Lanczos Bidiagonalization Method Research Articles

Thick-restarted joint Lanczos bidiagonalization for the GSVD

The computation of the partial generalized singular value decomposition (GSVD) of large-scale matrix pairs can be approached by means of iterative methods based on expanding subspaces, particularly Krylov subspaces. We consider the joint Lanczos bidiagonalization method, and analyze the feasibility of adapting the thick restart technique that is being used successfully in the context of other linear algebra problems. Numerical experiments illustrate the effectiveness of the proposed method. We also compare the new method with an alternative solution via equivalent eigenvalue problems, considering accuracy as well as computational performance. The analysis is done using a parallel implementation in the SLEPc library.

The improvement of sparsity gravity inversion using an adaptive lanczos bidiagonalization method

Inversion of gravity data is one the important steps in the interpretation of practical data. The detection of sharp boundaries between anomalous bodies and host rocks is an interesting point in the geological frameworks. The gravity inversion with sparsity constraint is a useful method to recover block subsurface density distribution, which is efficiently used for the quantitative interpretation of gravity data. The reweighted regularized method is a useful method to solve the inverse problem. However, in this type, we must face the updating gravity forward matrix and large matrix operation. The application of Lanczos bidiagonalization method can reduce the size of data and matrix in the inversion to resolve the large scale inversion problem. However, a very important problem is not resolved, which is update of reweighted forward matrix and new Lanczos bidiagonalization matrix. Here, an adaptive Lanczos bidiagonalization method is studied to select the Lanczos bidiagonalization factor. And a new projected method with adaptive Lanczos bidiagonalization method is study to avoid the updating sparsity reweighted function. We calculate the reweighted forward matrix and Lanczos bidiagonalization matrix only one time, which can essentially reduce the computational complexity. The inversion results of synthetic data show that the new improved method is faster and better than common reweight regularized Lanczos bidiagonalization method to produce an acceptable solution for focusing inverse problem. The improvement of adaptive Lanczos bidiagonalization in sparsity gravity inversion is also tested on gravity data collected over the Mobrun massive sulfide ore body in Noranda, Quebec, Canada. The inversion results indicate a remarkable correlation with true structure of the ore body that is achieved from drilling data.

Two-dimensional magnetotelluric data inversion using Lanczos bidiagonalization method with active constraint balancing

The magnetotelluric (MT) technique is an electromagnetic geophysical method, which is widely used as a complementary to seismic surveys for exploration of hydrocarbon reservoirs. In the inversion process, the method of matrix inverse calculation has a considerable effect on the speed of the inversion and the quality of obtained models. Lanczos Bidiagonalization (LB) method has been reported to be a fast and efficient approach for solving the inversion problems. In this study, we employ LB method for inverting large-scale 2D MT data. In LB algorithm, the full set of equations is replaced by a dimensionally reduced system of equations. As a result, the speed of the solution procedure is increased, while the original problem is solved with a high accuracy. In addition, we employ active constraint balancing approach for determining the optimum regularization parameter. The advantage of the method is that for highly resolvable parameters, a small value of the Lagrangian multiplier is assigned, and vice versa. The results of the synthetic data inversion show that both methods require equal computer memory but LB method is faster and more reliable than conjugate gradient method. The proposed approach is also applied to inverse real MT data collected from the Kashan area. The Kashan area is the most interesting area for oil and gas exploration of the Central Iran Basin. The inversion results obtained by LB are in a good agreement with the geological structure of the study area and the drilling data.

On computing the minimum singular value of a tensor sum

Abstract Recently, the Lanczos bidiagonalization method over tensor space has been proposed for computing the maximum and minimum singular values of a tensor sum T. The method over tensor space is practical in memory and has a simple implementation due to recent developments in tensor computations; however, there is still room for improvement in the convergence to the minimum singular value. This study reconstructed an invert Lanczos bidiagonalization method from vector space to tensor space. The resulting algorithm requires solving linear systems at each iteration step. Using standard direct methods, such as the LU decomposition for solving the linear systems requires a huge memory of O(n 6). Therefore, this paper proposes a tensor-structure-preserving direct methods of T whose memory requirements are of O(n 3), which is equivalent to the order of iterative methods. Numerical examples indicate that the number of iterations tends to be much smaller than that of the conventional method.

