Zexin Liu

Associated to a finite measure on the real line with finite moments are recurrence coefficients in a three-term formula for orthogonal polynomials with respect to this measure. These recurrence coefficients are frequently inputs to modern computational tools that facilitate evaluation and manipulation of polynomials with respect to the measure, and such tasks are foundational in numerical approximation and quadrature. Although the recurrence coefficients for classical measures are known explicitly, those for nonclassical measures must typically be numerically computed. We survey and review existing approaches for computing these recurrence coefficients for univariate orthogonal polynomial families and propose a novel “predictor–corrector” algorithm for a general class of continuous measures. We combine the predictor–corrector scheme with a stabilized Lanczos procedure for a new hybrid algorithm that computes recurrence coefficients for a fairly wide class of measures that can have both continuous and discrete parts. We evaluate the new algorithms against existing methods in terms of accuracy and efficiency.

Background: Computational biomedical simulations frequently contain parameters that model physical features, material coefficients, and physiological effects, whose values are typically assumed known a priori. Understanding the effect of variability in those assumed values is currently a topic of great interest. A general-purpose software tool that quantifies how variation in these parameters affects model outputs is not broadly available in biomedicine. For this reason, we developed the ‘UncertainSCI’ uncertainty quantification software suite to facilitate analysis of uncertainty due to parametric variability. Methods: We developed and distributed a new open-source Python-based software tool, UncertainSCI, which employs advanced parameter sampling techniques to build polynomial chaos (PC) emulators that can be used to predict model outputs for general parameter values. Uncertainty of model outputs is studied by modeling parameters as random variables, and model output statistics and sensitivities are then easily computed from the emulator. Our approaches utilize modern, near-optimal techniques for sampling and PC construction based on weighted Fekete points constructed by subsampling from a suitably randomized candidate set. Results: Concentrating on two test cases—modeling bioelectric potentials in the heart and electric stimulation in the brain—we illustrate the use of UncertainSCI to estimate variability, statistics, and sensitivities associated with multiple parameters in these models. Conclusion: UncertainSCI is a powerful yet lightweight tool enabling sophisticated probing of parametric variability and uncertainty in biomedical simulations. Its non-intrusive pipeline allows users to leverage existing software libraries and suites to accurately ascertain parametric uncertainty in a variety of applications.

We have developed UncertainSCI (UncertainSCI, 2020) as an open-source tool designed to make modern uncertainty quantification (UQ) techniques more accessible in biomedical simulation applications. UncertainSCI is implemented in Python with a noninvasive interface to meet our software design goals of 1) numerical accuracy, 2) simple application programming interface (API), 3) adaptability to many applications and methods, and 4) interfacing with diverse simulation software. Using a Python implementation in UncertainSCI allowed us to utilize the popularity and low barrier-to-entry of Python and its common packages and to leverage the built-in integration and support for Python in common simulation software packages and languages. Additionally, we used noninvasive UQ techniques and created a similarly noninvasive interface to external modeling software that can be called in diverse ways, depending on the complexity and level of Python integration in the external simulation pipeline. We have developed and included examples applying UncertainSCI to relatively simple 1D simulations implemented in Python, and to bioelectric field simulations implemented in external software packages, which demonstrate the use of UncertainSCI and the effectiveness of the architecture and implementation in achieving our design goals. UnceratainSCI differs from similar software, notably UQLab, Uncertainpy, and Simnibs, in that it can be efficiently and non-invasively used with external simulation software, specifically with high resolution 3D simulations often used in Bioelectric field simulations. Figure 1 illustrates the use of UncertainSCI in computing UQ with modeling pipelines for bioelectricity simulations.

Abstract. Orthogonal polynomials of several variables have a vector-valued three-term recurrence relation, much like the corresponding one-dimensional relation. This relation requires only knowledge of certain recurrence matrices, and allows simple and stable evaluation of multivariate orthogonal polynomials. In the univariate case, various algorithms can stably and accurately evaluate the recurrence coefficients given the ability to compute polynomial moments, but it is difficult to identify analogous procedures in multiple dimensions. We present a new Multivariate Stieltjes (MS) algorithm that fills this gap in the multivariate case, allowing computation of recurrence matrices assuming moments are available. The algorithm is essentially explicit in two and three dimensions, but requires the numerical solution to a nonconvex problem in more than three dimensions. Compared to direct Gram–Schmidt-type orthogonalization, we demonstrate on several examples in up to three dimensions that the MS algorithm is far more stable, and allows accurate computation of orthogonal bases in the multivariate setting, in contrast to direct orthogonalization approaches.