[Paper Review] Certified reduced basis methods for fractional Laplace equations via extension

Harbir Antil, Yanlai Chen+1 more|arXiv (Cornell University)|Aug 1, 2018

Fractional Differential Equations Solutions7 references3 citations

TL;DR

This paper presents a certified reduced basis method (RBM) for solving fractional Laplace equations via the Caffarelli-Silvestre extension, enabling rapid and accurate solution surrogates for multiple parameter queries—particularly the fractional exponent $ s $—with over two orders of magnitude speedup compared to standard finite element solvers, while maintaining rigorous error certification.

ABSTRACT

Fractional Laplace equations are becoming important tools for mathematical modeling and prediction. Recent years have shown much progress in developing accurate and robust algorithms to numerically solve such problems, yet most solvers for fractional problems are computationally expensive. Practitioners are often interested in choosing the fractional exponent of the mathematical model to match experimental and/or observational data; this requires the computational solution to the fractional equation for several values of the both exponent and other parameters that enter the model, which is a computationally expensive many-query problem. To address this difficulty, we present a model order reduction strategy for fractional Laplace problems utilizing the reduced basis method (RBM). Our RBM algorithm for this fractional partial differential equation (PDE) allows us to accomplish significant acceleration compared to a traditional PDE solver while maintaining accuracy. Our numerical results demonstrate this accuracy and efficiency of our RBM algorithm on fractional Laplace problems in two spatial dimensions.

Motivation & Objective

• Address the computational bottleneck in solving fractional Laplace equations when the fractional exponent $ s \in (0,1) $ is unknown and must be inferred from data.
• Develop a model order reduction strategy for parameterized fractional PDEs where $ s $ and other parameters vary across queries.
• Overcome the challenges of non-affine parameter dependence and nonlocal operators in RBM by leveraging the extension technique.
• Ensure rigorous error certification for the reduced basis surrogate across the parameter domain $ s \in (0,1) $.
• Demonstrate significant computational acceleration in both offline and online phases for many-query scenarios in two spatial dimensions.

Proposed method

• Use the Caffarelli-Silvestre extension to transform the nonlocal fractional Laplace problem into a local PDE on an extended domain $ \Omega \times (0,\infty) $, enabling standard finite element discretization.
• Apply a finite element method with truncation to the semi-infinite cylinder, exploiting exponential decay in the extended dimension for accurate approximation.
• Construct a truth approximation $ \mathcal{U}^\mathcal{N} $ using the extended finite element solution, which serves as the high-fidelity solution for RBM training.
• Implement an empirical interpolation method (EIM) to handle the non-affine dependence of the fractional operator on $ s $, enabling offline-online decomposition.
• Construct a reduced basis space $ \mathcal{U}_N $ from snapshots of the extended solution at selected $ s $ values, using greedy algorithms to ensure error bounds.
• Certify the reduced basis error via a certified error estimator derived from the inf-sup condition and the solution's parameter dependence.

Experimental results

Research questions

• RQ1 Can a certified reduced basis method be effectively applied to fractional Laplace equations with non-affine parameter dependence on the fractional exponent $ s $?
• RQ2 How can the extension technique be integrated into RBM to handle the nonlocal nature of the fractional Laplacian while maintaining computational efficiency?
• RQ3 What is the computational speedup and accuracy of the RBM surrogate compared to standard finite element solvers in many-query scenarios for varying $ s $?
• RQ4 How does the error of the RBM surrogate behave across the full parameter domain $ s \in (0,1) $, especially near singularities or boundary values?
• RQ5 Can the method be extended to handle additional parameters, such as source term variations, while preserving efficiency and certification?

Key findings

• The RBM achieves over two orders of magnitude speedup in cumulative computation time for $ M \approx 312 $ queries when solving for $ s \in D_1 $, with $ N = 7 $ reduced basis functions.
• For marginal online computation time, the speedup exceeds four orders of magnitude when using 10 reduced basis functions, demonstrating strong online efficiency.
• The offline phase cost is negligible, and the investment in offline computation becomes highly beneficial in many-query settings.
• RB error stagnates at low levels (e.g., $ \sim 10^{-4} $) for $ s \in D_2 $, indicating robustness despite limited $ y $-uniform convergence in the EIM approximation.
• The method maintains high accuracy across the entire parameter domain $ s \in (0.03, 0.97) $, with errors well below $ 10^{-4} $ for $ N = 7 $ in the one-parameter case.
• In the two-parameter case ($ s, \nu $), the RBM achieves accurate surrogates over $ 66,049 $ training points and $ 900 $ test points, with convergence observed in error metrics across the parameter space.