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doi: 10.3934/mcrf.2021004

Model reduction for fractional elliptic problems using Kato's formula

1. 

New York University, Warren Weaver Hall, 251 Mercer Street, New York, NY 10012, USA

2. 

George Mason University, 4400 University Drive, MS: 3F2, Fairfax, VA 22030, USA

3. 

University of Massachusetts Dartmouth, 285 Old Westport Road, North Dartmouth, MA 02747, USA

4. 

University of Utah, 155 South 1400 East, Salt Lake City, UT 84112-0090, USA

* Corresponding author: Akil Narayan

Received  August 2020 Revised  December 2020 Published  March 2021

We propose a novel numerical algorithm utilizing model reduction for computing solutions to stationary partial differential equations involving the spectral fractional Laplacian. Our approach utilizes a known characterization of the solution in terms of an integral of solutions to local (classical) elliptic problems. We reformulate this integral into an expression whose continuous and discrete formulations are stable; the discrete formulations are stable independent of all discretization parameters. We subsequently apply the reduced basis method to accomplish model order reduction for the integrand. Our choice of quadrature in discretization of the integral is a global Gaussian quadrature rule that we observe is more efficient than previously proposed quadrature rules. Finally, the model reduction approach enables one to compute solutions to multi-query fractional Laplace problems with orders of magnitude less cost than a traditional solver.

Citation: Huy Dinh, Harbir Antil, Yanlai Chen, Elena Cherkaev, Akil Narayan. Model reduction for fractional elliptic problems using Kato's formula. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021004
References:
[1]

M. Ainsworth and C. Glusa, Hybrid finite element-spectral method for the fractional Laplacian: Approximation theory and efficient solver, SIAM Journal on Scientific Computing, 40 (2018), A2383-A2405. doi: 10.1137/17M1144696.  Google Scholar

[2]

H. Antil and S. Bartels, Spectral approximation of fractional PDEs in image processing and phase field modeling, Comput. Methods Appl. Math., 17 (2017), 661-678.  doi: 10.1515/cmam-2017-0039.  Google Scholar

[3]

H. Antil, Y. Chen and A. Narayan, Kolmogorov widths and reduced order modeling for fractional elliptic operators, preprint, (2019). Google Scholar

[4]

H. Antil, Y. Chen and A. Narayan, Reduced basis methods for fractional Laplace equations via extension, SIAM J. Sci. Comput., 41 (2019), A3552-A3575. doi: 10.1137/18M1204802.  Google Scholar

[5]

H. Antil, R. Khatri and M. Warma, External optimal control of nonlocal PDEs, Inverse Problems, 35 (2019), 084003, 35 pp. doi: 10.1088/1361-6420/ab1299.  Google Scholar

[6]

H. Antil and J. Pfefferer, A short Matlab implementation of fractional Poisson equation with nonzero boundary conditions, Technical report, 2017. Google Scholar

[7]

H. AntilJ. Pfefferer and S. Rogovs, Fractional operators with inhomogeneous boundary conditions: Analysis, control, and discretization, Commun. Math. Sci., 16 (2018), 1395-1426.  doi: 10.4310/CMS.2018.v16.n5.a11.  Google Scholar

[8]

P. BinevA. CohenW. DahmenR. DeVoreG. Petrova and P. Wojtaszczyk, Convergence rates for greedy algorithms in reduced basis methods, SIAM J. Math. Anal., 43 (2011), 1457-1472.  doi: 10.1137/100795772.  Google Scholar

[9]

A. Bonito, D. Guignard and A. R. Zhang, Reduced basis approximations of the solutions to spectral fractional diffusion problems, J. Numer. Math., 28 (2020). doi: 10.1515/jnma-2019-0053.  Google Scholar

[10]

A. BonitoW. Lei and J. E. Pasciak, Numerical approximation of the integral fractional Laplacian, Numer. Math., 142 (2019), 235-278.  doi: 10.1007/s00211-019-01025-x.  Google Scholar

[11]

A. Bonito and J. Pasciak, Numerical approximation of fractional powers of elliptic operators, Math. Comp., 84 (2015), 2083-2110.  doi: 10.1090/S0025-5718-2015-02937-8.  Google Scholar

