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Register a new userKrylov Methods for Low-Rank Regularization
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Added by
Silvia Gazzola
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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This paper introduces new solvers for the computation of low-rank approximate solutions to large-scale linear problems, with a particular focus on the regularization of linear inverse problems. Although Krylov methods incorporating explicit projections onto low-rank subspaces are already used for well-posed systems that arise from discretizing stochastic or time-dependent PDEs, we are mainly concerned with algorithms that solve the so-called nuclear norm regularized problem, where a suitable nuclear norm penalization on the solution is imposed alongside a fit-to-data term expressed in the 2-norm: this has the effect of implicitly enforcing low-rank solutions. By adopting an iteratively reweighted norm approach, the nuclear norm regularized problem is reformulated as a sequence of quadratic problems, which can then be efficiently solved using Krylov methods, giving rise to an inner-outer iteration scheme. Our approach differs from the other solvers available in the literature in that: (a) Kronecker product properties are exploited to define the reweighted 2-norm penalization terms; (b) efficient preconditioned Krylov methods replace gradient (projection) methods; (c) the regularization parameter can be efficiently and adaptively set along the iterations. Furthermore, we reformulate within the framework of flexible Krylov methods both the new inner-outer methods for nuclear norm regularization and some of the existing Krylov methods incorporating low-rank projections. This results in an even more computationally efficient (but heuristic) strategy, that does not rely on an inner-outer iteration scheme. Numerical experiments show that our new solvers are competitive with other state-of-the-art solvers for low-rank problems, and deliver reconstructions of increased quality with respect to other classical Krylov methods.
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An approach is given for solving large linear systems that combines Krylov methods with use of two different grid levels. Eigenvectors are computed on the coarse grid and used to deflate eigenvalues on the fine grid. GMRES-type methods are first used on both the coarse and fine grids. Then another approach is given that has a restarted BiCGStab (or IDR) method on the fine grid. While BiCGStab is generally considered to be a non-restarted method, it works well in this context with deflating and restarting. Tests show this new approach can be very efficient for difficult linear equations problems.
Subspace recycling iterative methods and other subspace augmentation schemes are a successful extension to Krylov subspace methods in which a Krylov subspace is augmented with a fixed subspace spanned by vectors deemed to be helpful in accelerating convergence or conveying knowledge of the solution. Recently, a survey was published, in which a framework describing the vast majority of such methods was proposed [Soodhalter et al, GAMM-Mitt. 2020]. In many of these methods, the Krylov subspace is one generated by the system matrix composed with a projector that depends on the augmentation space. However, it is not a requirement that a projected Krylov subspace be used. There are augmentation methods built on using Krylov subspaces generated by the original system matrix, and these methods also fit into the general framework. In this note, we observe that one gains implementation benefits by considering such augmentation methods with unprojected Krylov subspaces in the general framework. We demonstrate this by applying the idea to the R$^3$GMRES method proposed in [Dong et al. ETNA 2014] to obtain a simplified implementation and to connect that algorithm to early augmentation schemes based on flexible preconditioning [Saad. SIMAX 1997].
The parallel strong-scaling of Krylov iterative methods is largely determined by the number of global reductions required at each iteration. The GMRES and Krylov-Schur algorithms employ the Arnoldi algorithm for nonsymmetric matrices. The underlying orthogonalization scheme is left-looking and processes one column at a time. Thus, at least one global reduction is required per iteration. The traditional algorithm for generating the orthogonal Krylov basis vectors for the Krylov-Schur algorithm is classical Gram Schmidt applied twice with reorthogonalization (CGS2), requiring three global reductions per step. A new variant of CGS2 that requires only one reduction per iteration is applied to the Arnoldi-QR iteration. Strong-scaling results are presented for finding eigenvalue-pairs of nonsymmetric matrices. A preliminary attempt to derive a similar algorithm (one reduction per Arnoldi iteration with a robust orthogonalization scheme) was presented by Hernandez et al.(2007). Unlike our approach, their method is not forward stable for eigenvalues.
Subspace recycling techniques have been used quite successfully for the acceleration of iterative methods for solving large-scale linear systems. These methods often work by augmenting a solution subspace generated iteratively by a known algorithm with a fixed subspace of vectors which are ``useful for solving the problem. Often, this has the effect of inducing a projected version of the original linear system to which the known iterative method is then applied, and this projection can act as a deflation preconditioner, accelerating convergence. Most often, these methods have been applied for the solution of well-posed problems. However, they have also begun to be considered for the solution of ill-posed problems. In this paper, we consider subspace augmentation-type iterative schemes applied to linear ill-posed problems in a continuous Hilbert space setting, based on a recently developed framework describing these methods. We show that under suitable assumptions, a recycling method satisfies the formal definition of a regularization, as long as the underlying scheme is itself a regularization. We then develop an augmented subspace version of the gradient descent method and demonstrate its effectiveness, both on an academic Gaussian blur model and on problems arising from the adaptive optics community for the resolution of large sky images by ground-based extremely large telescopes.
The orthogonal decomposition factorizes a tensor into a sum of an orthogonal list of rankone tensors. We present several properties of orthogonal rank. We find that a subtensor may have a larger orthogonal rank than the whole tensor and prove the lower semicontinuity of orthogonal rank. The lower semicontinuity guarantees the existence of low orthogonal rank approximation. To fit the orthogonal decomposition, we propose an algorithm based on the augmented Lagrangian method and guarantee the orthogonality by a novel orthogonalization procedure. Numerical experiments show that the proposed method has a great advantage over the existing methods for strongly orthogonal decompositions in terms of the approximation error.
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