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On The Choice Of Subspace For Iterative Methods For Linear Discrete IllPosed Problems
 Int. J. Appl. Math. Comput. Sci
, 2001
"... . Many iterative methods for the solution of linear discrete illposed problems with a large matrix require the computed approximate solutions to be orthogonal to the null space of the matrix. We show that it may be possible to determine a meaningful approximate solution with less computational work ..."
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Cited by 25 (20 self)
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. Many iterative methods for the solution of linear discrete illposed problems with a large matrix require the computed approximate solutions to be orthogonal to the null space of the matrix. We show that it may be possible to determine a meaningful approximate solution with less computational work when this requirement is not imposed. Key words. Minimal residual method, conjugate gradient method, linear illposed problems. 1. Introduction. This paper is concerned with the design of iterative methods for the computation of approximate solutions of linear systems of equations Ax = b, A # R mn , x # R n , b # R m , (1.1) with a large matrix A of illdetermined rank. Thus, A has many "tiny" singular values of di#erent orders of magnitude. In particular, A is severely illconditioned. Some of the singular values of A may be vanishing. We allow m # n or m < n. The righthand side vector b is not required to be in the range of A. Linear systems of equations of the fo...
A LargeScale TrustRegion Approach to the Regularization of Discrete IllPosed Problems
 RICE UNIVERSITY
, 1998
"... We consider the problem of computing the solution of largescale discrete illposed problems when there is noise in the data. These problems arise in important areas such as seismic inversion, medical imaging and signal processing. We pose the problem as a quadratically constrained least squares pro ..."
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Cited by 19 (8 self)
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We consider the problem of computing the solution of largescale discrete illposed problems when there is noise in the data. These problems arise in important areas such as seismic inversion, medical imaging and signal processing. We pose the problem as a quadratically constrained least squares problem and develop a method for the solution of such problem. Our method does not require factorization of the coefficient matrix, it has very low storage requirements and handles the high degree of singularities arising in discrete illposed problems. We present numerical results on test problems and an application of the method to a practical problem with real data.
Iterative Exponential Filtering for Large Discrete IllPosed Problems
"... We describe a new iterative method for the solution of large, very illconditioned linear systems of equations that arise when discretizing linear illposed problems. The righthand side vector represents the given data and is assumed to be contaminated by measurement errors. Our method applies a fi ..."
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Cited by 14 (7 self)
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We describe a new iterative method for the solution of large, very illconditioned linear systems of equations that arise when discretizing linear illposed problems. The righthand side vector represents the given data and is assumed to be contaminated by measurement errors. Our method applies a filter function of the form ' fi (t) := 1 \Gamma exp(\Gammafit ) with the purpose of reducing the influence of the errors in the righthand side vector on the computed approximate solution of the linear system. Here fi is a regularization parameter. The iterative method is derived by expanding ' fi (t) in terms of Chebyshev polynomials. The method requires only little computer memory and is well suited for the solution of largescale problems. We also show how a value of fi and an associated approximate solution that satisfies the Morozov discrepancy principle can be computed efficiently. An application to image restoration illustrates the performance of the method.
LanczosBased Exponential Filtering for Discrete IllPosed Problems
, 2002
"... We describe regularizing iterative methods for the solution of large illconditioned linear systems of equations that arise from the discretization of linear illposed problems. The regularization is specified by a filter function of Gaussian type. A parameter determines the amount of regularization ..."
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Cited by 13 (5 self)
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We describe regularizing iterative methods for the solution of large illconditioned linear systems of equations that arise from the discretization of linear illposed problems. The regularization is specified by a filter function of Gaussian type. A parameter determines the amount of regularization applied. The iterative methods are based on a truncated Lanczos decomposition and the filter function is approximated by a linear combination of Lanczos polynomials. A suitable value of the regularization parameter is determined by an Lcurve criterion. Computed examples that illustrate the performance of the methods are presented.
Smooth Or Abrupt: A Comparison of Regularization Methods
, 1998
"... In this paper we compare a new regularizing scheme based on the exponential filter function with two classical regularizing methods: Tikhonov regularization and a variant of truncated singular value regularization. The filter functions for the former methods are smooth, but for the latter discontinu ..."