[12]

A. Buhr, C. Engwer, M. Ohlberger and S. Rave, A numerically stable a posteriori error estimator for reduced basis approximations of elliptic equations, preprint, (2014), arXiv: 1407.8005. Google Scholar

[13]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[14]

F. Casenave, Accurate a posteriori error evaluation in the reduced basis method, C. R. Math. Acad. Sci. Paris, 350 (2012), 539-542.  doi: 10.1016/j.crma.2012.05.012.  Google Scholar

[15]

F. CasenaveA. Ern and T. Lelièvre, Accurate and online-efficient evaluation of the a posteriori error bound in the reduced basis method, ESAIM Math. Model. Numer. Anal., 48 (2014), 207-229.  doi: 10.1051/m2an/2013097.  Google Scholar

[16]

Y. ChenJ. Jiang and A. Narayan, A robust error estimator and a residual-free error indicator for reduced basis methods, Comput. Math. Appl., 77 (2019), 1963-1979.  doi: 10.1016/j.camwa.2018.11.032.  Google Scholar

[17]

A. Cohen and R. DeVore, Approximation of high-dimensional parametric PDEs, Acta Numer., 24 (2015), 1-159.  doi: 10.1017/S0962492915000033.  Google Scholar

[18]

T. Danczul and J. Schöberl, A Reduced Basis Method For Fractional Diffusion Operators I, preprint, (2019), arXiv: 1904.05599. Google Scholar

[19]

T. Danczul and J. Schöberl, A Reduced Basis Method For Fractional Diffusion Operators II, preprint, (2020), arXiv: 2005.03574. Google Scholar

[20]

R. DeVoreG. Petrova and P. Wojtaszczyk, Greedy algorithms for reduced bases in Banach spaces, Constr. Approx., 37 (2013), 455-466.  doi: 10.1007/s00365-013-9186-2.  Google Scholar

[21]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[22]

M. E. Farquhar, T. J. Moroney, Q. Yang, I. W. Turner and K. Burrage, Computational modelling of cardiac ischaemia using a variable-order fractional Laplacian, preprint, (2018), arXiv: 1809.07936. Google Scholar

[23]

F. Gesztesy and M. Mitrea, A description of all self-adjoint extensions of the Laplacian and Krein-type resolvent formulas on non-smooth domains, J. Anal. Math., 113 (2011), 53-172.  doi: 10.1007/s11854-011-0002-2.  Google Scholar

[24]

G. Grubb, Regularity of spectral fractional Dirichlet and Neumann problems, Math. Nachr., 289 (2016), 831-844.  doi: 10.1002/mana.201500041.  Google Scholar

[25]

Q. GuanM. GunzburgerC. G. Webster and G. Zhang, Reduced basis methods for nonlocal diffusion problems with random input data, Comput. Methods Appl. Mech. Engrg., 317 (2017), 746-770.  doi: 10.1016/j.cma.2016.12.019.  Google Scholar

[26]

J. S. Hesthaven, G. Rozza and B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations, SpringerBriefs in Mathematics, Springer International Publishing, Cham, 2016. doi: 10.1007/978-3-319-22470-1.  Google Scholar

[27]

M. Ilic, F. Liu, I. Turner and V. Anh, Numerical Approximation of a fractional-in-space diffusion equation, I, Fract. Calc. Appl. Anal., 8 (2005), 323-341. https://eudml.org/doc/11303  Google Scholar

[28]

M. Ilic, F. Liu, I. Turner and V. Anh, Numerical approximation of a fractional-in-space diffusion equation (II) - with nonhomogeneous boundary conditions, Fract. Calc. Appl. Anal., 9 (2006), 333-349. http://www.diogenes.bg/fcaa/  Google Scholar

[29]

T. Kato, Note on fractional powers of linear operators, Proc. Japan Acad., 36 (1960), 94-96.  doi: 10.3792/pja/1195524082.  Google Scholar

[30]

