Linka, Kevin; Schäfer, Amelie; Meng, Xuhui; Zou, Zongren; Karniadakis, George Em; Kuhl, Ellen Bayesian physics informed neural networks for real-world nonlinear dynamical systems. (English) Zbl 1507.68246 Comput. Methods Appl. Mech. Eng. 402, Article ID 115346, 22 p. (2022). MSC: 68T05 37M10 62F15 PDFBibTeX XMLCite \textit{K. Linka} et al., Comput. Methods Appl. Mech. Eng. 402, Article ID 115346, 22 p. (2022; Zbl 1507.68246) Full Text: DOI arXiv
Yin, Minglang; Zhang, Enrui; Yu, Yue; Karniadakis, George Em Interfacing finite elements with deep neural operators for fast multiscale modeling of mechanics problems. (English) Zbl 1507.74523 Comput. Methods Appl. Mech. Eng. 402, Article ID 115027, 26 p. (2022). MSC: 74S05 68T07 PDFBibTeX XMLCite \textit{M. Yin} et al., Comput. Methods Appl. Mech. Eng. 402, Article ID 115027, 26 p. (2022; Zbl 1507.74523) Full Text: DOI arXiv
Lu, Lu; Meng, Xuhui; Mao, Zhiping; Karniadakis, George Em DeepXDE: a deep learning library for solving differential equations. (English) Zbl 1459.65002 SIAM Rev. 63, No. 1, 208-228 (2021). MSC: 65-01 65-04 68T07 65L99 65M99 65N99 PDFBibTeX XMLCite \textit{L. Lu} et al., SIAM Rev. 63, No. 1, 208--228 (2021; Zbl 1459.65002) Full Text: DOI arXiv
Yu, Yue; Bargos, Fabiano F.; You, Huaiqian; Parks, Michael L.; Bittencourt, Marco L.; Karniadakis, George E. A partitioned coupling framework for peridynamics and classical theory: analysis and simulations. (English) Zbl 1440.74045 Comput. Methods Appl. Mech. Eng. 340, 905-931 (2018). MSC: 74A45 74S05 65N30 PDFBibTeX XMLCite \textit{Y. Yu} et al., Comput. Methods Appl. Mech. Eng. 340, 905--931 (2018; Zbl 1440.74045) Full Text: DOI
Babaee, Hessam; Choi, Minseok; Sapsis, Themistoklis P.; Karniadakis, George Em A robust bi-orthogonal/dynamically-orthogonal method using the covariance pseudo-inverse with application to stochastic flow problems. (English) Zbl 1380.76093 J. Comput. Phys. 344, 303-319 (2017). MSC: 76M25 35A35 35R60 76D06 35Q35 PDFBibTeX XMLCite \textit{H. Babaee} et al., J. Comput. Phys. 344, 303--319 (2017; Zbl 1380.76093) Full Text: DOI
Cho, H.; Yang, X.; Venturi, D.; Karniadakis, G. E. Algorithms for propagating uncertainty across heterogeneous domains. (English) Zbl 1347.60093 SIAM J. Sci. Comput. 37, No. 6, A3030-A3054 (2015). MSC: 60H35 60H15 65C30 60G60 PDFBibTeX XMLCite \textit{H. Cho} et al., SIAM J. Sci. Comput. 37, No. 6, A3030--A3054 (2015; Zbl 1347.60093) Full Text: DOI Link
Yu, Yue; Bittencourt, Marco L.; Karniadakis, George Em. A semi-local spectral/hp element solver for linear elasticity problems. (English) Zbl 1352.74476 Int. J. Numer. Methods Eng. 100, No. 5, 347-373 (2014). MSC: 74S25 74L15 76Z05 65M70 65M60 74B05 74S05 PDFBibTeX XMLCite \textit{Y. Yu} et al., Int. J. Numer. Methods Eng. 100, No. 5, 347--373 (2014; Zbl 1352.74476) Full Text: DOI
Yu, Yue; Baek, Hyoungsu; Bittencourt, Marco L.; Karniadakis, George Em Mixed spectral/\(hp\) element formulation for nonlinear elasticity. (English) Zbl 1243.74012 Comput. Methods Appl. Mech. Eng. 213-216, 42-57 (2012). MSC: 74B20 74S05 PDFBibTeX XMLCite \textit{Y. Yu} et al., Comput. Methods Appl. Mech. Eng. 213--216, 42--57 (2012; Zbl 1243.74012) Full Text: DOI
Foo, Jasmine; Karniadakis, George Em Multi-element probabilistic collocation method in high dimensions. (English) Zbl 1181.65014 J. Comput. Phys. 229, No. 5, 1536-1557 (2010). MSC: 65C30 PDFBibTeX XMLCite \textit{J. Foo} and \textit{G. E. Karniadakis}, J. Comput. Phys. 229, No. 5, 1536--1557 (2010; Zbl 1181.65014) Full Text: DOI
Wan, Xiaoliang; Karniadakis, George Em Solving elliptic problems with non-Gaussian spatially-dependent random coefficients. (English) Zbl 1227.65014 Comput. Methods Appl. Mech. Eng. 198, No. 21-26, 1985-1995 (2009). MSC: 65C30 PDFBibTeX XMLCite \textit{X. Wan} and \textit{G. E. Karniadakis}, Comput. Methods Appl. Mech. Eng. 198, No. 21--26, 1985--1995 (2009; Zbl 1227.65014) Full Text: DOI
Foo, Jasmine; Wan, Xiaoliang; Karniadakis, George Em The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications. (English) Zbl 1153.65008 J. Comput. Phys. 227, No. 22, 9572-9595 (2008). MSC: 65C30 60H15 60H35 35R60 65N35 65N12 65N15 65C05 PDFBibTeX XMLCite \textit{J. Foo} et al., J. Comput. Phys. 227, No. 22, 9572--9595 (2008; Zbl 1153.65008) Full Text: DOI
