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joss.04249/10.21105.joss.04249.crossref.xml | 398 ++++++++++++++++++++ 1 file changed, 398 insertions(+) create mode 100644 joss.04249/10.21105.joss.04249.crossref.xml diff --git a/joss.04249/10.21105.joss.04249.crossref.xml b/joss.04249/10.21105.joss.04249.crossref.xml new file mode 100644 index 0000000000..e2ff8cf19b --- /dev/null +++ b/joss.04249/10.21105.joss.04249.crossref.xml @@ -0,0 +1,398 @@ + + + + 20231027T152430-b71dbd3ba811ca57925ec8ba90ce0ab3da3f7053 + 20231027152430 + + JOSS Admin + admin@theoj.org + + The Open Journal + + + + + Journal of Open Source Software + JOSS + 2475-9066 + + 10.21105/joss + https://joss.theoj.org + + + + + 10 + 2023 + + + 8 + + 90 + + + + UncertainSCI: A Python Package for Noninvasive +Parametric Uncertainty Quantification of Simulation Pipelines + + + + Jess + Tate + https://orcid.org/0000-0002-2934-1453 + + + Zexin + Liu + https://orcid.org/0000-0003-3409-5709 + + + Jake A + Bergquist + https://orcid.org/0000-0002-4586-6911 + + + Sumientra + Rampersad + https://orcid.org/0000-0001-9860-4459 + + + Dan + White + + + Chantel + Charlebois + https://orcid.org/0000-0002-4139-3539 + + + Lindsay + Rupp + https://orcid.org/0000-0002-2688-7688 + + + Dana H + Brooks + https://orcid.org/0000-0003-3231-6715 + + + Rob S + MacLeod + https://orcid.org/0000-0002-0000-0356 + + + Akil + Narayan + https://orcid.org/0000-0002-5914-4207 + + + + 10 + 27 + 2023 + + + 4249 + + + 10.21105/joss.04249 + + + http://creativecommons.org/licenses/by/4.0/ + http://creativecommons.org/licenses/by/4.0/ + http://creativecommons.org/licenses/by/4.0/ + + + + Software archive + 10.5281/zenodo.8226383 + + + GitHub review issue + https://github.com/openjournals/joss-reviews/issues/4249 + + + + 10.21105/joss.04249 + https://joss.theoj.org/papers/10.21105/joss.04249 + + + https://joss.theoj.org/papers/10.21105/joss.04249.pdf + + + + + + UncertainSCI + 2020 + UncertainSCI. (2020). +https://www.sci.utah.edu/cibc-software/uncertainsci.html + + + Efficient sampling for polynomial chaos-based +uncertainty quantification and sensitivity analysis using weighted +approximate fekete points + Burk + International Journal for Numerical Methods +in Biomedical Engineering + 11 + 36 + 10.1002/cnm.3395 + 2020 + Burk, K. M., Narayan, A., & Orr, +J. A. (2020). Efficient sampling for polynomial chaos-based uncertainty +quantification and sensitivity analysis using weighted approximate +fekete points. International Journal for Numerical Methods in Biomedical +Engineering, 36(11), e3395. +https://doi.org/10.1002/cnm.3395 + + + Cardiac position sensitivity study in the +electrocardiographic forward problem using stochastic collocation and +BEM + Swenson + Annals of Biomedical +Engineering + 12 + 30 + 10.1007/s10439-011-0391-5 + 2011 + Swenson, D. J., Geneser, S. E., +Stinstra, J. G., Kirby, R. M., & MacLeod, R. S. (2011). Cardiac +position sensitivity study in the electrocardiographic forward problem +using stochastic collocation and BEM. Annals of Biomedical Engineering, +30(12), 2900–2910. +https://doi.org/10.1007/s10439-011-0391-5 + + + The influence of stochastic organ +conductivity in 2D ECG forward modeling: A stochastic finite element +study + Geneser + Proceedings of the IEEE engineering in +medicine and biology society 27th annual international +conference + 10.1109/iembs.2005.1615736 + 2005 + Geneser, S. E., Choe, S., Kirby, R. +M., & Macleod, R. S. (2005). The influence of stochastic organ +conductivity in 2D ECG forward modeling: A stochastic finite element +study. Proceedings of the IEEE Engineering in Medicine and Biology +Society 27th Annual International Conference, 5528–5531. +https://doi.org/10.1109/iembs.2005.1615736 + + + Uncertainty quantification of the effects of +segmentation variability in ECGI + Tate + Functional imaging and modeling of the +heart + 10.1007/978-3-030-78710-3_49 + 2019 + Tate, J. D., Good, W. W., Zemzemi, +N., Boonstra, M., Dam, P. van, Brooks, D. H., Narayan, A., & +MacLeod, R. S. (2019). Uncertainty quantification of the effects of +segmentation variability in ECGI. In Functional imaging and modeling of +the heart (pp. 515–522). Springer-Cham. +https://doi.org/10.1007/978-3-030-78710-3_49 + + + Quantification of uncertainty due to tissue +conductivity variability in simulations of brain +stimulation + Rampersad + 10th international IEEE EMBS conference on +neural engineering + 2021 + Rampersad, S., Charlebois, C., Tate, +J. D., MacLeod, R. S., Brooks, D. H., & Narayan, A. (2021, May). +Quantification of uncertainty due to tissue conductivity variability in +simulations of brain stimulation. 10th International IEEE EMBS +Conference on Neural Engineering. + + + Using UncertainSCI to quantify uncertainty in +cardiac simulations + Rupp + 2020 computing in cardiology + 10.22489/CinC.2020.275 + 2020 + Rupp, L. C., Liu, Z., Bergquist, J. +A., Rampersad, S., White, D., Tate, J. D., Brooks, D. H., Narayan, A., +& MacLeod, R. S. (2020). Using UncertainSCI to quantify uncertainty +in cardiac simulations. 