D. KumarJ. Singh and S. Kumar, A fractional model of Navier-Stokes equation arising in unsteady flow of a viscous fluid, Journal of the Association of Arab Universities for Basic and Applied Sciences, 17 (2015), 14-19.  doi: 10.1016/j.jaubas.2014.01.001.  Google Scholar

[31]

M. Kwaśnicki, Ten equivalent definitions of the fractional laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.  Google Scholar

[32]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. 1, Die Grundlehren der mathematischen Wissenschaften, Springer-Verlag, New York-Heidelberg, 1972. https://www.springer.com/gp/book/9783642651632  Google Scholar

[33]

D. MeidnerJ. PfeffererK. Schürholz and B. Vexler, $$\text{hp}$$-finite elements for fractional diffusion, SIAM J. Numer. Anal., 56 (2018), 2345-2374.  doi: 10.1137/17M1135517.  Google Scholar

[34]

S. Molchanov and E. Ostrovskii, Symmetric stable processes as traces of degenerate diffusion processes, Theory of Probability & Its Applications, 14 (1969), 127-130. https://epubs.siam.org/doi/abs/10.1137/1114012  Google Scholar

[35]

R. H. NochettoE. Otárola and A. J. Salgado, A PDE approach to fractional diffusion in general domains: A priori error analysis, Found. Comput. Math., 15 (2015), 733-791.  doi: 10.1007/s10208-014-9208-x.  Google Scholar

[36]

B. N. Parlett, The Symmetric Eigenvalue Problem, Classics in Applied Mathematics, vol. 20, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. doi: 10.1137/1.9781611971163.  Google Scholar

[37] A. T. Patera and G. Rozza, Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations, MIT Press, 2007.   Google Scholar
[38]

P. Perdikaris and G. E. Karniadakis, Fractional-order viscoelasticity in one-dimensional blood flow models, Annals of Biomedical Engineering, 42 (2014), 1012-1023.  doi: 10.1007/s10439-014-0970-3.  Google Scholar

[39]

A. Quarteroni, A. Manzoni and F. Negri, Reduced Basis Methods for Partial Differential Equations, UNITEXT, vol. 92, La Matematica per il 3+2. Springer, Cham, 2016. doi: 10.1007/978-3-319-15431-2.  Google Scholar

[40]

F. Song, C. Xu and G. E. Karniadakis, Computing fractional Laplacians on complex-geometry domains: Algorithms and simulations, SIAM J. Sci. Comput., 39 (2017), A1320-A1344. doi: 10.1137/16M1078197.  Google Scholar

[41]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680.  Google Scholar

[42]

C. J. WeissB. G. van Bloemen Waanders and H. Antil, Fractional operators applied to geophysical electromagnetics, Geophysical Journal International, 220 (2020), 1242-1259.  doi: 10.1093/gji/ggz516.  Google Scholar

[43]

C. J. Weiss, B. G. v. B. Waanders and H. Antil, Fractional Operators Applied to Geophysical Electromagnetics, preprint, arXiv: 1902.05096. Google Scholar

[44]

D. R. WitmanM. Gunzburger and J. Peterson, Reduced-order modeling for nonlocal diffusion problems, Internat. J. Numer. Methods Fluids, 83 (2017), 307-327.  doi: 10.1002/fld.4269.  Google Scholar

[45]

Q. YangI. TurnerF. Liu and M. Ilić, Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions, SIAM J. Sci. Comput., 33 (2011), 1159-1180.  doi: 10.1137/100800634.  Google Scholar

show all references

References:
[1]

M. Ainsworth and C. Glusa, Hybrid finite element-spectral method for the fractional Laplacian: Approximation theory and efficient solver, SIAM Journal on Scientific Computing, 40 (2018), A2383-A2405. doi: 10.1137/17M1144696.  Google Scholar

[2]

H. Antil and S. Bartels, Spectral approximation of fractional PDEs in image processing and phase field modeling, Comput. Methods Appl. Math., 17 (2017), 661-678.  doi: 10.1515/cmam-2017-0039.  Google Scholar

[3]

