Quadrature Strategies for Constructing Polynomial Approximations

Finding suitable points for multivariate polynomial interpolation and approximation is a challenging task. Yet, despite this challenge, there has been tremendous research dedicated to this singular cause. In this paper, we begin by reviewing classical methods for finding suitable quadrature points for polynomial approximation in both the univariate and multivariate setting. Then, we categorize recent advances into those that propose a new sampling approach and those centered on an optimization strategy. The sampling approaches yield a favorable discretization of the domain, while the optimization methods pick a subset of the discretized samples that minimize certain objectives. While not all strategies follow this two-stage approach, most do. Sampling techniques covered include subsampling quadratures, Christoffel, induced and Monte Carlo methods. Optimization methods discussed range from linear programming ideas and Newton's method to greedy procedures from numerical linear algebra. Our exposition is aided by examples that implement some of the aforementioned strategies

Quadrature Strategies for Constructing Polynomial Approximations

Finding suitable points for multivariate polynomial interpolation and approximation is a challenging task. Yet, despite this challenge, there has been tremendous research dedicated to this singular cause. In this paper, we begin by reviewing classical methods for finding suitable quadrature points for polynomial approximation in both the univariate and multivariate setting. Then, we categorize recent advances into those that propose a new sampling approach, and those centered on an optimization strategy. The sampling approaches yield a favorable discretization of the domain, while the optimization methods pick a subset of the discretized samples that minimize certain objectives. While not all strategies follow this two-stage approach, most do. Sampling techniques covered include subsampling quadratures, Christoffel, induced and Monte Carlo methods. Optimization methods discussed range from linear programming ideas and Newton's method to greedy procedures from numerical linear algebra. Our exposition is aided by examples that implement some of the aforementioned strategies

Dynamic kinetochore size regulation promotes microtubule capture and chromosome biorientation in mitosis

Faithful chromosome segregation depends on the ability of sister kinetochores to attach to spindle microtubules. The outer layer of kinetochores transiently expands in early mitosis to form a fibrous corona, and compacts following microtubule capture. Here we show that the dynein adaptor Spindly and the RZZ (ROD-Zwilch-ZW10) complex drive kinetochore expansion in a dynein-independent manner. C-terminal farnesylation and MPS1 kinase activity cause conformational changes of Spindly that promote oligomerization of RZZ-Spindly complexes into a filamentous meshwork in cells and in vitro. Concurrent with kinetochore expansion, Spindly potentiates kinetochore compaction by recruiting dynein via three conserved short linear motifs. Expanded kinetochores unable to compact engage in extensive, long-lived lateral microtubule interactions that persist to metaphase, and result in merotelic attachments and chromosome segregation errors in anaphase. Thus, dynamic kinetochore size regulation in mitosis is coordinated by a single, Spindly-based mechanism that promotes initial microtubule capture and subsequent correct maturation of attachments