He, Yanchen; Schwab, Christoph Analytic regularity and solution approximation for a semilinear elliptic partial differential equation in a polygon. (English) Zbl 07795707 Calcolo 61, No. 1, Paper No. 11, 23 p. (2024). MSC: 35J61 35A20 35B65 65N30 PDFBibTeX XMLCite \textit{Y. He} and \textit{C. Schwab}, Calcolo 61, No. 1, Paper No. 11, 23 p. (2024; Zbl 07795707) Full Text: DOI
Opschoor, Joost A. A.; Schwab, Christoph Exponential Expressivity of ReLU\(^k\) Neural Networks on Gevrey Classes with Point Singularities. arXiv:2403.02035 Preprint, arXiv:2403.02035 [math.NA] (2024). MSC: 65N30 41A25 BibTeX Cite \textit{J. A. A. Opschoor} and \textit{C. Schwab}, ``Exponential Expressivity of ReLU$^k$ Neural Networks on Gevrey Classes with Point Singularities'', Preprint, arXiv:2403.02035 [math.NA] (2024) Full Text: arXiv OA License
Marcati, Carlo; Opschoor, Joost A. A.; Petersen, Philipp C.; Schwab, Christoph Exponential ReLU neural network approximation rates for point and edge singularities. (English) Zbl 1519.35253 Found. Comput. Math. 23, No. 3, 1043-1127 (2023). MSC: 35Q40 41A25 41A46 65N30 68T07 35B65 35P30 35J15 PDFBibTeX XMLCite \textit{C. Marcati} et al., Found. Comput. Math. 23, No. 3, 1043--1127 (2023; Zbl 1519.35253) Full Text: DOI arXiv
Opschoor, Joost A. A.; Schwab, Christoph Deep ReLU networks and high-order finite element methods II: Chebyshev emulation. arXiv:2310.07261 Preprint, arXiv:2310.07261 [math.NA] (2023). MSC: 65N30 41A05 41A10 41A25 41A50 BibTeX Cite \textit{J. A. A. Opschoor} and \textit{C. Schwab}, ``Deep ReLU networks and high-order finite element methods II: Chebyshev emulation'', Preprint, arXiv:2310.07261 [math.NA] (2023) Full Text: arXiv OA License
Kazeev, Vladimir; Oseledets, Ivan; Rakhuba, Maxim V.; Schwab, Christoph Quantized tensor FEM for multiscale problems: diffusion problems in two and three dimensions. (English) Zbl 1496.65220 Multiscale Model. Simul. 20, No. 3, 893-935 (2022). MSC: 65N30 65N50 65N15 65N22 35B27 15A69 PDFBibTeX XMLCite \textit{V. Kazeev} et al., Multiscale Model. Simul. 20, No. 3, 893--935 (2022; Zbl 1496.65220) Full Text: DOI arXiv
Marcati, C.; Rakhuba, M.; Schwab, Ch. Tensor rank bounds for point singularities in \(\mathbb{R}^3\). (English) Zbl 1487.35019 Adv. Comput. Math. 48, No. 3, Paper No. 18, 57 p. (2022). MSC: 35A35 15A69 35J15 41A25 41A46 65N30 PDFBibTeX XMLCite \textit{C. Marcati} et al., Adv. Comput. Math. 48, No. 3, Paper No. 18, 57 p. (2022; Zbl 1487.35019) Full Text: DOI arXiv
Hoang, Viet Ha; Quek, Jia Hao; Schwab, Christoph Analysis of a multilevel Markov chain Monte Carlo finite element method for Bayesian inversion of log-normal diffusions. (English) Zbl 1491.65010 Inverse Probl. 36, No. 3, Article ID 035021, 46 p. (2020). MSC: 65C05 65N30 65N21 35R30 62F15 60H15 65N12 PDFBibTeX XMLCite \textit{V. H. Hoang} et al., Inverse Probl. 36, No. 3, Article ID 035021, 46 p. (2020; Zbl 1491.65010) Full Text: DOI
