scholarly journals Projecting onto rectangular matrices with prescribed row and column sums

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Heinz H. Bauschke ◽  
Shambhavi Singh ◽  
Xianfu Wang
Keyword(s):  
Linear Operator ◽  
General Problem ◽  
Projection Operator ◽  
Numerical Experiments

AbstractIn 1990, Romero presented a beautiful formula for the projection onto the set of rectangular matrices with prescribed row and column sums. Variants of Romero’s formula were rediscovered by Khoury and by Glunt, Hayden, and Reams for bistochastic (square) matrices in 1998. These results have found various generalizations and applications.In this paper, we provide a formula for the more general problem of finding the projection onto the set of rectangular matrices with prescribed scaled row and column sums. Our approach is based on computing the Moore–Penrose inverse of a certain linear operator associated with the problem. In fact, our analysis holds even for Hilbert–Schmidt operators, and we do not have to assume consistency. We also perform numerical experiments featuring the new projection operator.

THE MODIFIED TIKHONOV REGULARIZATION METHOD WITH THE SMOOTHED TOTAL VARIATION FOR SOLVING THE LINEAR ILL-POSED PROBLEMS

2021 ◽  
Vol 9 (2) ◽  
pp. 88-100
Author(s):  
Vladimir Vasin ◽  
◽  
Vladimir Belyaev
Keyword(s):  
Linear Operator ◽  
Total Variation ◽  
Operator Equation ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Numerical Experiments ◽  
Approximate Solutions ◽  
Tikhonov Regularization Method ◽  
Smooth Component ◽  
Ill Posed

We investigate a linear operator equation of the first kind that is ill-posed in the Hadamard sence. It is assumed that its solution is representable as a sum of smooth and discontinuous components. To construct a stable approximate solutions, we use the modified Tikhonov method with the stabilizing functional as a sum of the Lebesgue norm for the smooth component and a smoothed BV-norm for the discontinuous component. Theorems of exis- tence, uniqueness, and convergence both the regularized solutions and its finite-dimentional approximations are proved. Also, results of numerical experiments are presented.

Effective secondary task execution of redundant manipulators

Robotica ◽  
1998 ◽  
Vol 16 (4) ◽  
pp. 457-462 ◽  
Author(s):  
Jadran Lenarčič
Keyword(s):  
Projection Operator ◽  
Secondary Task ◽  
Numerical Experiments ◽  
Null Space ◽  
Numerical Procedure ◽  
Redundant Manipulators ◽  
Space Operator ◽  
Task Execution ◽  
Weighting Matrix ◽  
Space Projection

In standard pseudoinverse-based approaches to treat redundant manipulators, the vector of joint increments that corresponds to a desired motion in the space of the secondary task is projected in the Jacobian null space associated with the primary task. In general, this projection may distort the projected vector, so that the secondary task may not adequately be executed. A usual remedy is to rotate the null space projection operator by using a special-purpose weighting matrix. The problem, however, is that this rotation cannot be enforced arbitrarily since it influences the manipulator's performance. In our work we propose an algorithm that is independent on the chosen null space operator and always provides the best attainable motion in the space of the secondary task. Hence, the secondary task is executed more efficiently and the numerical procedure is more robust. A series of numerical experiments confirmed these results.

scholarly journals Computing low-rank rightmost eigenpairs of a class of matrix-valued linear operators

2021 ◽  
Vol 47 (5) ◽  
Author(s):  
Nicola Guglielmi ◽  
Daniel Kressner ◽  
Carmela Scalone
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Numerical Methods ◽  
Linear Operator ◽  
Numerical Experiments ◽  
Linear Operators ◽  
New Method ◽  
Low Rank ◽  
General Linear ◽  
Whole Space

AbstractIn this article, a new method is proposed to approximate the rightmost eigenpair of certain matrix-valued linear operators, in a low-rank setting. First, we introduce a suitable ordinary differential equation, whose solution allows us to approximate the rightmost eigenpair of the linear operator. After analyzing the behaviour of its solution on the whole space, we project the ODE on a low-rank manifold of prescribed rank and correspondingly analyze the behaviour of its solutions. For a general linear operator we prove that—under generic assumptions—the solution of the ODE converges globally to its leading eigenmatrix. The analysis of the projected operator is more subtle due to its nonlinearity; when ca is self-adjoint, we are able to prove that the associated low-rank ODE converges (at least locally) to its rightmost eigenmatrix in the low-rank manifold, a property which appears to hold also in the more general case. Two explicit numerical methods are proposed, the second being an adaptation of the projector splitting integrator proposed recently by Lubich and Oseledets. The numerical experiments show that the method is effective and competitive.

A new iterative method for the split feasibility problem

2018 ◽  
Vol 34 (3) ◽  
pp. 313-320
Author(s):  
QIAO-LI DONG ◽  
◽  
DAN JIANG ◽  
Keyword(s):  
Inverse Problems ◽  
Linear Operator ◽  
Iterative Method ◽  
Projection Method ◽  
Numerical Experiments ◽  
Split Feasibility Problem ◽  
Feasibility Problem ◽  
The Split Feasibility Problem ◽  
Split Feasibility ◽  
New Iterative Method

The split feasibility problem (SFP) has many applications, which can be a model for many inverse problems where constraints are imposed on the solutions in the domain of a linear operator as well as in the operator’s range. In this paper, we introduce a new projection method to solve the SFP and prove its convergence under standard assumptions. Our results improve previously known corresponding methods and results of this area. The preliminary numerical experiments illustrates the advantage of our proposed methods.

Influence of Magnetic Fields on Solar Hydrodynamics: Experimental Results

1977 ◽  
Vol 36 ◽  
pp. 143-180 ◽  
Author(s):  
J.O. Stenflo
Keyword(s):  
Solar Activity ◽  
Energy Balance ◽  
Magnetic Fields ◽  
Solar Atmosphere ◽  
General Problem ◽  
Solar Rotation ◽  
Active Regions ◽  
Physical Conditions ◽  
Dynamic Phenomena

It is well-known that solar activity is basically caused by the Interaction of magnetic fields with convection and solar rotation, resulting in a great variety of dynamic phenomena, like flares, surges, sunspots, prominences, etc. Many conferences have been devoted to solar activity, including the role of magnetic fields. Similar attention has not been paid to the role of magnetic fields for the overall dynamics and energy balance of the solar atmosphere, related to the general problem of chromospheric and coronal heating. To penetrate this problem we have to focus our attention more on the physical conditions in the ‘quiet’ regions than on the conspicuous phenomena in active regions.

A Better Solution to the General Problem of Creation

2017 ◽  
Vol 9 (1) ◽  
pp. 147-162
Author(s):  
Jeremy W. Skrzypek
Keyword(s):  
Recent Work ◽  
General Problem ◽  
The State ◽  
State Of Affairs ◽  
Better Than

It is often suggested that, since the state of affairs in which God creates a good universe is better than the state of affairs in which He creates nothing, a perfectly good God would have to create that good universe. Making use of recent work by Christine Korgaard on the relational nature of the good, I argue that the state of affairs in which God creates is actually not better, due to the fact that it is not better for anyone or anything in particular. Hence, even a perfectly good God would not be compelled to create a good universe.

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