2020 Computing in Cardiology, 1–4. +https://doi.org/10.22489/CinC.2020.275 + + + The role of myocardial fiber direction in +epicardial activation patterns via uncertainty +quantification + Rupp + Computing in cardiology + 48 + 10.23919/cinc53138.2021.9662950 + 2021 + Rupp, L. C., Bergquist, J. A., +Zenger, B., Gillette, K., Narayan, A., Tate, J. D., Plank, G., & +MacLeod, R. S. (2021). The role of myocardial fiber direction in +epicardial activation patterns via uncertainty quantification. Computing +in Cardiology, 48. +https://doi.org/10.23919/cinc53138.2021.9662950 + + + Uncertainty quantification in simulations of +myocardial ischemia + Bergquist + Computing in cardiology + 48 + 10.23919/cinc53138.2021.9662837 + 2021 + Bergquist, J. A., Zenger, B., Rupp, +L. C., Narayan, A., Tate, J. D., & MacLeod, R. S. (2021). +Uncertainty quantification in simulations of myocardial ischemia. +Computing in Cardiology, 48. +https://doi.org/10.23919/cinc53138.2021.9662837 + + + Examining the impact of prior models in +transmural electrophysiological imaging: A hierarchical multiple-model +bayesian approach + Rahimi + IEEE Trans. Med. Imag. + 1 + 35 + 10.1109/TMI.2015.2464315 + 0278-0062 + 2016 + Rahimi, A., Sapp, J., Xu, J., +Bajorski, P., Horáček, M., & Wang, L. (2016). Examining the impact +of prior models in transmural electrophysiological imaging: A +hierarchical multiple-model bayesian approach. IEEE Trans. Med. Imag., +35(1), 229–243. +https://doi.org/10.1109/TMI.2015.2464315 + + + Uncertainty quantification in brain +stimulation using UncertainSCI + Tate + Brain Stimulation: Basic, Translational, and +Clinical Research in Neuromodulation + 6 + 14 + 10.1016/j.brs.2021.10.226 + 2021 + Tate, J. D., Rampersad, S., +Charlebois, C., Liu, Z., Bergquist, J. A., White, D., Rupp, L. C., +Brooks, D. H., Narayan, A., & MacLeod, R. S. (2021). Uncertainty +quantification in brain stimulation using UncertainSCI. Brain +Stimulation: Basic, Translational, and Clinical Research in +Neuromodulation, 14(6), 1659–1660. +https://doi.org/10.1016/j.brs.2021.10.226 + + + Variational bayesian electrophysiological +imaging of myocardial infarction. + Xu + Med Image Comput Comput Assist +Interv + Pt 2 + 17 + 10.1007/978-3-319-10470-6_66 + 2014 + Xu, J., Sapp, J. L., Dehaghani, A. +R., Gao, F., & Wang, L. (2014). Variational bayesian +electrophysiological imaging of myocardial infarction. Med Image Comput +Comput Assist Interv, 17(Pt 2), 529–537. +https://doi.org/10.1007/978-3-319-10470-6_66 + + + Numerical Methods for Stochastic Computations: +A Spectral Method Approach + Xiu + 10.1007/978-3-319-10470-6_66 + 0-691-14212-2 + 2010 + Xiu, D. (2010). Numerical Methods for +Stochastic Computations: A Spectral Method Approach. Princeton +University Press. +https://doi.org/10.1007/978-3-319-10470-6_66 + + + Gaussian Processes in Machine +Learning + Rasmussen + Advanced Lectures on Machine +Learning + 10.1007/978-3-540-28650-9_4 + 978-3-540-28650-9 + 2004 + Rasmussen, C. E. (2004). Gaussian +Processes in Machine Learning. In Advanced Lectures on Machine Learning +(pp. 63–71). Springer, Berlin, Heidelberg. +https://doi.org/10.1007/978-3-540-28650-9_4 + + + Weighted Approximate Fekete Points: Sampling +for Least-Squares Polynomial Approximation + Guo + SIAM Journal on Scientific +Computing + 1 + 40 + 10.1137/17M1140960 + 1064-8275 + 2018 + Guo, L., Narayan, A., Yan, L., & +Zhou, T. (2018). Weighted Approximate Fekete Points: Sampling for +Least-Squares Polynomial Approximation. SIAM Journal on Scientific +Computing, 40(1), A366–A387. +https://doi.org/10.1137/17M1140960 + + + Optimal weighted least-squares +methods + Cohen + SMAI Journal of Computational +Mathematics + 3 + 10.5802/smai-jcm.24 + 2426-8399 + 2017 + Cohen, A., & Migliorati, G. +(2017). Optimal weighted least-squares methods. SMAI Journal of +Computational Mathematics, 3, 181–203. +https://doi.org/10.5802/smai-jcm.24 + + + Computation of induced orthogonal polynomial +distributions + Narayan + Electronic Transactions on Numerical +Analysis + 50 + 10.1553/etna_vol50s71 + 2018 + Narayan, A. (2018). Computation of +induced orthogonal polynomial distributions. Electronic Transactions on +Numerical Analysis, 50, 71–97. +https://doi.org/10.1553/etna_vol50s71 + + + + + + From 110fef8be392db3238f123fe176a466ae240091c Mon Sep 17 00:00:00 2001 From: The Open Journals editorial robot <89919391+editorialbot@users.noreply.github.com> Date: Fri, 27 Oct 2023 16:25:01 +0100 Subject: [PATCH 3/4] Creating 10.21105.joss.04249.jats --- joss.04249/10.21105.joss.04249.jats | 724 ++++++++++++++++++++++++++++ 1 file changed, 724 insertions(+) create mode 100644 joss.04249/10.21105.joss.04249.jats diff --git a/joss.04249/10.21105.joss.04249.jats b/joss.04249/10.21105.joss.04249.jats new file mode 100644 index 0000000000..b5d9f413b6 --- /dev/null +++ b/joss.04249/10.21105.joss.04249.jats @@ -0,0 +1,724 @@ + + +

