Algorithm 887

Chen, Yanqing; Davis, Timothy A.; Hager, William W.; Rajamanickam, Sivasankaran

doi:10.1145/1391989.1391995

Cited by 540 publications

(93 citation statements)

References 34 publications

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“…Some are designed exclusively for such systems (for example, CHOLMOD [6] and HSL MA87 [25]), while others can also be used to solve indefinite systems (notably, MA57 [11], HSL MA97 [27], MUMPS [38], WSMP [23], PARDISO [43], and SPRAL SSIDS [24]). We employ the packages HSL MA87 and HSL MA97 from the HSL Mathematical Software Library [31]; an overview of both packages together with a numerical comparison is provided by Hogg and Scott [29].…”

Section: Sparse Direct Solversmentioning

confidence: 99%

“…Here we set lsize = 200 and vary rsize from 0 to 1000; the drop tolerances are set to 0.0. For each value of rsize, the number of entries in L is nnz(L) = 6.63×10 6 . We see that increasing the intermediate memory stabilizes the factorization, reducing the shift and giving a higher quality preconditioner that requires fewer LSMR iterations and less time.…”

Section: Incomplete Factorization Of the Normal Matrix Cmentioning

confidence: 99%

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On Using Cholesky-Based Factorizations and Regularization for Solving Rank-Deficient Sparse Linear Least-Squares Problems

Scott¹

2017

SIAM J. Sci. Comput.

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Abstract. By examining the performance of modern parallel sparse direct solvers and exploiting our knowledge of the algorithms behind them, we perform numerical experiments to study how they can be used to efficiently solve rank-deficient sparse linear least-squares problems arising from practical applications. The Cholesky factorization of the normal equations breaks down when the least-squares problem is rank-deficient, while applying a symmetric indefinite solver to the augmented system can give an unacceptable level of fill in the factors. To try to resolve these difficulties, we consider a regularization procedure that modifies the diagonal of the unregularized matrix. This leads to matrices that are easier to factorize. We consider both the regularized normal equations and the regularized augmented system. We employ the computed factors of the regularized systems as preconditioners with an iterative solver to obtain the solution of the original (unregularized) problem. Furthermore, we look at using limited-memory incomplete Cholesky-based factorizations and how these can offer the potential to solve very large problems.Key words. least-squares problems, normal equations, augmented system, sparse matrices, direct methods, iterative methods, Cholesky factorizations, preconditioning, regularization AMS subject classifications. 65F05, 65F50 DOI. 10.1137/16M10653801. Introduction. In recent years, a number of methods have been proposed for preconditioning sparse linear least-squares problems; a brief overview with a comprehensive list of references is included in the introduction to the paper of Bru et al. [4]. The recent study of Gould and Scott [20,21] reviewed many of these methods (specifically those for which software has been made available) and then tested and compared their performance using a range of examples coming from practical applications. One of the outcomes of that study was some insight into which least-squares problems in the widely used sparse matrix collections CUTEst [19] and University of Florida [10] currently pose a real challenge for direct methods and/or iterative solvers. In particular, the study found that most of the available software packages were not reliable or efficient for rank-deficient least-squares problems (at least not when run with the recommended settings for the input parameters that were employed in the study). In this paper, we look further at such problems and focus on the effectiveness of both sparse direct solvers and iterative methods with incomplete factorization preconditioners. A key theme is the use of regularization (see, for example, [15,48,49]). We propose computing a factorization (either complete or incomplete) of a regularized problem and then using this as a preconditioner for an iterative solver to recover the solution of the original (unregularized) problem.The problem we are interested in is

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Section: Sparse Direct Solversmentioning

confidence: 99%

Section: Incomplete Factorization Of the Normal Matrix Cmentioning

confidence: 99%

On Using Cholesky-Based Factorizations and Regularization for Solving Rank-Deficient Sparse Linear Least-Squares Problems

Scott¹

2017

SIAM J. Sci. Comput.

