Rational filter functions can be used to improve convergence of contour-based eigensolvers, a popular family of algorithms for the solution of the interior eigenvalue problem. We present a framework for the optimization of rational filters based on a non-convex weighted Least-Squares scheme. When used in combination with the FEAST library, our filters out-perform existing ones on a large and representative set of benchmark problems. This work provides a detailed description of : (1) a set up of the optimization process that exploits symmetries of the filter function for Hermitian eigenproblems, (2) a formulation of the gradient descent and Levenberg-Marquardt algorithms that exploits the symmetries, (3) a method to select the starting position for the optimization algorithms that reliably produces effective filters, (4) a constrained optimization scheme that produces filter functions with specific properties that may be beneficial to the performance of the eigensolver that employs them.

Idealizations are omnipresent in science. However, to date, science education research has paid surprisingly little attention to the use of idealizations in fostering students’ model competence and understanding of the nature of science (NOS). The starting point for the theoretical reflection in this paper is that insufficient consideration of idealizations in the science classroom can lead to learning difficulties. The following discussions should help to clarify the terms idealization and model and their relationship to each other. An example is drawn from physics. At least two cases can apply when considering model usage in the classroom. In the first case, to understand an observed phenomenon, a model (as a representation) of the situation to be explained is constructed. At this point, it is necessary to perform idealization. Seemingly, this step is still neglected in much of the science education literature but is well addressed in the philosophy of science. In the second case, existing models to work with are introduced, perhaps alongside a real experimental situation. This approach is called working with models in science education. This paper focuses primarily on the first case. Against the background of model building, a positioning and conceptual approximation of idealizations take place. To organize the idealization process, a framework of several categories of idealization adopted from science philosophy is offered. The framework is intended to stimulate explicit reflection about how models are constructed. The construction of a model by idealization is illustrated through an example from geometrical optics. Finally, the considerations presented are discussed in the context of the literature, and suggested research topics are provided.

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Solving dense Hermitian eigenproblems arranged in a sequence with direct solvers fails to take advantage of those spectral properties that are pertinent to the entire sequence and not just to the single problem. When such features take the form of correlations between the eigenvectors of consecutive problems, as is the case in many real-world applications, the potential benefit of exploiting them can be substantial. We present the Chebyshev Accelerated Subspace iteration Eigensolver (ChASE), a modern algorithm and library based on subspace iteration with polynomial acceleration. Novel to ChASE is the computation of the spectral estimates that enter in the filter and an optimization of the polynomial degree that further reduces the necessary floating-point operations. ChASE is written in C++ using the modern software engineering concepts that favor a simple integration in application codes and a straightforward portability over heterogeneous platforms. When solving sequences of Hermitian eigenproblems for a portion of their extremal spectrum, ChASE greatly benefits from the sequence’s spectral properties and outperforms direct solvers in many scenarios. The library ships with two distinct parallelization schemes, supports execution over distributed GPUs, and is easily extensible to other parallel computing architectures.

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