H. Antil, Y. Chen and A. Narayan, Kolmogorov widths and reduced order modeling for fractional elliptic operators, preprint, (2019). Google Scholar

[4]

H. Antil, Y. Chen and A. Narayan, Reduced basis methods for fractional Laplace equations via extension, SIAM J. Sci. Comput., 41 (2019), A3552-A3575. doi: 10.1137/18M1204802.  Google Scholar

[5]

H. Antil, R. Khatri and M. Warma, External optimal control of nonlocal PDEs, Inverse Problems, 35 (2019), 084003, 35 pp. doi: 10.1088/1361-6420/ab1299.  Google Scholar

[6]

H. Antil and J. Pfefferer, A short Matlab implementation of fractional Poisson equation with nonzero boundary conditions, Technical report, 2017. Google Scholar

[7]

H. AntilJ. Pfefferer and S. Rogovs, Fractional operators with inhomogeneous boundary conditions: Analysis, control, and discretization, Commun. Math. Sci., 16 (2018), 1395-1426.  doi: 10.4310/CMS.2018.v16.n5.a11.  Google Scholar

[8]

P. BinevA. CohenW. DahmenR. DeVoreG. Petrova and P. Wojtaszczyk, Convergence rates for greedy algorithms in reduced basis methods, SIAM J. Math. Anal., 43 (2011), 1457-1472.  doi: 10.1137/100795772.  Google Scholar

[9]

A. Bonito, D. Guignard and A. R. Zhang, Reduced basis approximations of the solutions to spectral fractional diffusion problems, J. Numer. Math., 28 (2020). doi: 10.1515/jnma-2019-0053.  Google Scholar

[10]

A. BonitoW. Lei and J. E. Pasciak, Numerical approximation of the integral fractional Laplacian, Numer. Math., 142 (2019), 235-278.  doi: 10.1007/s00211-019-01025-x.  Google Scholar

[11]

A. Bonito and J. Pasciak, Numerical approximation of fractional powers of elliptic operators, Math. Comp., 84 (2015), 2083-2110.  doi: 10.1090/S0025-5718-2015-02937-8.  Google Scholar

[12]

A. Buhr, C. Engwer, M. Ohlberger and S. Rave, A numerically stable a posteriori error estimator for reduced basis approximations of elliptic equations, preprint, (2014), arXiv: 1407.8005. Google Scholar

[13]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[14]

F. Casenave, Accurate a posteriori error evaluation in the reduced basis method, C. R. Math. Acad. Sci. Paris, 350 (2012), 539-542.  doi: 10.1016/j.crma.2012.05.012.  Google Scholar

[15]

F. CasenaveA. Ern and T. Lelièvre, Accurate and online-efficient evaluation of the a posteriori error bound in the reduced basis method, ESAIM Math. Model. Numer. Anal., 48 (2014), 207-229.  doi: 10.1051/m2an/2013097.  Google Scholar

[16]

Y. ChenJ. Jiang and A. Narayan, A robust error estimator and a residual-free error indicator for reduced basis methods, Comput. Math. Appl., 77 (2019), 1963-1979.  doi: 10.1016/j.camwa.2018.11.032.  Google Scholar

[17]

A. Cohen and R. DeVore, Approximation of high-dimensional parametric PDEs, Acta Numer., 24 (2015), 1-159.  doi: 10.1017/S0962492915000033.  Google Scholar

[18]

T. Danczul and J. Schöberl, A Reduced Basis Method For Fractional Diffusion Operators I, preprint, (2019), arXiv: 1904.05599. Google Scholar

[19]

T. Danczul and J. Schöberl, A Reduced Basis Method For Fractional Diffusion Operators II, preprint, (2020), arXiv: 2005.03574. Google Scholar

[20]

R. DeVoreG. Petrova and P. Wojtaszczyk, Greedy algorithms for reduced bases in Banach spaces, Constr. Approx., 37 (2013), 455-466.  doi: 10.1007/s00365-013-9186-2.  Google Scholar

[21]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[22]

M. E. Farquhar, T. J. Moroney, Q. Yang, I. W. Turner and K. Burrage, Computational modelling of cardiac ischaemia using a variable-order fractional Laplacian, preprint, (2018), arXiv: 1809.07936. Google Scholar

