An ENO-Based Method for Second-Order Equations and Application to the Control of Dike Levels

Pijl, S.P. van der; Oosterlee, Cornelis W.

doi:10.1007/s10915-011-9493-3

Cited by 7 publications

(5 citation statements)

References 17 publications

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“…The time increment is denoted by t. The PDF p approximated at the node i and the time step n is denoted as p n i . The semi-discretized KFE (23) in the cell i (except at the boundary cells, 0, M and N) is Yoshioka and Unami (2013) [25] employed the fitting technique [14] for constructing fluxes on the cell interfaces. In this technique, fluxes on the cell interfaces is evaluated by using an exact solution to a two-point boundary value problem, leading to a stable discretization complying with the TVD property [21].…”

Section: Finite Volume Methodsmentioning

confidence: 99%

“…The stochastic impulse control problem aims at controlling a stochastic system, such as a diffusion process, through impulsive interventions so that a performance index is maximized or minimized [4,5,15,22,23]. Usually, stochastic impulse control problems lead to threshold-type optimal intervention strategies, such that the state variables are instantaneously transported from a threshold to another threshold.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Analysis and computation of probability density functions for a 1-D impulsively controlled diffusion process

Yaegashi

Yoshioka

Tsugihashi

et al. 2019

Comptes Rendus. Mathématique

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Section: Finite Volume Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Analysis and computation of probability density functions for a 1-D impulsively controlled diffusion process

Yaegashi

Yoshioka

Tsugihashi

et al. 2019

Comptes Rendus. Mathématique

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“…By Godunov's theorem [19], in the case of explicit linear schemes for the approximation of the linear advection equation, the monotonicity property restricts the scheme to be of order at most one. Also, in [34], a similar result is given for the approximation of a diffusion equation and order two. To the best of our knowledge, for general diffusions in more than one dimension, no monotone schemes of order higher than one are available in the literature.…”

Section: Introductionmentioning

confidence: 64%

High-order filtered schemes for time-dependent second order HJB equations

2018

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In this paper, we present and analyse a class of "filtered" numerical schemes for second order Hamilton-Jacobi-Bellman (HJB) equations. Our approach follows the ideas introduced in B.D. Froese and A.M. Oberman, Convergent filtered schemes for the Monge-Ampère partial differential equation, SIAM J. Numer. Anal., 51(1): 2013, and more recently applied by other authors to stationary or time-dependent first order Hamilton-Jacobi equations. For high order approximation schemes (where "high" stands for greater than one), the inevitable loss of monotonicity prevents the use of the classical theoretical results for convergence to viscosity solutions. The work introduces a suitable local modification of these schemes by "filtering" them with a monotone scheme, such that they can be proven convergent and still show an overall high order behaviour for smooth enough solutions. We give theoretical proofs of these claims and illustrate the behaviour with numerical tests from mathematical finance, focussing also on the use of backward differencing formulae for constructing the high order schemes. 1 arXiv:1611.04939v1 [math.NA] 15 Nov 2016 b, σ, f, are bounded, Lipschitz and Hölder continuous respectively in space and time, i.e. there is a constant K such that, for any ϕ ∈ {b, σ, f, }, sup a∈A sup (t,x) =(s,y)Moreover, the solution is then also Lipschitz continuous in space and locally 1/2-Hölder continuous in time [14]. The boundedness assumption can be removed, see also [14]. Moreover, in the paper we consider the equation on bounded numerical domains so that the boundedness assumption is automatically satisfied.In this article, we propose approximation schemes for (1.1) for which convergence is guaranteed in a general setting, and which exhibit high order convergence under sufficient regularity of the solution. As we will explain in the following, these are in a sense two conflicting goals, and we will meet them by application of a so-called "filter", an idea introduced in [18].The seminal work by Barles and Souganidis [6] establishes that a consistent and stable scheme converges to the viscosity solution of (1.1) if it is also monotone. That this is not simply a requirement of the proof, but can be crucial in practice, is demonstrated, e.g., by [30] (and also in Section 4.2 here). It is shown there empirically that for the uncertain volatility model from [25], perhaps the simplest non-trivial second order HJB equation there is, the (consistent and stable but) non-monotone Crank-Nicolson scheme fails to converge to the correct viscosity solution in the presence of Lipschitz initial data without higher regularity. This is in contrast to classical solutions where there is no such monotonicity requirement. As the setting of solutions above (i.e., Lipschitz in space and 1/2-Hölder in time) is the natural setting for HJB equations and often no higher global regularity is observed, a wide literature on monotone schemes has developed.By Godunov's theorem [19], in the case of explicit linear schemes for the approximation of the line...

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“…In part this is caused and exacerbated by the range of approaches falling under the ROA umbrella. Here, we distinguish between what we refer to as "traditional" ROA, which applies risk-based models from the financial options pricing literature (Dixit & Pindyck, 1994;Trigeorgis, 1995), and "scenario-based" ROA, which values investment flexibility within or across scenarios. Examples of the latter are scenario tree analysis (Conrad, 1980), which assign probabilities to climate scenarios as if these are probabilistic states of the world (e.g., Abadie, 2018).…”

Section: Origins Of Roamentioning

confidence: 99%

The added value of real options analysis for climate change adaptation

Wreford

Dittrich

Pol

2020

WIREs Climate Change

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Climate change adaptation investment decisions can be made more efficiently if uncertainty and new information are considered in their economic appraisal. Real options analysis (ROA) is a robust decision‐making tool that allows for the incorporation of both uncertainty and new information. In this opinion article, we argue that ROA is a valuable tool, providing the analysis is designed to reflect the real‐world characteristics of the decision context. We highlight the differences between traditional risk‐based ROA, and scenario‐based ROA, and discuss the relative merits of the approaches from the perspective of their assumptions and use of climate information. We also emphasize the need for increased co‐development of ROA design and applications with end‐users. Given the large climate uncertainties for long‐term adaptation planning, we suggest that an emerging strand of scenario‐based ROA methods offers ways to help identify and conditionally value flexibility without aggregating values into precise expected values across states of the world. This article is categorized under: Climate Economics > Iterative Risk‐Management Policy Portfolios

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An ENO-Based Method for Second-Order Equations and Application to the Control of Dike Levels

Cited by 7 publications

References 17 publications

Analysis and computation of probability density functions for a 1-D impulsively controlled diffusion process

Analysis and computation of probability density functions for a 1-D impulsively controlled diffusion process

High-order filtered schemes for time-dependent second order HJB equations

The added value of real options analysis for climate change adaptation

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