+ + + + +Journal of Open Source Software +JOSS + +2475-9066 + +Open Journals + + + +4249 +10.21105/joss.04249 + +UncertainSCI: A Python Package for Noninvasive Parametric +Uncertainty Quantification of Simulation Pipelines + + + +https://orcid.org/0000-0002-2934-1453 + +Tate +Jess + + + + +https://orcid.org/0000-0003-3409-5709 + +Liu +Zexin + + + + + +https://orcid.org/0000-0002-4586-6911 + +Bergquist +Jake A + + + + + + +https://orcid.org/0000-0001-9860-4459 + +Rampersad +Sumientra + + + + + + +White +Dan + + + + +https://orcid.org/0000-0002-4139-3539 + +Charlebois +Chantel + + + + + +https://orcid.org/0000-0002-2688-7688 + +Rupp +Lindsay + + + + + + +https://orcid.org/0000-0003-3231-6715 + +Brooks +Dana H + + + + +https://orcid.org/0000-0002-0000-0356 + +MacLeod +Rob S + + + + + + +https://orcid.org/0000-0002-5914-4207 + +Narayan +Akil + + + + + + +Scientific Computing and Imaging Institute, University of +Utah, Salt Lake City, UT, USA + + + + +Mathematics Department, University of Utah, Salt Lake City, +UT, USA + + + + +Biomedical Engineering Department , University of Utah, +Salt Lake City, UT, USA + + + + +Nora Eccles Cardiovascular Research and Training Institute, +University of Utah, Salt Lake City, UT, USA + + + + +Physics Department, University of Massachusetts, Boston, +MA, USA + + + + +Electrical and Computer Engineering Department, +Northeastern University, Boston, MA, USA + + + +8 +90 +4249 + +Authors of papers retain copyright and release the +work under a Creative Commons Attribution 4.0 International License (CC +BY 4.0) +2022 +The article authors + +Authors of papers retain copyright and release the work under +a Creative Commons Attribution 4.0 International License (CC BY +4.0) + + + +Python +uncertainty quantification +computer modeling +polynomial chaos +bioelectricity + + + + + + Summary +