Opschoor, Joost A. A.; Petersen, Philipp C.; Schwab, Christoph Deep ReLU networks and high-order finite element methods. (English) Zbl 1452.65354 Anal. Appl., Singap. 18, No. 5, 715-770 (2020). MSC: 65N30 65D07 65N12 41A25 41A46 35B65 35R02 68T07 92B20 PDFBibTeX XMLCite \textit{J. A. A. Opschoor} et al., Anal. Appl., Singap. 18, No. 5, 715--770 (2020; Zbl 1452.65354) Full Text: DOI
Zech, Jakob; Dũng, Dinh; Schwab, Christoph Multilevel approximation of parametric and stochastic PDES. (English) Zbl 1427.65386 Math. Models Methods Appl. Sci. 29, No. 9, 1753-1817 (2019). MSC: 65N30 35A20 35B30 32D05 65D30 35R60 65N15 65N12 65F15 PDFBibTeX XMLCite \textit{J. Zech} et al., Math. Models Methods Appl. Sci. 29, No. 9, 1753--1817 (2019; Zbl 1427.65386) Full Text: DOI
Devaud, Denis; Schwab, Christoph Space-time \(hp\)-approximation of parabolic equations. (English) Zbl 1404.65118 Calcolo 55, No. 3, Paper No. 35, 23 p. (2018). MSC: 65M12 65M15 65M60 PDFBibTeX XMLCite \textit{D. Devaud} and \textit{C. Schwab}, Calcolo 55, No. 3, Paper No. 35, 23 p. (2018; Zbl 1404.65118) Full Text: DOI
Dahmen, Wolfgang; Kutyniok, Gitta; Lim, Wang-Q; Schwab, Christoph; Welper, Gerrit Adaptive anisotropic Petrov-Galerkin methods for first order transport equations. (English) Zbl 1435.65197 J. Comput. Appl. Math. 340, 191-220 (2018). MSC: 65N30 65N12 65N15 PDFBibTeX XMLCite \textit{W. Dahmen} et al., J. Comput. Appl. Math. 340, 191--220 (2018; Zbl 1435.65197) Full Text: DOI arXiv
Kazeev, Vladimir; Schwab, Christoph Quantized tensor-structured finite elements for second-order elliptic PDEs in two dimensions. (English) Zbl 1382.65409 Numer. Math. 138, No. 1, 133-190 (2018). Reviewer: Calin Ioan Gheorghiu (Cluj-Napoca) MSC: 65N30 15A69 35J25 65N12 PDFBibTeX XMLCite \textit{V. Kazeev} and \textit{C. Schwab}, Numer. Math. 138, No. 1, 133--190 (2018; Zbl 1382.65409) Full Text: DOI Link
Bachmayr, Markus; Cohen, Albert; Dũng, Dinh; Schwab, Christoph Fully discrete approximation of parametric and stochastic elliptic PDEs. (English) Zbl 1377.65005 SIAM J. Numer. Anal. 55, No. 5, 2151-2186 (2017). MSC: 65C30 60H15 35R60 60H35 65N30 65T60 65N12 PDFBibTeX XMLCite \textit{M. Bachmayr} et al., SIAM J. Numer. Anal. 55, No. 5, 2151--2186 (2017; Zbl 1377.65005) Full Text: DOI arXiv
Dölz, J.; Harbrecht, H.; Schwab, Ch. Covariance regularity and \(\mathcal {H}\)-matrix approximation for rough random fields. (English) Zbl 1362.65124 Numer. Math. 135, No. 4, 1045-1071 (2017). Reviewer: Abdallah Bradji (Annaba) MSC: 65N30 35S15 60G10 58J32 PDFBibTeX XMLCite \textit{J. Dölz} et al., Numer. Math. 135, No. 4, 1045--1071 (2017; Zbl 1362.65124) Full Text: DOI
Kunoth, Angela; Schwab, Christoph Sparse adaptive tensor Galerkin approximations of stochastic PDE-constrained control problems. (English) Zbl 1398.65252 SIAM/ASA J. Uncertain. Quantif. 4, 1034-1059 (2016). MSC: 65M60 49N10 65K10 65M75 PDFBibTeX XMLCite \textit{A. Kunoth} and \textit{C. Schwab}, SIAM/ASA J. Uncertain. Quantif. 4, 1034--1059 (2016; Zbl 1398.65252) Full Text: DOI
