# American Institute of Mathematical Sciences

March  2022, 12(1): 115-146. 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 2022 Early access  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 and Related Fields, 2022, 12 (1) : 115-146. 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. [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. [3] H. Antil, Y. Chen and A. Narayan, Kolmogorov widths and reduced order modeling for fractional elliptic operators, preprint, (2019). [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. [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. [6] H. Antil and J. Pfefferer, A short Matlab implementation of fractional Poisson equation with nonzero boundary conditions, Technical report, 2017. [7] H. Antil, J. 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. [8] P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. 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. [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. [10] A. Bonito, W. 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. [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. [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. [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. [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. [15] F. Casenave, A. 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. [16] Y. Chen, J. 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. [17] A. Cohen and R. DeVore, Approximation of high-dimensional parametric PDEs, Acta Numer., 24 (2015), 1-159.  doi: 10.1017/S0962492915000033. [18] T. Danczul and J. Schöberl, A Reduced Basis Method For Fractional Diffusion Operators I, preprint, (2019), arXiv: 1904.05599. [19] T. Danczul and J. Schöberl, A Reduced Basis Method For Fractional Diffusion Operators II, preprint, (2020), arXiv: 2005.03574. [20] R. DeVore, G. 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. [21] E. Di Nezza, G. 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. [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. [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. [24] G. Grubb, Regularity of spectral fractional Dirichlet and Neumann problems, Math. Nachr., 289 (2016), 831-844.  doi: 10.1002/mana.201500041. [25] Q. Guan, M. Gunzburger, C. 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. [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. [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 [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/ [29] T. Kato, Note on fractional powers of linear operators, Proc. Japan Acad., 36 (1960), 94-96.  doi: 10.3792/pja/1195524082. [30] D. Kumar, J. 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. [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. [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 [33] D. Meidner, J. Pfefferer, K. Schürholz and B. Vexler, $$\text{hp}$$-finite elements for fractional diffusion, SIAM J. Numer. Anal., 56 (2018), 2345-2374.  doi: 10.1137/17M1135517. [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 [35] R. H. Nochetto, E. 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. [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. [37] A. T. Patera and G. Rozza, Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations, MIT Press, 2007. [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. [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. [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. [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. [42] C. J. Weiss, B. 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. [43] C. J. Weiss, B. G. v. B. Waanders and H. Antil, Fractional Operators Applied to Geophysical Electromagnetics, preprint, arXiv: 1902.05096. [44] D. R. Witman, M. 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. [45] Q. Yang, I. Turner, F. 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.

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. [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. [3] H. Antil, Y. Chen and A. Narayan, Kolmogorov widths and reduced order modeling for fractional elliptic operators, preprint, (2019). [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. [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. [6] H. Antil and J. Pfefferer, A short Matlab implementation of fractional Poisson equation with nonzero boundary conditions, Technical report, 2017. [7] H. Antil, J. 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. [8] P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. 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. [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. [10] A. Bonito, W. 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. [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. [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. [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. [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. [15] F. Casenave, A. 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. [16] Y. Chen, J. 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. [17] A. Cohen and R. DeVore, Approximation of high-dimensional parametric PDEs, Acta Numer., 24 (2015), 1-159.  doi: 10.1017/S0962492915000033. [18] T. Danczul and J. Schöberl, A Reduced Basis Method For Fractional Diffusion Operators I, preprint, (2019), arXiv: 1904.05599. [19] T. Danczul and J. Schöberl, A Reduced Basis Method For Fractional Diffusion Operators II, preprint, (2020), arXiv: 2005.03574. [20] R. DeVore, G. 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. [21] E. Di Nezza, G. 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. [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. [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. [24] G. Grubb, Regularity of spectral fractional Dirichlet and Neumann problems, Math. Nachr., 289 (2016), 831-844.  doi: 10.1002/mana.201500041. [25] Q. Guan, M. Gunzburger, C. 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. [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. [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 [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/ [29] T. Kato, Note on fractional powers of linear operators, Proc. Japan Acad., 36 (1960), 94-96.  doi: 10.3792/pja/1195524082. [30] D. Kumar, J. 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. [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. [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 [33] D. Meidner, J. Pfefferer, K. Schürholz and B. Vexler, $$\text{hp}$$-finite elements for fractional diffusion, SIAM J. Numer. Anal., 56 (2018), 2345-2374.  doi: 10.1137/17M1135517. [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 [35] R. H. Nochetto, E. 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. [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. [37] A. T. Patera and G. Rozza, Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations, MIT Press, 2007. [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. [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. [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. [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. [42] C. J. Weiss, B. 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. [43] C. J. Weiss, B. G. v. B. Waanders and H. Antil, Fractional Operators Applied to Geophysical Electromagnetics, preprint, arXiv: 1902.05096. [44] D. R. Witman, M. 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. [45] Q. Yang, I. Turner, F. 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.
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
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
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]
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 $0.9$
"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 $0.9$
Error indicators $\Delta_N(s)$ as a function of $N$, for $s = 0.2, 0.5, 0.8$
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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