Churchill, Victor; Chen, Yuan; Xu, Zhongshu; Xiu, Dongbin DNN modeling of partial differential equations with incomplete data. (English) Zbl 07748076 J. Comput. Phys. 493, Article ID 112502, 20 p. (2023). MSC: 68Txx 65Lxx 37Mxx PDFBibTeX XMLCite \textit{V. Churchill} et al., J. Comput. Phys. 493, Article ID 112502, 20 p. (2023; Zbl 07748076) Full Text: DOI
Churchill, Victor; Manns, Steve; Chen, Zhen; Xiu, Dongbin Robust modeling of unknown dynamical systems via ensemble averaged learning. (English) Zbl 07640582 J. Comput. Phys. 474, Article ID 111842, 12 p. (2023). MSC: 68Txx 65Lxx 35Qxx PDFBibTeX XMLCite \textit{V. Churchill} et al., J. Comput. Phys. 474, Article ID 111842, 12 p. (2023; Zbl 07640582) Full Text: DOI arXiv
Chen, Zhen; Churchill, Victor; Wu, Kailiang; Xiu, Dongbin Deep neural network modeling of unknown partial differential equations in nodal space. (English) Zbl 07524779 J. Comput. Phys. 449, Article ID 110782, 20 p. (2022). MSC: 68Txx 35Qxx 37Mxx PDFBibTeX XMLCite \textit{Z. Chen} et al., J. Comput. Phys. 449, Article ID 110782, 20 p. (2022; Zbl 07524779) Full Text: DOI arXiv
Chen, Zhen; Xiu, Dongbin On generalized residual network for deep learning of unknown dynamical systems. (English) Zbl 07505957 J. Comput. Phys. 438, Article ID 110362, 16 p. (2021). MSC: 68Txx 35Qxx 37Mxx PDFBibTeX XMLCite \textit{Z. Chen} and \textit{D. Xiu}, J. Comput. Phys. 438, Article ID 110362, 16 p. (2021; Zbl 07505957) Full Text: DOI arXiv
Wu, Kailiang; Xiu, Dongbin Data-driven deep learning of partial differential equations in modal space. (English) Zbl 07505629 J. Comput. Phys. 408, Article ID 109307, 22 p. (2020). MSC: 35-XX 65-XX PDFBibTeX XMLCite \textit{K. Wu} and \textit{D. Xiu}, J. Comput. Phys. 408, Article ID 109307, 22 p. (2020; Zbl 07505629) Full Text: DOI arXiv
Canuto, Claudio; Pieraccini, Sandra; Xiu, Dongbin Uncertainty quantification of discontinuous outputs via a non-intrusive bifidelity strategy. (English) Zbl 1453.62796 J. Comput. Phys. 398, Article ID 108885, 19 p. (2019). MSC: 62P35 76S05 PDFBibTeX XMLCite \textit{C. Canuto} et al., J. Comput. Phys. 398, Article ID 108885, 19 p. (2019; Zbl 1453.62796) Full Text: DOI
Qin, Tong; Wu, Kailiang; Xiu, Dongbin Data driven governing equations approximation using deep neural networks. (English) Zbl 1455.65125 J. Comput. Phys. 395, 620-635 (2019). MSC: 65L99 37C10 37M10 68T07 PDFBibTeX XMLCite \textit{T. Qin} et al., J. Comput. Phys. 395, 620--635 (2019; Zbl 1455.65125) Full Text: DOI arXiv
Wu, Kailiang; Xiu, Dongbin Numerical aspects for approximating governing equations using data. (English) Zbl 1451.65008 J. Comput. Phys. 384, 200-221 (2019). MSC: 65C20 37M10 34A55 34A09 62G08 PDFBibTeX XMLCite \textit{K. Wu} and \textit{D. Xiu}, J. Comput. Phys. 384, 200--221 (2019; Zbl 1451.65008) Full Text: DOI arXiv
Chou, Ching-Shan; Li, Yukun; Xiu, Dongbin Energy conserving Galerkin approximation of two dimensional wave equations with random coefficients. (English) Zbl 1451.65143 J. Comput. Phys. 381, 52-66 (2019). MSC: 65M60 65C30 35R60 PDFBibTeX XMLCite \textit{C.-S. Chou} et al., J. Comput. Phys. 381, 52--66 (2019; Zbl 1451.65143) Full Text: DOI arXiv
Shin, Yeonjong; Wu, Kailiang; Xiu, Dongbin Sequential function approximation with noisy data. (English) Zbl 1415.65035 J. Comput. Phys. 371, 363-381 (2018). MSC: 65D15 65C20 PDFBibTeX XMLCite \textit{Y. Shin} et al., J. Comput. Phys. 371, 363--381 (2018; Zbl 1415.65035) Full Text: DOI
Wu, Kailiang; Xiu, Dongbin Sequential function approximation on arbitrarily distributed point sets. (English) Zbl 1380.65034 J. Comput. Phys. 354, 370-386 (2018). MSC: 65D15 65C20 PDFBibTeX XMLCite \textit{K. Wu} and \textit{D. Xiu}, J. Comput. Phys. 354, 370--386 (2018; Zbl 1380.65034) Full Text: DOI