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. + [fig:pipeline] + illustrates the use of UncertainSCI in computing UQ with modeling + pipelines for bioelectricity simulations.

+ +

User pipeline for UncertainSCI. After the user inputs + parameter distributions, UncertainSCI will compute an efficient + sampling scheme. The parameter samples are run through the targeted + modeling pipeline, which can be implemented in external software + tools. The computed solutions are collected and compiled into + relevant statistics with UncertainSCI. +

+ +
+
+ + Statement of need +

Biomedical computer models include many input parameters that do + not have precisely defined values, for example because their value + defines physiological processes that are not uniform across patients. + As such, any simulation output necessarily has some uncertainty + associated with the uncertain value of the input parameter. + Exploration and quantification of this model output uncertainty is + challenging when more than a single parameter is present; biomedical + computer models often have 5-20 such parameters. Quantification of + this uncertainty through UQ techniques provides statistics and + sensitivity information, a critical component when evaluating the + relative impact of parameter variation on the solution accuracy. While + the need and importance of UQ in clinical modeling is generally + accepted, automated tools for implementing UQ techniques remain + evasive for many researchers. UncertainSCI has been used to quantify + uncertainty in multiple modeling pipelines, including: cardiac tissue + modeling + (Bergquist + et al., 2021; + Rupp + et al., 2020, + 2021), + electrocardiographic (ECG) simulation + (Geneser + et al., 2005; + Swenson + et al., 2011), ECG imaging (ECGI) + (Tate + et al., 2019), transcranial current stimulation (tCS) modeling + (Rampersad + et al., 2021; + Tate + et al., 2021), and electrocorticography (ECoG) stimulation + (Rampersad + et al., 2021; + Tate + et al., 2021).

+
+ + Mathematics +

In UncertainSCI, we quantify forward parametric uncertainty in + cardiac simulations using polynomial chaos expansions (PCE) + (Xiu, + 2010). Although we also have an implementation of Monte Carlo + sampling, and we intend to expand to UncertianSCI to include other UQ + methods, such as Guassian process emulators + (Rasmussen, + 2004), Markov Chain Monte Carlo + (Rahimi + et al., 2016) and Bayesian + inference(Xu + et al., 2014), we primarily focused on PCE due to its + non-invasive formulation and computational efficiency. PCE + approximates the dependence of a quantity of interest (QoI) that is + the model output from the forward simulation on a finite number of + random parameters via a multivariate polynomial function of those + parameters. With + + u + the QoI (scalar-valued for simplicity), and + + + pd + the vector of uncertain parameters, the PCE approach builds the + function + + uN, + given as, + + u(p)uN(p)=j=1Ncjϕj(p), + where the + + {ϕj}j=1N + functions are multivariate polynomials, and the coefficients + + + {cj}j=1N + are learned through an ensemble of collected data, + + + {pj}j=1MForward model u{u(pj)}j=1MEmulator training{cj}j=1N. + Typically one seeks convergence of + + uN + to + + u + in an + + L2-type + norm weighted by the probability density of the parameters + + + p. + The polynomial function + + uN + constitutes an emulator for QoI + + u, + from which statistics of the QoI, including the mean, variance, and + parameter sensitivities, are efficiently computed from the polynomial. + UncertainSCI uses a particular type of pseudo-random sampling for the + parameter design + + {pj}j=1M, + along with a particular weighted least-squares approach for emulator + training, both of which are recently developed advances in + high-dimensional approximation. The entire procedure + 1 is non-intrusive, since + the forward model need only be queried at a particular set of + samples.

