RESUMO
The Poisson-Boltzmann equation is a widely used model to study 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 allow 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% and come out to be more efficient than uniform refinement when computing error estimations. This result sets the basis toward efficient automatic mesh refinement schemes that produce optimal meshes for solvation energy calculations.
RESUMO
The use of computer models as a tool for the study and understanding of the complex phenomena of cardiac electrophysiology has attained increased importance nowadays. At the same time, the increased complexity of the biophysical processes translates into complex computational and mathematical models. To speed up cardiac simulations and to allow more precise and realistic uses, 2 different techniques have been traditionally exploited: parallel computing and sophisticated numerical methods. In this work, we combine a modern parallel computing technique based on multicore and graphics processing units (GPUs) and a sophisticated numerical method based on a new space-time adaptive algorithm. We evaluate each technique alone and in different combinations: multicore and GPU, multicore and GPU and space adaptivity, multicore and GPU and space adaptivity and time adaptivity. All the techniques and combinations were evaluated under different scenarios: 3D simulations on slabs, 3D simulations on a ventricular mouse mesh, ie, complex geometry, sinus-rhythm, and arrhythmic conditions. Our results suggest that multicore and GPU accelerate the simulations by an approximate factor of 33×, whereas the speedups attained by the space-time adaptive algorithms were approximately 48. Nevertheless, by combining all the techniques, we obtained speedups that ranged between 165 and 498. The tested methods were able to reduce the execution time of a simulation by more than 498× for a complex cellular model in a slab geometry and by 165× in a realistic heart geometry simulating spiral waves. The proposed methods will allow faster and more realistic simulations in a feasible time with no significant loss of accuracy.