Wu, Kailiang; Tang, Huazhong; Xiu, Dongbin A stochastic Galerkin method for first-order quasilinear hyperbolic systems with uncertainty. (English) Zbl 1380.65315 J. Comput. Phys. 345, 224-244 (2017). MSC: 65M75 65M60 35L40 PDFBibTeX XMLCite \textit{K. Wu} et al., J. Comput. Phys. 345, 224--244 (2017; Zbl 1380.65315) Full Text: DOI arXiv
Zhu, Xueyu; Linebarger, Erin M.; Xiu, Dongbin Multi-fidelity stochastic collocation method for computation of statistical moments. (English) Zbl 1378.65040 J. Comput. Phys. 341, 386-396 (2017). MSC: 65C50 PDFBibTeX XMLCite \textit{X. Zhu} et al., J. Comput. Phys. 341, 386--396 (2017; Zbl 1378.65040) Full Text: DOI
Shin, Yeonjong; Xiu, Dongbin On a near optimal sampling strategy for least squares polynomial regression. (English) Zbl 1384.62249 J. Comput. Phys. 326, 931-946 (2016). MSC: 62J05 65D15 41A10 PDFBibTeX XMLCite \textit{Y. Shin} and \textit{D. Xiu}, J. Comput. Phys. 326, 931--946 (2016; Zbl 1384.62249) Full Text: DOI
He, Yanyan; Xiu, Dongbin Numerical strategy for model correction using physical constraints. (English) Zbl 1349.65202 J. Comput. Phys. 313, 617-634 (2016). MSC: 65K99 62J02 PDFBibTeX XMLCite \textit{Y. He} and \textit{D. Xiu}, J. Comput. Phys. 313, 617--634 (2016; Zbl 1349.65202) Full Text: DOI
Jin, Shi; Xiu, Dongbin; Zhu, Xueyu Asymptotic-preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings. (English) Zbl 1352.65407 J. Comput. Phys. 289, 35-52 (2015). MSC: 65M75 PDFBibTeX XMLCite \textit{S. Jin} et al., J. Comput. Phys. 289, 35--52 (2015; Zbl 1352.65407) Full Text: DOI
Zhou, Tao; Narayan, Akil; Xiu, Dongbin Weighted discrete least-squares polynomial approximation using randomized quadratures. (English) Zbl 1349.60086 J. Comput. Phys. 298, 787-800 (2015). MSC: 60G99 41A10 65C20 65D15 65D30 PDFBibTeX XMLCite \textit{T. Zhou} et al., J. Comput. Phys. 298, 787--800 (2015; Zbl 1349.60086) Full Text: DOI
Jakeman, John D.; Narayan, Akil; Xiu, Dongbin Minimal multi-element stochastic collocation for uncertainty quantification of discontinuous functions. (English) Zbl 1311.65158 J. Comput. Phys. 242, 790-808 (2013). MSC: 65N75 65C50 PDFBibTeX XMLCite \textit{J. D. Jakeman} et al., J. Comput. Phys. 242, 790--808 (2013; Zbl 1311.65158) Full Text: DOI
Chen, Xiaoxiao; Park, Eun-Jae; Xiu, Dongbin A flexible numerical approach for quantification of epistemic uncertainty. (English) Zbl 1305.65013 J. Comput. Phys. 240, 211-224 (2013). MSC: 65C30 68T37 PDFBibTeX XMLCite \textit{X. Chen} et al., J. Comput. Phys. 240, 211--224 (2013; Zbl 1305.65013) Full Text: DOI
Narayan, Akil; Marzouk, Youssef; Xiu, Dongbin Sequential data assimilation with multiple models. (English) Zbl 1284.62801 J. Comput. Phys. 231, No. 19, 6401-6418 (2012). MSC: 62P35 60G25 62M20 PDFBibTeX XMLCite \textit{A. Narayan} et al., J. Comput. Phys. 231, No. 19, 6401--6418 (2012; Zbl 1284.62801) Full Text: DOI
Li, Jing; Li, Jinglai; Xiu, Dongbin An efficient surrogate-based method for computing rare failure probability. (English) Zbl 1370.65005 J. Comput. Phys. 230, No. 24, 8683-8697 (2011). MSC: 65C50 65C20 62G32 62N05 PDFBibTeX XMLCite \textit{J. Li} et al., J. Comput. Phys. 230, No. 24, 8683--8697 (2011; Zbl 1370.65005) Full Text: DOI
Jakeman, John D.; Archibald, Richard; Xiu, Dongbin Characterization of discontinuities in high-dimensional stochastic problems on adaptive sparse grids. (English) Zbl 1218.65010 J. Comput. Phys. 230, No. 10, 3977-3997 (2011). MSC: 65C30 60H15 60H35 35R60 PDFBibTeX XMLCite \textit{J. D. Jakeman} et al., J. Comput. Phys. 230, No. 10, 3977--3997 (2011; Zbl 1218.65010) Full Text: DOI