[23]

F. Gesztesy and M. Mitrea, A description of all self-adjoint extensions of the Laplacian and Krein-type resolvent formulas on non-smooth domains, J. Anal. Math., 113 (2011), 53-172.  doi: 10.1007/s11854-011-0002-2.  Google Scholar

[24]

G. Grubb, Regularity of spectral fractional Dirichlet and Neumann problems, Math. Nachr., 289 (2016), 831-844.  doi: 10.1002/mana.201500041.  Google Scholar

[25]

Q. GuanM. GunzburgerC. G. Webster and G. Zhang, Reduced basis methods for nonlocal diffusion problems with random input data, Comput. Methods Appl. Mech. Engrg., 317 (2017), 746-770.  doi: 10.1016/j.cma.2016.12.019.  Google Scholar

[26]

J. S. Hesthaven, G. Rozza and B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations, SpringerBriefs in Mathematics, Springer International Publishing, Cham, 2016. doi: 10.1007/978-3-319-22470-1.  Google Scholar

[27]

M. Ilic, F. Liu, I. Turner and V. Anh, Numerical Approximation of a fractional-in-space diffusion equation, I, Fract. Calc. Appl. Anal., 8 (2005), 323-341. https://eudml.org/doc/11303  Google Scholar

[28]

M. Ilic, F. Liu, I. Turner and V. Anh, Numerical approximation of a fractional-in-space diffusion equation (II) - with nonhomogeneous boundary conditions, Fract. Calc. Appl. Anal., 9 (2006), 333-349. http://www.diogenes.bg/fcaa/  Google Scholar

[29]

T. Kato, Note on fractional powers of linear operators, Proc. Japan Acad., 36 (1960), 94-96.  doi: 10.3792/pja/1195524082.  Google Scholar

[30]

D. KumarJ. Singh and S. Kumar, A fractional model of Navier-Stokes equation arising in unsteady flow of a viscous fluid, Journal of the Association of Arab Universities for Basic and Applied Sciences, 17 (2015), 14-19.  doi: 10.1016/j.jaubas.2014.01.001.  Google Scholar

[31]

M. Kwaśnicki, Ten equivalent definitions of the fractional laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.  Google Scholar

[32]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. 1, Die Grundlehren der mathematischen Wissenschaften, Springer-Verlag, New York-Heidelberg, 1972. https://www.springer.com/gp/book/9783642651632  Google Scholar

[33]

D. MeidnerJ. PfeffererK. Schürholz and B. Vexler, $$\text{hp}$$-finite elements for fractional diffusion, SIAM J. Numer. Anal., 56 (2018), 2345-2374.  doi: 10.1137/17M1135517.  Google Scholar

[34]

S. Molchanov and E. Ostrovskii, Symmetric stable processes as traces of degenerate diffusion processes, Theory of Probability & Its Applications, 14 (1969), 127-130. https://epubs.siam.org/doi/abs/10.1137/1114012  Google Scholar

[35]

R. H. NochettoE. Otárola and A. J. Salgado, A PDE approach to fractional diffusion in general domains: A priori error analysis, Found. Comput. Math., 15 (2015), 733-791.  doi: 10.1007/s10208-014-9208-x.  Google Scholar

[36]

B. N. Parlett, The Symmetric Eigenvalue Problem, Classics in Applied Mathematics, vol. 20, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. doi: 10.1137/1.9781611971163.  Google Scholar

[37] A. T. Patera and G. Rozza, Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations, MIT Press, 2007.   Google Scholar
[38]

P. Perdikaris and G. E. Karniadakis, Fractional-order viscoelasticity in one-dimensional blood flow models, Annals of Biomedical Engineering, 42 (2014), 1012-1023.  doi: 10.1007/s10439-014-0970-3.  Google Scholar

[39]

A. Quarteroni, A. Manzoni and F. Negri, Reduced Basis Methods for Partial Differential Equations, UNITEXT, vol. 92, La Matematica per il 3+2. Springer, Cham, 2016. doi: 10.1007/978-3-319-15431-2.  Google Scholar

