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# Model reduction for fractional elliptic problems using Kato's formula

• * Corresponding author: Akil Narayan
• 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.

Mathematics Subject Classification: Primary: 35R11, 65M12.

 Citation: • • 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

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 

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 $0.9$

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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