Fast 3D inversion of gravity data using solution space priorconditioned lanczos bidiagonalization

Inversion of gravity data is one of the most important steps in the quantitative interpretation of practical data. Inversion is a mathematical technique that automatically constructs a subsurface geophysical model from measured data, incorporating some priori information. Inversion of gravity data is time consuming because of increase in data and model parameters. Some efforts have been made to deal with this problem, one of them is using fast algorithms for solving system of equations in inverse problem. Lanczos bidiagonalization method is a fast algorithm that works based on Krylov subspace iterations and projection method, but cannot always provide a good basis for a projection method. So in this study, we combined the Krylov method with a regularization method applied to the low-dimensional projected problem. To achieve the goal, the orthonormal basis vectors of the discrete cosine transform (DCT) were used to build the low-dimensional subspace. The forward operator matrix replaced with a matrix of lower dimension, thus, the required memory and running time of the inverse modeling is decreased by using the proposed algorithm. It is shown that this algorithm can be appropriate to solve a Tikhonov cost function for inversion of gravity data. The proposed method has been applied on a noise-corrupted synthetic data and field gravity data (Mobrun gravity data) to demonstrate its reliability for three dimensional (3D) gravity inversion. The obtained results of 3D inversion both synthetic and field gravity data (Mobrun gravity data) indicate the proposed inversion algorithm could produce density models consistent with true structures.

Fast 3D Focusing Inversion of Gravity Data Using Reweighted Regularized Lanczos Bidiagonalization Method

Inversion of gravity data is one of the important steps in the interpretation of practical data. One of the most interesting geological frameworks for gravity data inversion is the detection of sharp boundaries between orebody and host rocks. The focusing inversion is able to reconstruct a sharp image of the geological target. This technique can be efficiently applied for the quantitative interpretation of gravity data. In this study, a new reweighted regularized method for the 3D focusing inversion technique based on Lanczos bidiagonalization method is developed. The inversion results of synthetic data show that the new method is faster than common reweighted regularized conjugate gradient method to produce an acceptable solution for focusing inverse problem. The new developed inversion scheme is also applied for inversion of the gravity data collected over the San Nicolas Cu-Zn orebody in Zacatecas State, Mexico. The inversion results indicate a remarkable correlation with the true structure of the orebody that is achieved from drilling data.

Fast 3D inversion of airborne gravity-gradiometry data using Lanczos bidiagonalization method

We developed a new fast inversion method for to process and interpret airborne gravity gradiometry data, which was based on Lanczos bidiagonalization algorithm. Here, we describe the application of this new 3D gravity gradiometry inversion method to recover a subsurface density distribution model from the airborne measured gravity gradiometry anomalies. For this purpose, the survey area is divided into a large number of rectangular cells with each cell possessing a constant unknown density. It is well known that the solution of large linear gravity gradiometry is an ill-posed problem since using the smoothest inversion method is considerably time consuming. We demonstrate that the Lanczos bidiagonalization method can be an appropriate algorithm to solve a Tikhonov solver time cost function for resolving the large equations within a short time. Lanczos bidiagonalization is designed to make the very large gravity gradiometry forward modeling matrices to become low-rank, which will considerably reduce the running time of the inversion method. We also use a weighted generalized cross validation method to choose the appropriate Tikhonov parameter to improve inversion results. The inversion incorporates a model norm that allows us to attain the smoothing and depth of the solution; in addition, the model norm counteracts the natural decay of the kernels, which concentrate at shallow depths. The method is applied on noise-contaminated synthetic gravity gradiometry data to demonstrate its suitability for large 3D gravity gradiometry data inversion. The airborne gravity gradiometry data from the Vinton Salt Dome, USE, were considered as a case study. The validity of the new method on real data is discussed with reference to the Vinton Dome inversion result. The intermediate density values in the constructed model coincide well with previous results and geological information. This demonstrates the validity of the gravity gradiometry inversion method.

Improving Approximate Singular Triplets in Lanczos Bidiagonalization Method

Lanczos bidiagonalization method is the most popular method for computing some largest singular triplets of large matrices. In this method, $2m+1$ base vectors are generated from the $m$-step Lanczos bidiagonalization process, but only $2m$ of them are used to form the approximate singular vectors and one of them is not used. In this paper, we make two improvements on the classical Lanczos bidiagonalization method. Firstly, following Jia and Elsner's idea for eigenproblems [9], we form the new approximate singular vectors by minimizing the corresponding residual norms in subspaces generated by $2m+1$ base vectors to replace the old approximate singular vectors. Secondly, in the process of implicit restarting, we replace the classical exact shifts by new shifts based on the information of the new approximate singular vectors. The total extra cost of the new method can be neglected. Numerical experiments show that, after two improvements, the new method proposed in this paper performs much better than the classical Lanczos bidiagonalization method. It uses less restarts and CPU time to reach the desired convergence.