[40]

F. Song, C. Xu and G. E. Karniadakis, Computing fractional Laplacians on complex-geometry domains: Algorithms and simulations, SIAM J. Sci. Comput., 39 (2017), A1320-A1344. doi: 10.1137/16M1078197.  Google Scholar

[41]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680.  Google Scholar

[42]

C. J. WeissB. G. van Bloemen Waanders and H. Antil, Fractional operators applied to geophysical electromagnetics, Geophysical Journal International, 220 (2020), 1242-1259.  doi: 10.1093/gji/ggz516.  Google Scholar

[43]

C. J. Weiss, B. G. v. B. Waanders and H. Antil, Fractional Operators Applied to Geophysical Electromagnetics, preprint, arXiv: 1902.05096. Google Scholar

[44]

D. R. WitmanM. Gunzburger and J. Peterson, Reduced-order modeling for nonlocal diffusion problems, Internat. J. Numer. Methods Fluids, 83 (2017), 307-327.  doi: 10.1002/fld.4269.  Google Scholar

[45]

Q. YangI. TurnerF. Liu and M. Ilić, Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions, SIAM J. Sci. Comput., 33 (2011), 1159-1180.  doi: 10.1137/100800634.  Google Scholar

Figure 1.  Values of $ \log_{10} G_{\pm} $ defined in (30) as a function of $ (M,s) $. We show $ s $-dependence as $ \log_{10}(1/s) $ since the error behavior for small $ s $ is the most restrictive. We observe that, for fixed $ M $, $ G_{\pm} $ has large values when $ s_\pm $ is small
Figure 2.  Values of $ \widetilde{M} $ and $ \widetilde{M}_\pm $ defined in (33) for various values of the tolerance $ \delta $. We show $ s $-dependence as $ \log_{10}(1/s) $ since the error behavior for small $ s $ is most restrictive. For visual reference, a $ 1/s $ curve is also plotted. We see that for small values of $ s_{\pm} $, the corresponding value of $ \widetilde{M}_{\pm} $ is large
6]">Figure 3.  Convergence of SQ (solid blue) and GQ methods (red dashed) as the spatial mesh is refined. Each used a dyadic mesh along the spatial variable with increasing resolution. A parameter value of $ s = 0.2 $ was used and similar results were seen for value of $ s $ between $ 0.1 $ and $ 0.9 $. We used the number of quadrature points for the integral suggested by current methods [6]
Figure 4.  Accuracy comparison of the SQ (red, dotted and dashed) and GQ methods (blue, dashed), with fractional order $ s = 0.2 $ (top plots) and $ s = 0.5 $ (bottom plots). Similar results where observed for value of $ s $ between $ 0.1 $ and $ 0.9 $
$ s = 0.2 $. These experiments compare both the direct (dotted and dashed) and reduced basis methods (solid) using the gaussian quadrature. Each used a dyadic mesh with $ 7 $ levels. Similar results where seen for value of $ s $ between $ 0.1 $ and $ 0.9 $">Figure 5.  "Offline" (i.e., one-time) computational investment for a single solve of (5) with a fixed value of $ s = 0.2 $. These experiments compare both the direct (dotted and dashed) and reduced basis methods (solid) using the gaussian quadrature. Each used a dyadic mesh with $ 7 $ levels. Similar results where seen for value of $ s $ between $ 0.1 $ and $ 0.9 $
Figure 6.  Error indicators $ \Delta_N(s) $ as a function of $ N $, for $ s = 0.2, 0.5, 0.8 $
Figure 7.  Accuracy of the RBM algorithm over a range of values of $ s $ (left and center). The Sine example is plotted in a red dot-dashed, the Mixed Modes in a blue solid line, and the Square Bump case in black crosses. In the right pane we show the cumulative computational time required by the GQ algorithm (blue) versus the RBM algorithm (red). Each query refers to an evaluation of the map $ s \mapsto u(s) $. In particular this cumulative time for the RBM solver includes the one-time offline cost required by OfflineFracLapRBM in Algorithm 2
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