Li, Jing; Xiu, Dongbin Evaluation of failure probability via surrogate models. (English) Zbl 1204.65010 J. Comput. Phys. 229, No. 23, 8966-8980 (2010). MSC: 65C50 65C05 60H10 65C30 60H35 60H15 PDFBibTeX XMLCite \textit{J. Li} and \textit{D. Xiu}, J. Comput. Phys. 229, No. 23, 8966--8980 (2010; Zbl 1204.65010) Full Text: DOI
Jakeman, John; Eldred, Michael; Xiu, Dongbin Numerical approach for quantification of epistemic uncertainty. (English) Zbl 1204.65008 J. Comput. Phys. 229, No. 12, 4648-4663 (2010). Reviewer: R. E. Maiboroda (Kyïv) MSC: 65C30 60H15 35R60 65M60 35K05 PDFBibTeX XMLCite \textit{J. Jakeman} et al., J. Comput. Phys. 229, No. 12, 4648--4663 (2010; Zbl 1204.65008) Full Text: DOI Link
Li, Jia; Xiu, Dongbin A generalized polynomial chaos based ensemble Kalman filter with high accuracy. (English) Zbl 1280.93084 J. Comput. Phys. 228, No. 15, 5454-5469 (2009). MSC: 93E11 60G99 62M20 65P20 PDFBibTeX XMLCite \textit{J. Li} and \textit{D. Xiu}, J. Comput. Phys. 228, No. 15, 5454--5469 (2009; Zbl 1280.93084) Full Text: DOI
Xiu, Dongbin; Shen, Jie Efficient stochastic Galerkin methods for random diffusion equations. (English) Zbl 1161.65008 J. Comput. Phys. 228, No. 2, 266-281 (2009). MSC: 65C30 60H15 60H35 35R60 35K05 65M60 65M12 PDFBibTeX XMLCite \textit{D. Xiu} and \textit{J. Shen}, J. Comput. Phys. 228, No. 2, 266--281 (2009; Zbl 1161.65008) Full Text: DOI
Archibald, Rick; Gelb, Anne; Saxena, Rishu; Xiu, Dongbin Discontinuity detection in multivariate space for stochastic simulations. (English) Zbl 1161.65307 J. Comput. Phys. 228, No. 7, 2676-2689 (2009). MSC: 65C30 65C20 PDFBibTeX XMLCite \textit{R. Archibald} et al., J. Comput. Phys. 228, No. 7, 2676--2689 (2009; Zbl 1161.65307) Full Text: DOI
Xiu, Dongbin; Sherwin, Spencer J. Parametric uncertainty analysis of pulse wave propagation in a model of a human arterial network. (English) Zbl 1121.92023 J. Comput. Phys. 226, No. 2, 1385-1407 (2007). MSC: 92C35 60H30 92C50 PDFBibTeX XMLCite \textit{D. Xiu} and \textit{S. J. Sherwin}, J. Comput. Phys. 226, No. 2, 1385--1407 (2007; Zbl 1121.92023) Full Text: DOI
Tartakovsky, Daniel M.; Xiu, Dongbin Stochastic analysis of transport in tubes with rough walls. (English) Zbl 1146.76651 J. Comput. Phys. 217, No. 1, 248-259 (2006). MSC: 76R50 76D07 76M35 PDFBibTeX XMLCite \textit{D. M. Tartakovsky} and \textit{D. Xiu}, J. Comput. Phys. 217, No. 1, 248--259 (2006; Zbl 1146.76651) Full Text: DOI
Sirisup, Sirod; Karniadakis, George Em; Xiu, Dongbin; Kevrekidis, Ioannis G. Equation-free/Galerkin-free POD-assisted computation of incompressible flows. (English) Zbl 1213.76146 J. Comput. Phys. 207, No. 2, 568-587 (2005). MSC: 76M25 37N10 65M99 PDFBibTeX XMLCite \textit{S. Sirisup} et al., J. Comput. Phys. 207, No. 2, 568--587 (2005; Zbl 1213.76146) Full Text: DOI
Xiu, Dongbin; Karniadakis, George Em Modeling uncertainty in flow simulations via generalized polynomial chaos. (English) Zbl 1047.76111 J. Comput. Phys. 187, No. 1, 137-167 (2003). MSC: 76M35 76D05 PDFBibTeX XMLCite \textit{D. Xiu} and \textit{G. E. Karniadakis}, J. Comput. Phys. 187, No. 1, 137--167 (2003; Zbl 1047.76111) Full Text: DOI
Xiu, Dongbin; Karniadakis, George Em A semi-Lagrangian high-order method for Navier-Stokes equations. (English) Zbl 1028.76026 J. Comput. Phys. 172, No. 2, 658-684 (2001). MSC: 76M10 76M22 76D05 PDFBibTeX XMLCite \textit{D. Xiu} and \textit{G. E. Karniadakis}, J. Comput. Phys. 172, No. 2, 658--684 (2001; Zbl 1028.76026) Full Text: DOI