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The Inversion Problem for Computable Linear Operators
[chapter]

2003
*
Lecture Notes in Computer Science
*

This is

doi:10.1007/3-540-36494-3_35
fatcat:hgmqdhdyovcgnbpm4hs65eps2m
*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*. ...##
###
PyLops—A linear-operator Python library for scalable algebra and optimization

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. ...

##
###
A compiler for linear algebra operations

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

doi:10.1145/2984043.2998539
dblp:conf/oopsla/Barthels16
fatcat:moxdr47onba3rpccxphibbine4
*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. ...##
###
hIPPYlib: An Extensible Software Framework for Large-Scale Inverse Problems

2018
*
Journal of Open Source Software
*

A

doi:10.21105/joss.00940
fatcat:z54j7jee7rbbpigas66lp7tpqq
*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. ...##
###
Improving computational efficiency in large linear inverse problems: an example from carbon dioxide flux estimation

2013
*
Geoscientific Model Development
*

This work proposes two approaches to lower

doi:10.5194/gmd-6-583-2013
fatcat:c46rnus3q5gntoeanyanjhsr3q
*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. ...##
###
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]

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. ...

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

2021
*
arXiv
*
pre-print

We present

arXiv:2005.12998v3
fatcat:x2aeba2l4fh7ph6vlhw5zfpqke
*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. ...##
###
A multilevel ℌ-matrix-based approximate matrix inversion algorithm for vectorless power grid verification

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*...

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

2012
*
Geoscientific Model Development Discussions
*

This work proposes two approaches to lower

doi:10.5194/gmdd-5-3325-2012
fatcat:qlptmz6urzdndlxtu3snjndyei
*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 ...##
###
Invariant imbedding and inverse problems with applications to radiative transfer

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*. ...

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

2020
*
arXiv
*
pre-print

algorithms

arXiv:1909.03948v2
fatcat:mslipsg4onhtlcpe7evjqi2j6y
*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. ...##
###
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

2003
*
Automatica
*

An analytic expression is provided

doi:10.1016/s0005-1098(02)00280-7
fatcat:uecosjtwenetvldl2c3ccwxjvm
*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. ...##
###
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

2016
*
Applied Mathematics
*

in complex

doi:10.4236/am.2016.711108
fatcat:vqf3zrfcebasbcsxc427h4e5lu
*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. ...
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