Lang, Annika; Schwab, Christoph Isotropic Gaussian random fields on the sphere: regularity, fast simulation and stochastic partial differential equations. (English) Zbl 1328.60126 Ann. Appl. Probab. 25, No. 6, 3047-3094 (2015). MSC: 60G60 60G15 60G17 60H15 60H35 33C55 41A25 65C30 65T50 65N30 PDFBibTeX XMLCite \textit{A. Lang} and \textit{C. Schwab}, Ann. Appl. Probab. 25, No. 6, 3047--3094 (2015; Zbl 1328.60126) Full Text: DOI arXiv Euclid
Hiptmair, Ralf; Moiola, Andrea; Perugia, Ilaria; Schwab, Christoph Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz \(hp\)-dGFEM. (English) Zbl 1295.31004 ESAIM, Math. Model. Numer. Anal. 48, No. 3, 727-752 (2014). Reviewer: H. P. Dikshit (Bhopal) MSC: 31A05 30E10 65N30 PDFBibTeX XMLCite \textit{R. Hiptmair} et al., ESAIM, Math. Model. Numer. Anal. 48, No. 3, 727--752 (2014; Zbl 1295.31004) Full Text: DOI
Hoang, Viet Ha; Schwab, Christoph \(N\)-term Wiener chaos approximation rates for elliptic PDEs with lognormal Gaussian random inputs. (English) Zbl 1294.65010 Math. Models Methods Appl. Sci. 24, No. 4, 797-826 (2014). Reviewer: Grigori N. Milstein (Yekaterinburg) MSC: 65C30 60H15 60H35 35R60 35J25 65N30 60G60 PDFBibTeX XMLCite \textit{V. H. Hoang} and \textit{C. Schwab}, Math. Models Methods Appl. Sci. 24, No. 4, 797--826 (2014; Zbl 1294.65010) Full Text: DOI
Chernov, Alexey; Schwab, Christoph First-order \(k\)-th moment finite element analysis of nonlinear operator equations with stochastic data. (English) Zbl 1281.65008 Math. Comput. 82, No. 284, 1859-1888 (2013). Reviewer: Gong Guanglu (Beijing) MSC: 65C30 65J15 47J25 60H25 60H35 65N30 PDFBibTeX XMLCite \textit{A. Chernov} and \textit{C. Schwab}, Math. Comput. 82, No. 284, 1859--1888 (2013; Zbl 1281.65008) Full Text: DOI
Hansen, Markus; Schwab, Christoph Analytic regularity and nonlinear approximation of a class of parametric semilinear elliptic PDEs. (English) Zbl 1273.35134 Math. Nachr. 286, No. 8-9, 832-860 (2013). MSC: 35J61 35B30 65N30 41A58 PDFBibTeX XMLCite \textit{M. Hansen} and \textit{C. Schwab}, Math. Nachr. 286, No. 8--9, 832--860 (2013; Zbl 1273.35134) Full Text: DOI Link
Khoromskij, Boris N.; Schwab, Christoph Tensor-structured Galerkin approximation of parametric and stochastic elliptic PDEs. (English) Zbl 1243.65009 SIAM J. Sci. Comput. 33, No. 1, 364-385 (2011). Reviewer: Kevin Burrage (Brisbane) MSC: 65C30 65F30 65F50 65N35 60H15 60H35 35R60 65F08 PDFBibTeX XMLCite \textit{B. N. Khoromskij} and \textit{C. Schwab}, SIAM J. Sci. Comput. 33, No. 1, 364--385 (2011; Zbl 1243.65009) Full Text: DOI Link
Dijkema, Tammo Jan; Schwab, Christoph; Stevenson, Rob An adaptive wavelet method for solving high-dimensional elliptic PDEs. (English) Zbl 1205.65313 Constr. Approx. 30, No. 3, 423-455 (2009). Reviewer: Gerlind Plonka-Hoch (Göttingen) MSC: 65N30 46B28 65T60 35J25 65F35 65Y20 65N12 PDFBibTeX XMLCite \textit{T. J. Dijkema} et al., Constr. Approx. 30, No. 3, 423--455 (2009; Zbl 1205.65313) Full Text: DOI
