Deflation Techniques for Finding Distinct Solutions of Nonlinear Partial Differential Equations

By Patrick E. Farrell, 'Asgeir Birkisson, Simon W. Funke. 2014

173 citations recorded by the source corpus.

Nonlinear systems of partial differential equations (PDEs) may permit several distinct solutions. The typical current approach to finding distinct solutions is to start Newton's method with many different initial guesses, hoping to find starting points that lie in different basins of attraction. In this paper, we present an infinite-dimensional deflation algorithm for systematically modifying the residual of a nonlinear PDE problem to eliminate known solutions from consideration. This enables the Newton--Kantorovitch iteration to converge to several different solutions, even starting from the same initial guess. The deflated Jacobian is dense, but an efficient preconditioning strategy is devised, and the number of Krylov iterations is observed not to grow as solutions are deflated. The power of the approach is demonstrated on several problems from special functions, phase separation, differential geometry and fluid mechanics that permit distinct solutions.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Numerical Analysis

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Citations in the corpus

Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data (2019, background): “Moreover, many PDEs in physics have multiple solutions [50] which cannot be captured with a deterministic model.”
Bayesian Probabilistic Numerical Methods (2017, methodology): “This system was recently presented in Farrell et al. [2015] as an example of a boundary value problem with multiple solutions. Two distinct solutions are known, and are illustrated in Figure 4. These model solutions were obtained using the deflation technique described in Farrell et al. [2015]. The ”