+

The efficiency of PCE for analysis of UQ in a forward simulation + depends on efficient selection of parameter samples + + + {pj}j=1M. + The goal is to use as few samples + + M + as possible while still ensuring that + + uN + is an accurate emulator for + + u. + UncertainSCI uses a two-step approach to strategically sample the + parameter space: 1. A discrete candidate set is generated via random + sampling with respect to a ‘’biased’’ probability measure + + + μ, + that is distinct from (but related to) the probability distribution of + the parameter + + p. + 2. A weighted D-optimal design is sought by subsampling this discrete + candidate set. UncertainSCI uses a greedy approach to produce an + approximation to such a design.

+

The probability measure + + μ + that must be randomly sampled is a distribution that exploits a + concentration of measurable phenomena to provably increase the quality + of the candidate set + (Cohen + & Migliorati, 2017). Sampling from this distribution when + the components of the parameter vector + + p + are independent is computationally efficient, having complexity that + is linear in the number of parameters + + d + (Narayan, + 2018). The relatively large candidate set generated from this + random sampling is pruned via subsampling using a weighted D-optimal + design optimization. UncertainSCI’s algorithm for this approach + approximately computes a weighted D-optimal design via the weighted + approximate Fekete points (WAFP) procedure + (Burk + et al., 2020; + Guo + et al., 2018), which greedily maximizes a weighted matrix + determinant. The result is a geometrically unstructured parameter + design of + + M + samples for use in the pipeline + 1.

+

Once the experimental design is created through the WAFP procedure, + an ensemble of forward simulations + + {u(pj)}j=1M + is collected from the simulation software, and UncertainSCI produces a + PCE emulator + + uN + through a (weighted) least-squares procedure. From this emulator, + UncertainSCI can compute statistics, sensitivities, residuals, and + cross-validation metrics, and can adaptively tune the complexity of + the PCE emulator based on a user-prescribed tolerance and/or + computational budget.

+
+ + Acknowledgements +

This project was supported by grants from the National Institute of + Biomedical Imaging and Bioengineering (U24EB029012) from the National + Institutes of Health.