Todor, Radu Alexandru; Schwab, Christoph Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. (English) Zbl 1120.65004 IMA J. Numer. Anal. 27, No. 2, 232-261 (2007). Reviewer: Grigori N. Milstein (Ekaterinburg) MSC: 65C30 60H15 35R60 60H35 65N30 65N35 PDFBibTeX XMLCite \textit{R. A. Todor} and \textit{C. Schwab}, IMA J. Numer. Anal. 27, No. 2, 232--261 (2007; Zbl 1120.65004) Full Text: DOI Link
von Petersdorff, Tobias; Schwab, Christoph Sparse finite element methods for operator equations with stochastic data. (English) Zbl 1164.65300 Appl. Math., Praha 51, No. 2, 145-180 (2006). MSC: 65C05 65N30 60H25 60H15 65C30 65T60 65F10 65F35 PDFBibTeX XMLCite \textit{T. von Petersdorff} and \textit{C. Schwab}, Appl. Math., Praha 51, No. 2, 145--180 (2006; Zbl 1164.65300) Full Text: DOI EuDML Link
Schwab, Christoph; Todor, Radu Alexandru Karhunen-Loève approximation of random fields by generalized fast multipole methods. (English) Zbl 1104.65008 J. Comput. Phys. 217, No. 1, 100-122 (2006). Reviewer: Silvia Curteanu (Iaşi) MSC: 65C50 65M60 60G60 65C30 60H15 65N25 PDFBibTeX XMLCite \textit{C. Schwab} and \textit{R. A. Todor}, J. Comput. Phys. 217, No. 1, 100--122 (2006; Zbl 1104.65008) Full Text: DOI
Matache, Ana-Maria; Schwab, Christoph Finite dimensional approximations for elliptic problems with rapidly oscillating coefficients. (English) Zbl 1165.35307 Antonić, Nenad (ed.) et al., Multiscale problems in science and technology. Challenges to mathematical analysis and perspectives. Proceedings of the conference on multiscale problems in science and technology, Dubrovnik, Croatia, September 3–9, 2000. Berlin: Springer (ISBN 3-540-43584-0/pbk). 203-242 (2002). Reviewer: Dan Polisevski (Bucureşti) MSC: 35A35 35J15 65N30 35B27 PDFBibTeX XMLCite \textit{A.-M. Matache} and \textit{C. Schwab}, in: Multiscale problems in science and technology. Challenges to mathematical analysis and perspectives. Proceedings of the conference on multiscale problems in science and technology, Dubrovnik, Croatia, September 3--9, 2000. Berlin: Springer. 203--242 (2002; Zbl 1165.35307)
Schwab, Christoph \(hp\)-FEM for fluid flow simulation. (English) Zbl 0937.76037 Barth, Timothy J. (ed.) et al., High-order methods for computational physics. Berlin: Springer. Lect. Notes Comput. Sci. Eng. 9, 325-438 (1999). MSC: 76M10 76R50 65M60 76D10 PDFBibTeX XMLCite \textit{C. Schwab}, Lect. Notes Comput. Sci. Eng. 9, 325--438 (1999; Zbl 0937.76037)
Schwab, Christoph; Suri, Manil; Xenophontos, C. The \(hp\) finite element method for problems in mechanics with boundary layers. (English) Zbl 0959.74073 Comput. Methods Appl. Mech. Eng. 157, No. 3-4, 311-333 (1998). MSC: 74S05 74K20 74K25 PDFBibTeX XMLCite \textit{C. Schwab} et al., Comput. Methods Appl. Mech. Eng. 157, No. 3--4, 311--333 (1998; Zbl 0959.74073) Full Text: DOI
Gerdes, Klaus; Schwab, Christoph Hierarchic models of Helmholtz problems on thin domains. (English) Zbl 0957.74026 Math. Models Methods Appl. Sci. 8, No. 1, 1-35 (1998). MSC: 74K20 74G10 74S05 35J05 PDFBibTeX XMLCite \textit{K. Gerdes} and \textit{C. Schwab}, Math. Models Methods Appl. Sci. 8, No. 1, 1--35 (1998; Zbl 0957.74026) Full Text: DOI