Control Perspectives on Numerical Algorithms and Matrix Problems

Bhaya, Amit; Kaszkurewicz, E.

doi:10.1137/1.9780898718669

Cited by 88 publications

(101 citation statements)

References 0 publications

Supporting

Mentioning

100

Contrasting

Unclassified

Order By: Relevance

“…As discussed above, researchers have tried to manage these dynamics through the time step (Tocci et al, 1997;Kavetski et al, 2002;D'Haese et al, 2007) and/or linearization technique (Paniconi and Putti, 1994;Lehmann and Ackerer, 1998;Lipnikov et al, 2016). Alternative linearization approaches (e.g., List and Radu, 2016) and better algorithms for managing the complex dynamics of adaptive solvers taken, for example, from control-theory (Bhaya and Kaszkurewicz, 2006) could directly improve existing production codes and also lower the barrier to adoption of more accurate spatial approximations.…”

Section: Discussion and Alternativesmentioning

confidence: 99%

Numerical Solution of Richards' Equation: A Review of Advances and Challenges

Farthing

Ogden

2017

Soil Science Soc of Amer J

245

138

View full text Add to dashboard Cite

The flow of water in partially saturated porous media is of importance in fields such as hydrology, agriculture, environment and waste management. It is also one of the most complex flows in nature. The Richards' equation describes the flow of water in an unsaturated porous medium due to the actions of gravity and capillarity neglecting the flow of the non-wetting phase, usually air. Analytical solutions of Richards' equation exist only for simplified cases, so most practical situations require a numerical solution in one-two-or threedimensions, depending on the problem and complexity of the flow situation. Despite the fact that the first reasonably complete conservative numerical solution method was published in the early 1990s, the numerical solution of the Richards' equation remains computationally expensive and in certain circumstances, unreliable. A universally robust and accurate solution methodology has not yet been identified that is applicable across the range of soils, initial and boundary conditions found in practice. Existing solution codes have been modified over years to attempt to increase robustness. Despite theoretical results on the existence of solutions given sufficiently regular data and constitutive relations, our numerical methods often fail to demonstrate reliable convergence behavior in practice, especially for higher-order methods. Because of robustness, the lack of higher-order accuracy and computational expense, alternative solution approaches or methods are needed. There is also a need for better documentation of improved solution methodologies and benchmark test problems to facilitate consistent advances and avoid re-inventing of the wheel.T his review paper is intended to serve as a touchstone on the state of the science of calculating the flow of water through unsaturated porous media. As active researchers we believe it is good to occasionally take stock in what has been accomplished up to date, where things may be headed, and what important issues remain. This review concentrates on computational modeling of flow through the unsaturated zone via Richards' equation.This effort can help provide some focus to the research community and serve to help propel new research forward, particularly by those just joining the field. There are many practical problems that require calculation of one-, two-, and threedimensional fluxes in the unsaturated zone. For a much broader review of modeling soil processes, we recommend the recent paper from Vereecken et al. (2016). A detailed review of mathematical models of infiltration can be found (Assouline, 2013), while Paniconi and Putti (2015) provide historical perspective and an overview of modeling Richards' equation in the context of catchment hydrology.The equation attributed to Richards (1931) that describes the flow of water through unsaturated porous media under the action of capillarity and gravity was first published by the English mathematician and physicist Lewis Fry Richardson in 1922(Richardson, 1922. Therefore, the equation would r...

show abstract

Section: Discussion and Alternativesmentioning

confidence: 99%

Numerical Solution of Richards' Equation: A Review of Advances and Challenges

Farthing

Ogden

2017

Soil Science Soc of Amer J

245

138

View full text Add to dashboard Cite

show abstract

“…See [39,1,12,71,47,37,55], as well as the review paper [25] and the feedback control interpretation in [10,11].…”

Section: Iterative Methods and Forward Eulermentioning

confidence: 99%

Artificial time integration

2007

View full text Add to dashboard Cite

Abstract.Many recent algorithmic approaches involve the construction of a differential equation model for computational purposes, typically by introducing an artificial time variable. The actual computational model involves a discretization of the now time-dependent differential system, usually employing forward Euler. The resulting dynamics of such an algorithm is then a discrete dynamics, and it is expected to be "close enough" to the dynamics of the continuous system (which is typically easier to analyze) provided that small -hence many -time steps, or iterations, are taken. Indeed, recent papers in inverse problems and image processing routinely report results requiring thousands of iterations to converge. This makes one wonder if and how the computational modeling process can be improved to better reflect the actual properties sought.In this article we elaborate on several problem instances that illustrate the above observations. Algorithms may often lend themselves to a dual interpretation, in terms of a simply discretized differential equation with artificial time and in terms of a simple optimization algorithm; such a dual interpretation can be advantageous. We show how a broader computational modeling approach may possibly lead to algorithms with improved efficiency.

show abstract

“…It is also appropriate for use with large, sparse matrices, since it uses only matrix-vector products in its computations. Notice, however, that the CG algorithm works with a state space of dimension twice that of the vector x, since it uses coupled iterations in both r and p. This is interpreted from a control perspective in [9,10]. Note also that the coupling between the r and p iterations (5) is of the Gauss-Seidel type, i.e, the updated value r k+1 is used immediately after it is computed in the p update.…”

Section: Preliminariesmentioning

confidence: 99%

Cooperative concurrent asynchronous computation of the solution of symmetric linear systems

2016

Self Cite

View full text Add to dashboard Cite

This paper extends the idea of Brezinski's hybrid acceleration procedure, for the solution of a system of linear equations with a symmetric coefficient matrix of dimension n, to a new context called cooperative computation, involving m agents (m n), each one concurrently computing the solution of the whole system, using an iterative method. Cooperation occurs between the agents through the communication, periodic or probabilistic, of the estimate of each agent to one randomly chosen agent, thus characterizing the computation as concurrent and asynchronous. Every time one agent receives solution estimates from the others, it carries out a least squares computation, involving a small linear system of dimension m, in order to replace its current solution estimate by an affine combination of the received estimates, and the algorithm continues until a stopping criterion is met. In addition, an autocooperative algorithm, in which estimates are updated using affine combinations of current and past estimates, is also proposed. The proposed algorithms are shown to be efficient for certain matrices, specifically in relation to the popular Barzilai-Borwein algorithm, through numerical experiments.

show abstract

Control Perspectives on Numerical Algorithms and Matrix Problems

Cited by 88 publications

References 0 publications

Numerical Solution of Richards' Equation: A Review of Advances and Challenges

Numerical Solution of Richards' Equation: A Review of Advances and Challenges

Artificial time integration

Cooperative concurrent asynchronous computation of the solution of symmetric linear systems

Contact Info

Product

Resources

About