Probabilistic Bounds for the Matrix Condition Number with Extended Lanczos Bidiagonalization

Reliable estimates for the condition number of a large, sparse, real matrix $A$ are important in many applications. To get an approximation for the condition number $\kappa(A)$, an approximation for the smallest singular value is needed. Standard Krylov subspaces are usually unsuitable for finding a good approximation to the smallest singular value. Therefore, we study extended Krylov subspaces which turn out to be ideal for the simultaneous approximation of both the smallest and largest singular value of a matrix. First, we develop a new extended Lanczos bidiagonalization method. With this method we obtain a lower bound for the condition number. Moreover, the method also yields probabilistic upper bounds for $\kappa(A)$. The user can select the probability with which the upper bound holds, as well as the ratio of the probabilistic upper bound and the lower bound.

A Preconditioned Hybrid SVD Method for Accurately Computing Singular Triplets of Large Matrices

The computation of a few singular triplets of large, sparse matrices is a challenging task, especially when the smallest magnitude singular values are needed in high accuracy. Most recent efforts try to address this problem through variations of the Lanczos bidiagonalization method, but they are still challenged even for medium matrix sizes due to the difficulty of the problem. We propose a novel SVD approach that can take advantage of preconditioning and of any well designed eigensolver to compute both largest and smallest singular triplets. Accuracy and efficiency is achieved through a hybrid, two-stage meta-method, PHSVDS. In the first stage, PHSVDS solves the normal equations up to the best achievable accuracy. If further accuracy is required, the method switches automatically to an eigenvalue problem with the augmented matrix. Thus it combines the advantages of the two stages, faster convergence and accuracy, respectively. For the augmented matrix, solving the interior eigenvalue is facilitated by a proper use of the good initial guesses from the first stage and an efficient implementation of the refined projection method. We also discuss how to precondition PHSVDS and to cope with some issues that arise. Numerical experiments illustrate the efficiency and robustness of the method.

An implicitly restarted Lanczos bidiagonalization method with refined harmonic shifts for computing smallest singular triplets

E. Kokiopoulou et al. developed an implicitly restarted Lanczos bidiagonalization method (named IRLANB) for computing the smallest singular triplets. They use the wanted harmonic Ritz values and th...

An implicitly restarted Lanczos bidiagonalization method with refined harmonic shifts for computing smallest singular triplets

E. Kokiopoulou et al. developed an implicitly restarted Lanczos bidiagonalization method (named IRLANB) for computing the smallest singular triplets. They use the wanted harmonic Ritz values and the corresponding refined harmonic Ritz vectors to approximate the desired smallest singular triplets. However, they use the unwanted harmonic Ritz values as shifts, named the harmonic shifts, to implicitly restart their algorithm. Therefore, although they replace the harmonic Ritz vectors by the refined harmonic Ritz vectors, which are always better than the harmonic Ritz vectors, to improve the convergence of IRLANB, the subspace after implicitly restarting is the same as that without this replacement. In this paper, using the information of the refined harmonic Ritz vectors, we present a new shift strategy, named the refined harmonic shifts, to replace the harmonic shifts used in IRLANB and obtain a new method, called IRRLANB. The refined harmonic shifts are better than the harmonic shifts and can be computed cheaply and reliably. The numerical experiments are reported to indicate that IRRLANB is superior than IRLANB.

Approximate resistivity and susceptibility mapping from airborne electromagnetic and magnetic data, a case study for a geologically plausible porphyry copper unit in Iran

This paper describes the application of approximate methods to invert airborne magnetic data as well as helicopter-borne frequency domain electromagnetic data in order to retrieve a joint model of magnetic susceptibility and electrical resistivity. The study area located in Semnan province of Iran consists of an arc-shaped porphyry andesite covered by sedimentary units which may have potential of mineral occurrences, especially porphyry copper. Based on previous studies, which assume a homogenous half-space earth model, two approximate methods involving the Siemon and the Mundry approaches are used in this study to generate a resistivity-depth image of underground geologically plausible porphyry unit derived from airborne electromagnetic data. The 3D visualization of the 1D inverted resistivity models along all flight lines provides a resistive geological unit which corresponds to the desired porphyry andesite. To reduce uncertainty arising from single geophysical model, i.e., the resistivity model acquired from the frequency domain electromagnetic data, a fast implementable approach for 3D inversion of magnetic data called the Lanczos bidiagonalization method is also applied to the large scale airborne magnetic data in order to construct a 3D distribution model of magnetic susceptibility, by which the obtained model consequently confirms the extension of an arc-shaped porphyry andesite at depth. The susceptible-resistive porphyry andesite model provided by integrated geophysical data indicates a thicker structure than what is shown on the geological map while extends down at depth. As a result, considering simultaneous interpretation of airborne magnetic and frequency domain electromagnetic data certainly yield lower uncertainty in the modeling of andesite unit as a potential source of copper occurrences.

