Burak Aksoylu

Burak Aksoylu, Stephen Bond, Eric Cyr et al.

In this article, we develop goal-oriented error indicators to drive adaptive refinement algorithms for the Poisson-Boltzmann equation. Empirical results for the solvation free energy linear functional demonstrate that goal-oriented indicators are not sufficient on their own to lead to a superior refinement algorithm. To remedy this, we propose a problem-specific marking strategy using the solvation free energy computed from the solution of the linear regularized Poisson-Boltzmann equation. The convergence of the solvation free energy using this marking strategy, combined with goal-oriented refinement, compares favorably to adaptive methods using an energy-based error indicator. Due to the use of adaptive mesh refinement, it is critical to use multilevel preconditioning in order to maintain optimal computational complexity. We use variants of the classical multigrid method, which can be viewed as generalizations of the hierarchical basis multigrid and Bramble-Pasciak-Xu (BPX) preconditioners.

Burak Aksoylu, Stephen Bond, Michael Holst

In this paper, we examine a number of additive and multiplicative multilevel iterative methods and preconditioners in the setting of two-dimensional local mesh refinement. While standard multilevel methods are effective for uniform refinement-based discretizations of elliptic equations, they tend to be less effective for algebraic systems which arise from discretizations on locally refined meshes, losing their optimal behavior in both storage and computational complexity. Our primary focus here is on BPX-style additive and multiplicative multilevel preconditioners, and on various stabilizations of the additive and multiplicative hierarchical basis method (HB), and their use in the local mesh refinement setting. In this article, we describe in detail the implementation of these types of algorithms, including detailed discussions of the datastructures and traversal algorithms we employ for obtaining optimal storage and computational complexity in our implementations. We show how each of the algorithms can be implemented using standard datatypes available in languages such as C and FORTRAN, so that the resulting algorithms have optimal (linear) storage requirements, and so that the resulting multilevel method or preconditioner can be applied with optimal (linear) computational costs. Our implementations are performed in both C and MATLAB using the Finite Element ToolKit (FETK), an open source finite element software package. We finish the paper with a sequence of numerical experiments illustrating the effectiveness of a number of BPX and stabilized HB variants for several examples requiring local refinement.

Suraj Pawar, Omer San, Burak Aksoylu et al.

Recent applications of machine learning, in particular deep learning, motivate the need to address the generalizability of the statistical inference approaches in physical sciences. In this letter, we introduce a modular physics guided machine learning framework to improve the accuracy of such data-driven predictive engines. The chief idea in our approach is to augment the knowledge of the simplified theories with the underlying learning process. To emphasise on their physical importance, our architecture consists of adding certain features at intermediate layers rather than in the input layer. To demonstrate our approach, we select a canonical airfoil aerodynamic problem with the enhancement of the potential flow theory. We include features obtained by a panel method that can be computed efficiently for an unseen configuration in our training procedure. By addressing the generalizability concerns, our results suggest that the proposed feature enhancement approach can be effectively used in many scientific machine learning applications, especially for the systems where we can use a theoretical, empirical, or simplified model to guide the learning module.

Burak Aksoylu, Michael L. Parks

In this article we present the first results on domain decomposition methods for nonlocal operators. We present a nonlocal variational formulation for these operators and establish the well-posedness of associated boundary value problems, proving a nonlocal Poincaré inequality. To determine the conditioning of the discretized operator, we prove a spectral equivalence which leads to a mesh size independent upper bound for the condition number of the stiffness matrix. We then introduce a nonlocal two-domain variational formulation utilizing nonlocal transmission conditions, and prove equivalence with the single-domain formulation. A nonlocal Schur complement is introduced. We establish condition number bounds for the nonlocal stiffness and Schur complement matrices. Supporting numerical experiments demonstrating the conditioning of the nonlocal one- and two-domain problems are presented.