Wan, Xiaoliang; Karniadakis, George Em Long-term behavior of polynomial chaos in stochastic flow simulations. (English) Zbl 1123.76058 Comput. Methods Appl. Mech. Eng. 195, No. 41-43, 5582-5596 (2006). MSC: 76M35 PDFBibTeX XMLCite \textit{X. Wan} and \textit{G. E. Karniadakis}, Comput. Methods Appl. Mech. Eng. 195, No. 41--43, 5582--5596 (2006; Zbl 1123.76058) Full Text: DOI
Lin, Guang; Su, Chau-Hsing; Karniadakis, G. E. Predicting shock dynamics in the presence of uncertainties. (English) Zbl 1146.76613 J. Comput. Phys. 217, No. 1, 260-276 (2006). MSC: 76L05 65P20 76J20 76M10 76M35 PDFBibTeX XMLCite \textit{G. Lin} et al., J. Comput. Phys. 217, No. 1, 260--276 (2006; Zbl 1146.76613) Full Text: DOI
Wan, Xiaoliang; Karniadakis, George Em Beyond Wiener-Askey expansions: handling arbitrary PDFs. (English) Zbl 1102.65006 J. Sci. Comput. 27, No. 1-3, 455-464 (2006). MSC: 65C30 60H15 60H35 PDFBibTeX XMLCite \textit{X. Wan} and \textit{G. E. Karniadakis}, J. Sci. Comput. 27, No. 1--3, 455--464 (2006; Zbl 1102.65006) Full Text: DOI
Wan, Xiaoliang; Karniadakis, George Em An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. (English) Zbl 1078.65008 J. Comput. Phys. 209, No. 2, 617-642 (2005). Reviewer: Melvin D. Lax (Long Beach) MSC: 65C30 60H10 34F05 60H15 35R60 60H35 PDFBibTeX XMLCite \textit{X. Wan} and \textit{G. E. Karniadakis}, J. Comput. Phys. 209, No. 2, 617--642 (2005; Zbl 1078.65008) Full Text: DOI
Kirby, Robert M.; Karniadakis, George Em Selecting the numerical flux in discontinuous Galerkin methods for diffusion problems. (English) Zbl 1104.76060 J. Sci. Comput. 22-23, 385-411 (2005). Reviewer: Titus Petrila (Cluj-Napoca) MSC: 76M10 76R50 PDFBibTeX XMLCite \textit{R. M. Kirby} and \textit{G. E. Karniadakis}, J. Sci. Comput. 22--23, 385--411 (2005; Zbl 1104.76060) 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 Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. (English) Zbl 1016.65001 Comput. Methods Appl. Mech. Eng. 191, No. 43, 4927-4948 (2002). MSC: 65C30 35R60 35J05 65C05 60H15 65N30 65F10 60H35 PDFBibTeX XMLCite \textit{D. Xiu} and \textit{G. E. Karniadakis}, Comput. Methods Appl. Mech. Eng. 191, No. 43, 4927--4948 (2002; Zbl 1016.65001) Full Text: DOI
Evangelinos, Constantinos; Sherwin, Spencer J.; Karniadakis, George Em Parallel DNS algorithm on unstructured grids. (English) Zbl 0957.76021 Comput. Methods Appl. Mech. Eng. 184, No. 2-4, 401-425 (2000). MSC: 76F65 76M22 65Y05 76M10 PDFBibTeX XMLCite \textit{C. Evangelinos} et al., Comput. Methods Appl. Mech. Eng. 184, No. 2--4, 401--425 (2000; Zbl 0957.76021) Full Text: DOI
Lomtev, I.; Kirby, R. M.; Karniadakis, G. E. A discontinuous Galerkin ALE method for compressible viscous flows in moving domains. (English) Zbl 0956.76046 J. Comput. Phys. 155, No. 1, 128-159 (1999). MSC: 76M10 76N15 PDFBibTeX XMLCite \textit{I. Lomtev} et al., J. Comput. Phys. 155, No. 1, 128--159 (1999; Zbl 0956.76046) Full Text: DOI Link
Warburton, T. C.; Karniadakis, G. E. A discontinuous Galerkin method for the viscous MHD equations. (English) Zbl 0954.76051 J. Comput. Phys. 152, No. 2, 608-641 (1999). MSC: 76M10 76W05 76X05 PDFBibTeX XMLCite \textit{T. C. Warburton} and \textit{G. E. Karniadakis}, J. Comput. Phys. 152, No. 2, 608--641 (1999; Zbl 0954.76051) Full Text: DOI Link
Sherwin, Spencer J.; Karniadakis, George Em. A new triangular and tetrahedral basis for high-order \((hp)\) finite element methods. (English) Zbl 0837.73075 Int. J. Numer. Methods Eng. 38, No. 22, 3775-3802 (1995). MSC: 74S05 65N30 PDFBibTeX XMLCite \textit{S. J. Sherwin} and \textit{G. Em. Karniadakis}, Int. J. Numer. Methods Eng. 38, No. 22, 3775--3802 (1995; Zbl 0837.73075) Full Text: DOI
Sherwin, S. J.; Karniadakis, G. E. A triangular spectral element method; applications to the incompressible Navier-Stokes equations. (English) Zbl 1075.76621 Comput. Methods Appl. Mech. Eng. 123, No. 1-4, 189-229 (1995). MSC: 76M22 76M10 76D05 PDFBibTeX XMLCite \textit{S. J. Sherwin} and \textit{G. E. Karniadakis}, Comput. Methods Appl. Mech. Eng. 123, No. 1--4, 189--229 (1995; Zbl 1075.76621) Full Text: DOI