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Cited by 4 (1 self)
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In this paper we compare a new regularizing scheme based on the exponential filter function with two classical regularizing methods: Tikhonov regularization and a variant of truncated singular value regularization. The filter functions for the former methods are smooth, but for the latter discontinuous. These regularization methods are applied to the restoration of images degraded by blur and noise. The norm of the noise is assumed to be known, and this allows application of the Morozov discrepancy principle to determine the amount of regularization. We compare the restored images produced by the three regularization methods with optimal values of the regularization parameter. This comparison sheds light on the how these different approaches are related. Keywords: exponential filter, Tikhonov regularization, truncated singular value decomposition 1. INTRODUCTION This paper compares three regularization methods for the solution of linear systems of equations Ax = ~ g; A 2 R n\Thetan...
Approximate Solution of LargeScale Linear Inverse Problems with Monte Carlo Simulation
, 2009
"... We consider the approximate solution of linear illposed inverse problems of high dimension with a simulationbased algorithm that approximates the solution within a lowdimensional subspace. The algorithm uses Tikhonov regularization, regression, and lowdimensional linear algebra calculations and ..."
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Cited by 1 (1 self)
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We consider the approximate solution of linear illposed inverse problems of high dimension with a simulationbased algorithm that approximates the solution within a lowdimensional subspace. The algorithm uses Tikhonov regularization, regression, and lowdimensional linear algebra calculations and storage. For sampling efficiency, we use variance reduction/importance sampling schemes, specially tailored to the structure of inverse problems. We demonstrate the implementation of our algorithm in a series of practical largescale examples arising from Fredholm integral equations of the first kind.
SimulationBased Approximate Solution of LargeScale Linear Least Squares Problems and Applications
, 2010
"... We consider linear least squares problems, or linear systems that can be formulated into least squares problems, of very large dimension, such as those arising for example in dynamic programming (DP) and inverse problems. We introduce an associated approximate problem, within a subspace spanned by ..."
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We consider linear least squares problems, or linear systems that can be formulated into least squares problems, of very large dimension, such as those arising for example in dynamic programming (DP) and inverse problems. We introduce an associated approximate problem, within a subspace spanned by a relatively small number of basis functions, and solution methods that use simulation, importance sampling, and lowdimensional calculations. The main components of this methodology are a regression/regularization approach that can deal with nearly singular problems, and an importance sampling design approach that exploits existing continuity structures in the underlying models, and allows the solution of very large problems. We also investigate the use of our regression/regularization approach in temporal differencetype methods in the context of approximate DP. Finally we demonstrate the application of our methodology in a series of practical largescale examples arising from Fredholm integral equations of the first kind.
Summary
, 2009
"... A plant growthpromoting rhizobacterium (Azospirillum brasilense strain Az) and a biocontrol fungus (Trichoderma harzianum strain T24) have been evaluated for their individual and combined production of hydrolytic enzymes, nitrogen fixation and their possible role in growth promotion of tomato seed ..."
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A plant growthpromoting rhizobacterium (Azospirillum brasilense strain Az) and a biocontrol fungus (Trichoderma harzianum strain T24) have been evaluated for their individual and combined production of hydrolytic enzymes, nitrogen fixation and their possible role in growth promotion of tomato seedlings. The studied organisms were inoculated as free or calcium alginateencapsulated cells. All freshly prepared macrobeads showed high encapsulation capacity (EC/%) of inocula compared with dry macrobeads. Results of enzyme production did not exhibit consistent pattern of the effect of encapsulation process on enzyme production. Beads entrapping bacterial and/or fungal cells were used successfully in 3 repeated cycles in the presence of fresh sterile culture medium in each growth cycle. Enzyme production by immobilized bacterial and/or fungal cells increased as the growth cycles were repeated. Coculturing of A. brasilense with T. harzianum (free or immobilized) in semisolid nitrogen deficient medium (Nfree medium) enabled A. brasilense to fix nitrogen on pectin, chitin and carboxymethyl cellulose. The activity of nitrogen fixation by A. brasilense in the case of single and combined cultures with Trichoderma (using dry encapsulated beads) into the sterile soil increased with the addition of carbon source. Most of ino