstrated through two- and three-dimensional test problems.
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.
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