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“…In both cases, the system can be solved efficiently as a sparse linear system, using a sparse direct Cholesky method [6]. Figure 7c shows how this capability can be used to preserve continuity and smoothness in the right fore-leg of the elephant when the foot (marked by the green patch) is moved, while the yellow thin-plate spline serves as an anchor.…”

Section: Smooth Surface Patchesmentioning

confidence: 99%

Interactive Inverse 3D Modeling

Andrews

Jin

Séquin

2012

Computer-Aided Design and Applications

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ABSTRACT"Interactive Inverse 3D Modeling" is a user-guided approach to shape construction and redesign that extracts well-structured, parameterized, procedural descriptions from unstructured, hierarchically flat input data, such as point clouds, boundary representation meshes, or even multiple pictorial views of a given inspirational prototype. This approach combines traditional "forward" 3D modeling tools with a system of user-guided extraction modules and optimization routines. With a few cursor strokes users can express their preferences of the type of modeling primitives to be used in a particular area of the given prototype to be approximated, and they can also select the degree of parameterization associated with each modeling routine. The results are then pliable, structured descriptions that are well suited to implement the particular design modifications intended by the user.

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“…As DIVA (Data-Interpolating Variational Analysis), the ndimensional tool divand also uses norm splines to define the background error covariance in combination with a direct solver which does not require preconditioning. In the case of DIVA the skyline method (Dhatt and Touzot, 1984) is used, while divand uses the supernodal sparse Cholesky factorization (Chen et al, 2008;Davis and Hager, 2009). Other iterative methods (which require preconditioning) have also been implemented.…”

Section: A Barth Et Al: Divandmentioning

confidence: 99%

“…The solver based on SuiteSparse (Chen et al, 2008;Davis and Hager, 2009) can be used directly if the matrix R \ H is sparse. This is in particular the case when R is diagonal.…”

Section: Primal Formulationmentioning

confidence: 99%

divand-1.0: <i>n</i>-dimensional variational data analysis for ocean observations

et al. 2014

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Abstract.A tool for multidimensional variational analysis (divand) is presented. It allows the interpolation and analysis of observations on curvilinear orthogonal grids in an arbitrary high dimensional space by minimizing a cost function. This cost function penalizes the deviation from the observations, the deviation from a first guess and abruptly varying fields based on a given correlation length (potentially varying in space and time). Additional constraints can be added to this cost function such as an advection constraint which forces the analysed field to align with the ocean current. The method decouples naturally disconnected areas based on topography and topology. This is useful in oceanography where disconnected water masses often have different physical properties. Individual elements of the a priori and a posteriori error covariance matrix can also be computed, in particular expected error variances of the analysis. A multidimensional approach (as opposed to stacking two-dimensional analysis) has the benefit of providing a smooth analysis in all dimensions, although the computational cost is increased.Primal (problem solved in the grid space) and dual formulations (problem solved in the observational space) are implemented using either direct solvers (based on Cholesky factorization) or iterative solvers (conjugate gradient method). In most applications the primal formulation with the direct solver is the fastest, especially if an a posteriori error estimate is needed. However, for correlated observation errors the dual formulation with an iterative solver is more efficient.The method is tested by using pseudo-observations from a global model. The distribution of the observations is based on the position of the Argo floats. The benefit of the threedimensional analysis (longitude, latitude and time) compared to two-dimensional analysis (longitude and latitude) and the role of the advection constraint are highlighted. The tool divand is free software, and is distributed under the terms of the General Public Licence (GPL)

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Algorithm 887

Cited by 540 publications

References 34 publications

On Using Cholesky-Based Factorizations and Regularization for Solving Rank-Deficient Sparse Linear Least-Squares Problems

On Using Cholesky-Based Factorizations and Regularization for Solving Rank-Deficient Sparse Linear Least-Squares Problems

Interactive Inverse 3D Modeling

divand-1.0: <i>n</i>-dimensional variational data analysis for ocean observations

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Algorithm 887

Cited by 540 publications

References 34 publications

On Using Cholesky-Based Factorizations and Regularization for Solving Rank-Deficient Sparse Linear Least-Squares Problems

On Using Cholesky-Based Factorizations and Regularization for Solving Rank-Deficient Sparse Linear Least-Squares Problems

Interactive Inverse 3D Modeling

divand-1.0: &lt;i&gt;n&lt;/i&gt;-dimensional variational data analysis for ocean observations

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divand-1.0: <i>n</i>-dimensional variational data analysis for ocean observations