+
+ + + + + + UncertainSCI + 2020 + 20211202 + https://www.sci.utah.edu/cibc-software/uncertainsci.html + + + + + + BurkK. M. + NarayanA. + OrrJ. A. + + Efficient sampling for polynomial chaos-based uncertainty quantification and sensitivity analysis using weighted approximate fekete points + International Journal for Numerical Methods in Biomedical Engineering + 2020 + 36 + 11 + 10.1002/cnm.3395 + e3395 + + + + + + + SwensonD. J. + GeneserS. E. + StinstraJ. G. + KirbyR. M. + MacLeodR. S. + + Cardiac position sensitivity study in the electrocardiographic forward problem using stochastic collocation and BEM + Annals of Biomedical Engineering + 2011 + 30 + 12 + 10.1007/s10439-011-0391-5 + 2900 + 2910 + + + + + + GeneserS. E. + ChoeS. + KirbyR. M. + MacleodR. S. + + The influence of stochastic organ conductivity in 2D ECG forward modeling: A stochastic finite element study + Proceedings of the IEEE engineering in medicine and biology society 27th annual international conference + Shanghai, China + 200509 + 10.1109/iembs.2005.1615736 + 5528 + 5531 + + + + + + TateJ. D. + GoodW. W. + ZemzemiN. + BoonstraM. + DamP. van + BrooksD. H. + NarayanA. + MacLeodR. S. + + Uncertainty quantification of the effects of segmentation variability in ECGI + Functional imaging and modeling of the heart + Springer-Cham + Palo Alto, USA + 2019 + 10.1007/978-3-030-78710-3_49 + 515 + 522 + + + + + + RampersadS. + CharleboisC. + TateJ. D. + MacLeodR. S. + BrooksD. H. + NarayanA. + + Quantification of uncertainty due to tissue conductivity variability in simulations of brain stimulation + 10th international IEEE EMBS conference on neural engineering + 202105 + + + + + + RuppL. C. + LiuZ. + BergquistJ. A. + RampersadS. + WhiteD. + TateJ. D. + BrooksD. H. + NarayanA. + MacLeodR. S. + + Using UncertainSCI to quantify uncertainty in cardiac simulations + 2020 computing in cardiology + 2020 + + 10.22489/CinC.2020.275 + 1 + 4 + + + + + + RuppL. C. + BergquistJ. A. + ZengerB. + GilletteK. + NarayanA. + TateJ. D. + PlankG. + MacLeodR. S. + + The role of myocardial fiber direction in epicardial activation patterns via uncertainty quantification + Computing in cardiology + 202109 + 48 + 10.23919/cinc53138.2021.9662950 + + + + + + BergquistJ. A. + ZengerB. + RuppL. C. + NarayanA. + TateJ. D. + MacLeodR. S. + + Uncertainty quantification in simulations of myocardial ischemia + Computing in cardiology + 202109 + 48 + 10.23919/cinc53138.2021.9662837 + + + + + + RahimiA. + SappJ. + XuJ. + BajorskiP. + HoráčekM. + WangL. + + Examining the impact of prior models in transmural electrophysiological imaging: A hierarchical multiple-model bayesian approach + IEEE Trans. Med. Imag. + 201601 + 35 + 1 + 0278-0062 + 10.1109/TMI.2015.2464315 + 229 + 243 + + + + + + TateJ. D. + RampersadS. + CharleboisC. + LiuZexin + BergquistJ. A. + WhiteD. + RuppL. C. + BrooksD. H. + NarayanA. + MacLeodR. S. + + Uncertainty quantification in brain stimulation using UncertainSCI + Brain Stimulation: Basic, Translational, and Clinical Research in Neuromodulation + Elsevier + 2021 + 14 + 6 + https://doi.org/10.1016/j.brs.2021.10.226 + 10.1016/j.brs.2021.10.226 + 1659 + 1660 + + + + + + XuJingjia + SappJohn L + DehaghaniAzar Rahimi + GaoFei + WangLinwei + + Variational bayesian electrophysiological imaging of myocardial infarction. + Med Image Comput Comput Assist Interv + 2014 + 17 + Pt 2 + 10.1007/978-3-319-10470-6_66 + 529 + 537 + + + + + + XiuD. + + Numerical Methods for Stochastic Computations: A Spectral Method Approach + Princeton University Press + 201007 + 0-691-14212-2 + 10.1007/978-3-319-10470-6_66 + + + + + + RasmussenC. E. + + Gaussian Processes in Machine Learning + Advanced Lectures on Machine Learning + Springer, Berlin, Heidelberg + 2004 + 978-3-540-28650-9 + 10.1007/978-3-540-28650-9_4 + 63 + 71 + + + + + + GuoL. + NarayanA. + YanL. + ZhouT. + + Weighted Approximate Fekete Points: Sampling for Least-Squares Polynomial Approximation + SIAM Journal on Scientific Computing + 2018 + 40 + 1 + 1064-8275 + http://epubs.siam.org/doi/abs/10.1137/17M1140960 + 10.1137/17M1140960 + A366 + A387 + + + + + + CohenAlbert + MiglioratiGiovanni + + Optimal weighted least-squares methods + SMAI Journal of Computational Mathematics + 2017 + 3 + 2426-8399 + http://smai-jcm.cedram.org/item?id=SMAI-JCM_2017__3__181_0 + 10.5802/smai-jcm.24 + 181 + 203 + + + + + + NarayanA. + + Computation of induced orthogonal polynomial distributions + Electronic Transactions on Numerical Analysis + 2018 + 50 + https://epub.oeaw.ac.at?arp=0x003a184e + 10.1553/etna_vol50s71 + 71 + 97 + + + + +
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