Multilevel algorithms for ill-posed problems

King, J. Thomas

doi:10.1007/bf01385512

Cited by 40 publications

(61 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, for operators with smoothing properties such as H h β , the aforementioned iterators remove the low-frequency components from the error, thus leaving an error that cannot be represented accurately on a coarser mesh. In [25], frequently 1 2 , under appropriate and reasonable assumptions on the parabolic equation (cf. Picard's criterion, see [16]).…”

mentioning

confidence: 99%

Optimal order multilevel preconditioners for regularized ill-posed problems

Drăgănescu

Dupont

2008

Math. Comp.

View full text Add to dashboard Cite

Abstract. In this article we design and analyze multilevel preconditioners for linear systems arising from regularized inverse problems. Using a scaleindependent distance function that measures spectral equivalence of operators, it is shown that these preconditioners approximate the inverse of the operator to optimal order with respect to the spatial discretization parameter h. As a consequence, the number of preconditioned conjugate gradient iterations needed for solving the system will decrease when increasing the number of levels, with the possibility of performing only one fine-level residual computation if h is small enough. The results are based on the previously known two-level preconditioners of Rieder (1997) (see also Hanke and Vogel (1999)), and on applying Newton-like methods to the operator equation X −1 − A = 0. We require that the associated forward problem has certain smoothing properties; however, only natural stability and approximation properties are assumed for the discrete operators. The algorithm is applied to a reverse-time parabolic equation, that is, the problem of finding the initial value leading to a given final state. We also present some results on constructing restriction operators with preassigned approximating properties that are of independent interest.

show abstract

mentioning

confidence: 99%

Optimal order multilevel preconditioners for regularized ill-posed problems

Drăgănescu

Dupont

2008

Math. Comp.

View full text Add to dashboard Cite

show abstract

“…In this regard, we discuss the construction of smoothers based on the idea of decomposing the finite-dimensional space into relatively high-frequency and low-frequency subspaces given in [27] and [26]. This idea was further studied in [24] and [25].…”

Section: Smoothersmentioning

confidence: 99%

“…This implies that β cannot be fixed as we refine the mesh because we will not be able to recover the sought frequencies. There has been little work on a mesh-independent scheme for vanishing β [27].…”

mentioning

confidence: 99%

See 1 more Smart Citation

Multigrid Algorithms for Inverse Problems with Linear Parabolic PDE Constraints

Adavani¹,

Biros²

2008

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

We present a multigrid algorithm for the solution of source identification inverse problems constrained by variable-coefficient linear parabolic partial differential equations. We consider problems in which the inversion variable is a function of space only. We consider the case of L-2 Tikhonov regularization. The convergence rate of our algorithm is mesh-independent-even in the case of no regularization. This feature makes the method algorithmically robust to the value of the regularization parameter, and thus useful for the cases in which we seek high-fidelity reconstructions. The inverse problem is formulated as a PDE-constrained optimization. We use a reduced-space approach in which we eliminate the state and adjoint variables, and we iterate in the inversion parameter space using conjugate gradients. We precondition the Hessian with a V-cycle multigrid scheme. The multigrid smoother is a two-step stationary iterative solver that inexactly inverts an approximate Hessian by iterating exclusively in the high-frequency subspace (using a high-pass filter). We analyze the performance of the scheme for the constant coefficient case with full observations; we analytically calculate the spectrum of the reduced Hessian and the smoothing factor for the multigrid scheme. The forward and adjoint problems are discretized using a backward-Euler finite-difference scheme. The overall complexity of our inversion algorithm is O(NtN + N log(2) N), where N is the number of grid points in space and N-t is the number of time steps. We provide numerical experiments that demonstrate the effectiveness of the method for different diffusion coefficients and values of the regularization parameter. We also provide heuristics, and we conduct numerical experiments for the case with variable coefficients and partial observations. We observe the same complexity as in the constant-coefficient case. Finally, we examine the effectiveness of using the reduced-space solver as a preconditioner for a full-space solver. Abstract. We present a multigrid algorithm for the solution of source identification inverse problems constrained by variable-coefficient linear parabolic partial differential equations. We consider problems in which the inversion variable is a function of space only. We consider the case of L 2 Tikhonov regularization. The convergence rate of our algorithm is mesh-independent-even in the case of no regularization. This feature makes the method algorithmically robust to the value of the regularization parameter, and thus useful for the cases in which we seek high-fidelity reconstructions. The inverse problem is formulated as a PDE-constrained optimization. We use a reduced-space approach in which we eliminate the state and adjoint variables, and we iterate in the inversion parameter space using conjugate gradients. We precondition the Hessian with a V-cycle multigrid scheme. The multigrid smoother is a two-step stationary iterative solver that inexactly inverts an approximate Hessian by iterating exclusively in the high-freq...

show abstract

“…Multi-level regularization methods are widely studied in the mathematical literature, see e.g. [37].…”

Section: Recommendationsmentioning

confidence: 99%

Acoustic source localization exploring theory and practice

Wind¹

View full text Add to dashboard Cite

Multilevel algorithms for ill-posed problems

Cited by 40 publications

References 21 publications

Optimal order multilevel preconditioners for regularized ill-posed problems

Optimal order multilevel preconditioners for regularized ill-posed problems

Multigrid Algorithms for Inverse Problems with Linear Parabolic PDE Constraints

Acoustic source localization exploring theory and practice

Contact Info

Product

Resources

About