Burak Aksoylu, Michael Holst

In this article, we establish optimality of the Bramble-Pasciak-Xu (BPX) norm equivalence and optimality of the wavelet modified (or stabilized) hierarchical basis (WHB) preconditioner in the setting of local 3D mesh refinement. In the analysis of WHB methods, a critical first step is to establish the optimality of BPX norm equivalence for the refinement procedures under consideration. While the available optimality results for the BPX norm have been constructed primarily in the setting of uniformly refined meshes, a notable exception is the local 2D red-green result due to Dahmen and Kunoth. The purpose of this article is to extend this original 2D optimality result to the local 3D red-green refinement procedure introduced by Bornemann-Erdmann-Kornhuber (BEK), and then to use this result to extend the WHB optimality results from the quasiuniform setting to local 2D and 3D red-green refinement scenarios. The BPX extension is reduced to establishing that locally enriched finite element subspaces allow for the construction of a scaled basis which is formally Riesz stable. It is possible to show that the number of degrees of freedom used for smoothing is bounded by a constant times the number of degrees of freedom introduced at that level of refinement, indicating that a practical implementable version of the resulting BPX preconditioner for the BEK refinement setting has provably optimal (linear) computational complexity per iteration. An interesting implication of the optimality of the WHB preconditioner is the a priori H1-stability of the L2-projection. The theoretical framework employed supports arbitrary spatial dimension d >= 1 and requires no coefficient smoothness assumptions beyond those required for well-posedness in H1.

Burak Aksoylu, Zuhal Yeter

We study the high-contrast biharmonic plate equation with HCT and Morley discretizations. We construct a preconditioner that is robust with respect to contrast size and mesh size simultaneously based on the preconditioner proposed by Aksoylu et al. (2008, Comput. Vis. Sci. 11, pp. 319--331). By extending the devised singular perturbation analysis from linear finite element discretization to the above discretizations, we prove and numerically demonstrate the robustness of the preconditioner. Therefore, we accomplish a desirable preconditioning design goal by using the same family of preconditioners to solve elliptic family of PDEs with varying discretizations. We also present a strategy on how to generalize the proposed preconditioner to cover high-contrast elliptic PDEs of order $2k, k>2$. Moreover, we prove a fundamental qualitative property of solution of the high-contrast biharmonic plate equation. Namely, the solution over the highly-bending island becomes a linear polynomial asymptotically. The effectiveness of our preconditioner is largely due to the integration of this qualitative understanding of the underlying PDE into its construction.

Burak Aksoylu, Zuhal Yeter

We study a conservative 5-point cell-centered finite volume discretization of the high-contrast diffusion equation. We aim to construct preconditioners that are robust with respect to the magnitude of the coefficient contrast and the mesh size simultaneously. For that, we prove and numerically demonstrate the robustness of the preconditioner proposed by Aksoylu et al. (2008, Comput. Vis. Sci. 11, pp. 319--331) by extending the devised singular perturbation analysis from linear finite element discretization to the above discretization. The singular perturbation analysis is more involved than that of finite element because all the subblocks in the discretization matrix depend on the diffusion coefficient. However, that dependence is eliminated asymptotically. This allows the same preconditioner to be utilized due to similar limiting behaviours of the submatrices; leading to a narrowing family of preconditioners that can be used for different discretizations--a desirable preconditioner design goal. We compare our numerical results to standard cell-centered multigrid and observe that performance of our preconditioner is independent of the utilized prolongation operators and smoothers. As a side result, we also prove that the solution over the highly-diffusive island becomes constant asymptotically. Integration of this qualitative understanding of the underlying PDE to our preconditioner is the main reason behind its superior performance. Diagonal scaling is probably the most basic preconditioner for high-contrast coefficients. Extending the matrix entry based spectral analysis introduced by Graham and Hagger, we rigorously show that the number of small eigenvalues of the diagonally scaled matrix depends on the number of isolated islands comprising the highly-diffusive region.