Efficient preconditioning of -FEM matrix sequences with slowly-varying coefficients: An application to topology optimization
Introduction
In this work, we apply the construction introduced in [1] to discretizations of problems with strong material discontinuities and time-varying coefficients. The core of the construction rests on the observation that, for a large class of problems, the dense Schur complement matrices that arise in the nested dissection method display rank-deficient off-diagonal blocks. In the case of well-behaved elliptic problems, this property can be traced to the separability of the underlying Green’s function, see, e.g., [2], [3]. To the contrary, this argument does not apply to wave-propagation problems, and, in the high-frequency limit, this property indeed ceases to hold, see [4]. Nevertheless, in the case of moderate frequencies, the off-diagonal blocks of the Schur complements can, for all practical purposes, be treated as low-rank. The intuitive explanation of this phenomenon is that, when considered on sufficiently small subdomains, the solution of wave problems resembles that of elliptic problems. By exploiting this property, approximate matrix decompositions can be constructed cheaply, and turn out to be excellent preconditioners.
A number of compressed-rank formats have appeared in the literature. Hierarchical matrices, or -matrices, were first defined in the seminal work of Hackbusch, see [5]. Subsequently, the subclass of -matrices was introduced in [6]. Those matrices are attractive because allow traditionally expensive operations to be carried out within linear complexity, see, e.g., [7] for a detailed discussion. Hierarchically Semi-Separable (HSS) matrices were proposed by Chandrasekaran et al. in [8], and are closely related to - and -matrices. Fast algorithms for their manipulation have been proposed, among others, by Sheng et al. [9], Xia et al. [10], Martinsson [11], and Gillman and Martinsson [12].
As described in detail in [1], the preconditioner construction combines a classical method, i.e., an factorization, with an approximation scheme for the resulting dense Schur complements. In fact, this approach is well established in the literature, see, e.g., Grasedyck et al. [13], and Xia et al. [14]. While our construction was largely inspired by the work of Gillman and Martinsson done in the context of finite difference discretizations, see [15], its main novelty is that we exploited the geometric flexibility provided by the factorization to accommodate for finite elements discretizations and unstructured meshes. In fact, while the main body of the literature deals with finite difference approximations, in the context of finite element discretizations we are only aware of the work by Aminfar and Darve, see [16], [17]. Although their approach is similar since dense frontal matrices are approximated by low-rank compression, they do not take advantage of the geometric partitioning versatility allowed by the factorization. In the context of finite difference approximations, the recent work of Ho and Ying, see [18], addresses the case of highly discontinuous coefficients and moderately indefinite problems.
In this work we describe the application of the preconditioner to high-contrast, time-varying Helmholtz problems arising in the context of acoustic Topology Optimization, see [19]. Topology Optimization is an iterative method that creates highly optimized designs by determining a distribution of material that fulfills a specific task. Typically, the method requires 100 through 1000 iterations to recover a locally optimal and physically admissible design. Consequently, the governing equations for the problem under consideration must be solved a large number of times for a slowly changing material distribution. When systems of several millions of degrees of freedom are considered, as is often the case for real world applications, their solutions through traditional direct techniques become infeasible. Naturally, this raises the interest in using highly scalable parallel iterative techniques. For physical problems governed by the Helmholtz equation, like acoustic, electromagnetic and structural vibration problems, no general scalable parallel iterative techniques currently exist. Therefore it is of interest to investigate the performance of the preconditioner in the context of Topology Optimization.
The paper is organized as follows. In Section 2 we present an improved analytical apparatus that allows us to better characterize the Schur complements as solution operators. In Section 3 we recall the construction of the preconditioner in order to make this work self-contained. Section 4 is devoted to numerical results. Finally, in Section 5, we draw conclusions from this work, and point towards future directions of research.
Section snippets
Analytical apparatus
We provide an insight into the rank-structure of the Schur complements that arise in the construction of the preconditioner. Let be a finite element discretization of a boundary value problem posed on a domain with boundary . Since is sparse, we can reorder its degrees of freedom (dof’s) to expose the following block-structure, and define the (aggregated) sub-matrices : The diagonal
Preconditioner construction and update
The construction relies on a variant of the well-known nested dissection algorithm, see George [23], and extends the work of Gillman and Martinsson, see [15], developed in the context of finite difference approximations.
Numerical results
As anticipated in Section 1, Topology Optimization is an iterative method, used mainly for PDE’s constrained optimization problems, to create highly optimized designs for specific purposes. The objective is to determine a distribution of material that fulfills a specific task in a locally optimal manner, without the need to enforce any a priori restriction on the design topology, see [19]. Although the use of gradient-based techniques results in a significant reduction in the number of
Conclusions and outlook
In this work, we review the construction of the preconditioner proposed in [1], sharpen our theoretical understanding of the methodology, and provide accurate cost estimates for different scenarios. The main novel contribution is a strategy to apply the preconditioner to time-varying discretizations. Specifically, we discuss an application to Topology Optimization of an acoustic cavity, which requires multiple solutions of a notoriously challenging Helmholtz problem with strong material
Acknowledgment
The work of R.E. Christiansen was financially supported by Villum Fonden through the research project Topology Optimization–the Next Generation (NextOpt) .
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