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The Inversion Problem for Computable Linear Operators [chapter]

Vasco Brattka
2003 Lecture Notes in Computer Science  
This is the inversion problem for computable linear operators on Banach spaces in its non-uniform and uniform formulation, respectively.  ...  Given a program of a linear bounded and bijective operator T , does there exist a program for the inverse operator T −1 ?  ...  Let Conclusion We have investigated the non-uniform and the uniform version of the inversion problem for computable linear operators.  ... 
doi:10.1007/3-540-36494-3_35 fatcat:hgmqdhdyovcgnbpm4hs65eps2m

PyLops—A linear-operator Python library for scalable algebra and optimization

Matteo Ravasi, Ivan Vasconcelos
2020 SoftwareX  
Linear operators and optimization are at the core of many algorithms used in signal and image processing, remote sensing, and inverse problems.  ...  However, many real-application, large-scale operators do not lend themselves to explicit matrix representations, usually forcing practitioners to forego the convenient linear-algebra syntax available for  ...  The authors cannot be held liable for any inappropriate use of this software library.  ... 
doi:10.1016/j.softx.2019.100361 fatcat:2lq4wzab2bfgvgb3ertis7uv2u

A compiler for linear algebra operations

Henrik Barthels
2016 Companion Proceedings of the 2016 ACM SIGPLAN International Conference on Systems, Programming, Languages and Applications: Software for Humanity - SPLASH Companion 2016  
In this paper we present a compiler that translates arithmetic expressions containing matrices to efficient sequences of calls to basic linear algebra kernels.  ...  The Problem Linear algebra problems appear in fields as diverse as computational biology, finite element methods and control theory.  ...  To deal with the inverse operator, in addition to computing X T X directly, different factorizations are applied to X.  ... 
doi:10.1145/2984043.2998539 dblp:conf/oopsla/Barthels16 fatcat:moxdr47onba3rpccxphibbine4

hIPPYlib: An Extensible Software Framework for Large-Scale Inverse Problems

Umberto Villa, Noemi Petra, Omar Ghattas
2018 Journal of Open Source Software  
A computational framework for infinite-dimensional Bayesian inverse problems: Part II. Stochastic Newton MCMC with application to ice sheet inverse problems.  ...  HIPPYlib: An extensible software framework for large-scale inverse problems; Part I: Deterministic inversion and linearized Bayesian inference. In preparation.  ... 
doi:10.21105/joss.00940 fatcat:z54j7jee7rbbpigas66lp7tpqq

Improving computational efficiency in large linear inverse problems: an example from carbon dioxide flux estimation

V. Yadav, A. M. Michalak
2013 Geoscientific Model Development  
This work proposes two approaches to lower the computational costs and memory requirements for large linear space–time inverse problems, taking the Bayesian approach for estimating carbon dioxide (CO<sub  ...  For an examined benchmark problem, the two algorithms yielded massive savings in floating point operations relative to direct computation of the same quantities.  ...  This material is based upon work supported by the National Aeronautics and Space Administration under Grant No. NNX12AB90G and the National Science Foundation under Grant No. 1047871. Edited by: L.  ... 
doi:10.5194/gmd-6-583-2013 fatcat:c46rnus3q5gntoeanyanjhsr3q

Page 3925 of Mathematical Reviews Vol. , Issue 91G [page]

1991 Mathematical Reviews  
(D-TUB) A mollifier method for linear operator equations of the first kind. Inverse Problems 6 (1990), no. 3, 427-440.  ...  Summary: “In this paper an inversion method for the solution of ill-posed linear problems is presented.  ... 

On Linear Representation, Complexity and Inversion of maps over finite fields [article]

Ramachandran Anantharaman, Virendra Sule
2022 arXiv   pre-print
The paper primarily addressed the problem of linear representation, invertibility, and construction of the compositional inverse for non-linear maps over finite fields.  ...  The linear representation, defined over the space of functions, naturally defines a notion of linear complexity for non-linear maps, which can be viewed as a measure of computational complexity associated  ...  The formula for inverse of a map F solves the problem of computing the inverse image x given y = F (x). Hence the problem of computing the inverse image is weaker than computing the inverse formula.  ... 
arXiv:2010.14601v4 fatcat:hvyat7spb5gi3c3sf7mixhcuoe

Optimal Experimental Design for Infinite-dimensional Bayesian Inverse Problems Governed by PDEs: A Review [article]

Alen Alexanderian
2021 arXiv   pre-print
We present the mathematical foundations of OED in this context and survey the computational methods for the class of OED problems under study.  ...  We present a review of methods for optimal experimental design (OED) for Bayesian inverse problems governed by partial differential equations with infinite-dimensional parameters.  ...  Thanks also to Noémi Petra for helpful discussions and feedback. I am grateful to Karina Koval for carefully reading through a draft of this article and for help with proofreading.  ... 
arXiv:2005.12998v3 fatcat:x2aeba2l4fh7ph6vlhw5zfpqke

