TY - JOUR

T1 - Weakly intrusive low-rank approximation method for nonlinear parameter-dependent equations

AU - Giraldi, Loic

AU - Nouy, Anthony

N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: This work was supported by the French National Research Agency (grant ANR CHORUS MONU-0005).

PY - 2019/5/23

Y1 - 2019/5/23

N2 - This paper presents a weakly intrusive strategy for computing a low-rank approximation of the solution of a system of nonlinear parameter-dependent equations. The proposed strategy relies on a Newton-like iterative solver which only requires evaluations of the residual of the parameter-dependent equation and of a preconditioner (such as the differential of the residual) for instances of the parameters independently. The algorithm provides an approximation of the set of solutions associated with a possibly large number of instances of the parameters, with a computational complexity which can be orders of magnitude lower than when using the same Newton-like solver for all instances of the parameters. The reduction of complexity requires efficient strategies for obtaining low-rank approximations of the residual, of the preconditioner, and of the increment at each iteration of the algorithm. For the approximation of the residual and the preconditioner, weakly intrusive variants of the empirical interpolation method are introduced, which require evaluations of entries of the residual and the preconditioner. Then, an approximation of the increment is obtained by using a greedy algorithm for low-rank approximation, and a low-rank approximation of the iterate is finally obtained by using a truncated singular value decomposition. When the preconditioner is the differential of the residual, the proposed algorithm is interpreted as an inexact Newton solver for which a detailed convergence analysis is provided. Numerical examples illustrate the efficiency of the method.

AB - This paper presents a weakly intrusive strategy for computing a low-rank approximation of the solution of a system of nonlinear parameter-dependent equations. The proposed strategy relies on a Newton-like iterative solver which only requires evaluations of the residual of the parameter-dependent equation and of a preconditioner (such as the differential of the residual) for instances of the parameters independently. The algorithm provides an approximation of the set of solutions associated with a possibly large number of instances of the parameters, with a computational complexity which can be orders of magnitude lower than when using the same Newton-like solver for all instances of the parameters. The reduction of complexity requires efficient strategies for obtaining low-rank approximations of the residual, of the preconditioner, and of the increment at each iteration of the algorithm. For the approximation of the residual and the preconditioner, weakly intrusive variants of the empirical interpolation method are introduced, which require evaluations of entries of the residual and the preconditioner. Then, an approximation of the increment is obtained by using a greedy algorithm for low-rank approximation, and a low-rank approximation of the iterate is finally obtained by using a truncated singular value decomposition. When the preconditioner is the differential of the residual, the proposed algorithm is interpreted as an inexact Newton solver for which a detailed convergence analysis is provided. Numerical examples illustrate the efficiency of the method.

UR - http://hdl.handle.net/10754/626565

UR - https://epubs.siam.org/doi/10.1137/17M1136924

UR - http://www.scopus.com/inward/record.url?scp=85071857399&partnerID=8YFLogxK

U2 - 10.1137/17M1136924

DO - 10.1137/17M1136924

M3 - Article

VL - 41

SP - A1777-A1792

JO - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

SN - 1064-8275

IS - 3

ER -