Fast inversion of magnetic data using Lanczos bidiagonalization method

Abstract This paper describes application of a fast inversion method to recover a 3D susceptibility model from magnetic anomalies. For this purpose, the survey area is divided into a large number of rectangular prisms in a mesh with unknown susceptibilities. Solving the full set of equations is substantially time consuming, and applying an algorithm to solve it approximately can reduce the time significantly. It is shown that the Lanczos bidiagonalization method can be an appropriate algorithm to solve a Tikhonov cost function for this purpose. Running time of the inverse modeling significantly decreases by replacing the forward operator matrix with a matrix of lower dimension. A weighted generalized cross validation method is implemented to choose an optimum value of a regularization parameter. To avoid the natural tendency of magnetic structures to concentrate at shallow depth, a depth weighting is applied. This study assumes that there is no remanent magnetization. The method is applied on a noise-corrupted synthetic data to demonstrate its suitability for 3D inversion. A case study including ground based measurement of magnetic anomalies over a porphyry-Cu deposit located in Kerman providence of Iran, Now Chun deposit, is provided to show the performance of the new algorithm on real data. 3D distribution of Cu concentration is used to evaluate the obtained results. The intermediate susceptibility values in the constructed model coincide with the known location of copper mineralization.

An implicitly restarted block Lanczos bidiagonalization method using Leja shifts

In this paper, we propose an implicitly restarted block Lanczos bidiagonalization (IRBLB) method for computing a few extreme or interior singular values and associated right and left singular vectors of a large matrix A. Our method combines the advantages of a block routine, implicit shifting, and the application of Leja points as shifts in the accelerating polynomial. The method neither requires factorization of A nor of a matrix that contains A. This makes the method well suited for very large matrices. For many matrices, the method presented in this paper requires less computational work and computer storage than other available approaches. Computed examples illustrate the performance of the method described.

A harmonic Lanczos bidiagonalization method for computing interior singular triplets of large matrices

We propose a harmonic Lanczos bidiagonalization method for computing some interior singular triplets of large matrices. We prove that the approximate singular triplet is convergent if the norm of a certain Rayleigh quotient matrix is uniformly bounded and the approximate singular values are well separated. Combining with the implicit restarting technique, we develop an implicitly restarted harmonic Lanczos bidiagonalization algorithm and suggest a strategy to select shifts. Numerical experiments show that one can use this algorithm to compute the interior singular triplets efficiently.

A Refined Harmonic Lanczos Bidiagonalization Method and an Implicitly Restarted Algorithm for Computing the Smallest Singular Triplets of Large Matrices

The harmonic Lanczos bidiagonalization method can be used to compute the smallest singular triplets of a large matrix $A$. We prove that for good enough projection subspaces harmonic Ritz values converge if the columns of $A$ are strongly linearly independent. On the other hand, harmonic Ritz values may miss some desired singular values when the columns of $A$ almost linearly dependent. Furthermore, harmonic Ritz vectors may converge irregularly and even may fail to converge. Based on the refined projection principle for large matrix eigenproblems due to the first author, we propose a refined harmonic Lanczos bidiagonalization method that takes the Rayleigh quotients of the harmonic Ritz vectors as approximate singular values and extracts the best approximate singular vectors, called the refined harmonic Ritz approximations, from the given subspaces in the sense of residual minimizations. The refined approximations are shown to converge to the desired singular vectors once the subspaces are sufficiently good and the Rayleigh quotients converge. An implicitly restarted refined harmonic Lanczos bidiagonalization algorithm (IRRHLB) is developed. We study how to select the best possible shifts, and suggest refined harmonic shifts that are theoretically better than the harmonic shifts used within the implicitly restarted Lanczos bidiagonalization algorithm (IRHLB). We propose a novel procedure that can numerically compute the refined harmonic shifts efficiently and accurately. Numerical experiments are reported that compare IRRHLB with five other algorithms based on the Lanczos bidiagonalization process. It appears that IRRHLB is at least competitive with them and can be considerably more efficient when computing the smallest singular triplets.

Augmented Implicitly Restarted Lanczos Bidiagonalization Methods

New restarted Lanczos bidiagonalization methods for the computation of a few of the largest or smallest singular values of a large matrix are presented. Restarting is carried out by augmentation of Krylov subspaces that arise naturally in the standard Lanczos bidiagonalization method. The augmenting vectors are associated with certain Ritz or harmonic Ritz vectors. Computed examples show the new methods to be competitive with available schemes.