A multilevel ℌ-matrix-based approximate matrix inversion algorithm for vectorless power grid verification

Wei Zhao, Yici Cai, Jianlei Yang
2013 2013 18th Asia and South Pacific Design Automation Conference (ASP-DAC)  
For most of the existing vectorless verification algorithms, the subproblem of linear system solution which computes the inverse of the power grid matrix takes up a large part of the computation time and  ...  Experimental results have shown that the proposed algorithm can obtain an almost linear complexity both in runtime and memory consumption for efficient vectorless power grid verification.  ...  If we use preconditioned conjugate gradient (PCG) methods to compute the corresponding row of the inverse matrix for each node, the whole linear system solution may take up about 80% of the computation  ... 
doi:10.1109/aspdac.2013.6509590 dblp:conf/aspdac/ZhaoCY13 fatcat:bb3vhyqmjzg4ticch2nnawtef4

Technical Note: Improving computational efficiency in large linear inverse problems: an example from carbon dioxide flux estimation

V. Yadav, A. M. Michalak
2012 Geoscientific Model Development Discussions  
This work proposes two approaches to lower the computational costs and memory requirements for large linear space-time inverse problems, taking the Bayesian approach for estimating car-10 bon dioxide (  ...  Such estimation problems often represent inverse problems that are formulated as linear optimization problems.  ...  This material is based upon work supported by the National Aeronautics  ... 
doi:10.5194/gmdd-5-3325-2012 fatcat:qlptmz6urzdndlxtu3snjndyei

Invariant imbedding and inverse problems with applications to radiative transfer

A.P. Wang, S. Ueno
1999 Computers and Mathematics with Applications  
The first method involves two auxiliary integro-differential equations. The inverse problem is solved by a sequence of approximation solutions of linear equations.  ...  ln this paper, we developed two methods to solve the inverse problem of a nonlinear integro-differential equation. Both methods are based on the principle of invariant imbedding.  ...  And, in Section 4, the problem is reduced to computing inverses of a 2n x 2n matrix and computing under operators. All of the above procedures can be carried out on a large or parallel computer.  ... 
doi:10.1016/s0898-1221(99)00149-2 fatcat:3efdhqlk2vb57hwj3huob2nxby

hIPPYlib: An Extensible Software Framework for Large-Scale Inverse Problems Governed by PDEs; Part I: Deterministic Inversion and Linearized Bayesian Inference [article]

Umberto Villa and Noemi Petra and Omar Ghattas
2020 arXiv   pre-print
algorithms for PDE-based inverse problems that exploit the structure of the underlying operators, notably the Hessian of the log-posterior.  ...  The key property of the algorithms implemented in hIPPYlib is that the solution of the deterministic and linearized Bayesian inverse problem is computed at a cost, measured in linearized forward PDE solves  ...  ACKNOWLEDGMENTS e authors would like to thank Georg Stadler for providing very helpful comments on an earlier version of this paper.  ... 
arXiv:1909.03948v2 fatcat:mslipsg4onhtlcpe7evjqi2j6y

Page 4781 of Mathematical Reviews Vol. , Issue 86j [page]

1986 Mathematical Reviews  
This paper deals with iterative methods for the linear operator equation (*) Az = b, where A is a linear, bounded, selfadjoint and nonnegative operator (i.e., (Az, z) > 0 for all z) in a Hilbert space  ...  This paper is a continuation of part I [see the preceding review]. The authors establish lower bounds for iterative procedures for solving linear ill-posed operator equations.  ... 

Regularization networks for inverse problems: A state-space approach

Giuseppe De Nicolao, Giancarlo Ferrari-Trecate
2003 Automatica  
An analytic expression is provided for the basis functions as well as for the entries of the matrix of the linear system used to compute the weights.  ...  Linear inverse problems with discrete data are equivalent to the estimation of the continuous-time input of a linear dynamical system from samples of its output.  ...  The main aim of the present paper is to provide an algorithm for the computation of the activation functions η i (t) that holds for linear inverse problems admitting a state-space representation.  ... 
doi:10.1016/s0005-1098(02)00280-7 fatcat:uecosjtwenetvldl2c3ccwxjvm

A Class of Generalized Approximate Inverse Solvers for Unsymmetric Linear Systems of Irregular Structure Based on Adaptive Algorithmic Modelling for Solving Complex Computational Problems in Three Space Dimensions

Anastasia-Dimitra Lipitakis
2016 Applied Mathematics  
in complex computational problems in three space dimensions.  ...  A class of general inverse matrix techniques based on adaptive algorithmic modelling methodologies is derived yielding iterative methods for solving unsymmetric linear systems of irregular structure arising  ...  problems in parallel computer environments.  ... 
doi:10.4236/am.2016.711108 fatcat:vqf3zrfcebasbcsxc427h4e5lu
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