Accuracy of multivalue methods during change of time‐step size: Adams‐Moulton

Multivalue methods are a class of time‐stepping procedures for numerical solution of differential equations that progress to a new time level using the approximate solution for the function of interest and its derivatives at a single time level. The methods differ from multistep procedures, which make use of solutions to the differential equation at multiple time levels to advance to the new time level. Multistep methods are difficult to employ when a change in time‐step is desired, because the standard formulas (e.g., Adams‐Moulton or Gear) must be modified to accommodate the change. Multivalue methods seem to possess the desirable feature that the time‐step may be changed arbitrarily as one proceeds, since the solution proceeds from a single time level. However, in practice, changes in the time‐step introduce lower order errors or alter the coefficient in the truncation error term. Here, the multivalue Adams‐Moulton method is presented based on a general interpolation procedure. Modifications required to retain the high‐order accuracy of these methods during a change in time‐step are developed. Additionally, a formula for the unknown initial derivatives is presented. Finally, two examples are provided to illustrate the potential merit of the modification to the standard multivalue methods. © 2000 John Wiley & Sons, Inc. Numer Methods Partials Differential Eq 16: 312–326, 2000     

Convergence of goal‐oriented adaptive finite element methods for nonsymmetric problems

In this article, we develop convergence theory for a class of goal‐oriented adaptive finite element algorithms for second‐order nonsymmetric linear elliptic equations. In particular, we establish contraction results for a method of this type for Dirichlet problems involving the elliptic operator with A Lipschitz, symmetric positive definite, with b divergence‐free, and with . We first describe the problem class and review some standard facts concerning conforming finite element discretization and error‐estimate‐driven adaptive finite element methods (AFEM). We then describe a goal‐oriented variation of standard AFEM. Following the recent work of Mommer and Stevenson for symmetric problems, we establish contraction and convergence of the goal‐oriented method in the sense of the goal function. Our analysis approach is signficantly different from that of Mommer and Stevenson, combining the recent contraction frameworks developed by Cascon, Kreuzer, Nochetto, and Siebert; by Nochetto, Siebert, and Veeser; and by Holst, Tsogtgerel, and Zhu. We include numerical results, demonstrating performance of our method with standard goal‐oriented strategies on a convection problem. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 479–509, 2016     

A Characterization of the degree sequences of 2‐trees

A graph G is a 2‐tree if G = K₃, or G has a vertex v of degree 2, whose neighbors are adjacent, and G/ v is a 2‐ tree. A characterization of the degree sequences of 2‐trees is given. This characterization yields a linear‐time algorithm for recognizing and realizing degree sequences of 2‐trees. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:191‐209, 2008     

Knowledge‐based scenario tree generation methods and application in multiperiod portfolio selection problem

A scenario tree is an efficient way to represent a stochastic data process in decision problems under uncertainty. This paper addresses how to efficiently generate appropriate scenario trees. A knowledge‐based scenario tree generation method is proposed; the new method is further improved by accounting for subjective judgements or expectations about the random future. Compared with existing approaches, complicated mathematical models and time‐consuming estimation, simulation and optimization problem solution are avoided in our knowledge‐based algorithms, and large‐scale scenario trees can be quickly generated. To show the advantages of the new algorithms, a multiperiod portfolio selection problem is considered, and a dynamic risk measure is adopted to control the intermediate risk, which is superior to the single‐period risk measure used in the existing literature. A series of numerical experiments are carried out by using real trading data from the Shanghai stock market. The results show that the scenarios generated by our algorithms can properly represent the underlying distribution; our algorithms have high performance, say, a scenario tree with up to 10,000 scenarios can be generated in less than a half minute. The applications in the multiperiod portfolio management problem demonstrate that our scenario tree generation methods are stable, and the optimal trading strategies obtained with the generated scenario tree are reasonable, efficient and robust. Copyright © 2013 John Wiley & Sons, Ltd.     

Modulus‐based synchronous multisplitting methods for solving horizontal linear complementarity problems on parallel computers

In this article, we generalize modulus‐based synchronous multisplitting methods to horizontal linear complementarity problems. In particular, first we define the methods of our concern and prove their convergence under suitable smoothness assumptions. Particular attention is devoted also to modulus‐based multisplitting accelerated overrelaxation methods. Then, as multisplitting methods are well‐suited for parallel computations, we analyze the parallel behavior of the proposed procedures. In particular, we do so by solving various test problems by a parallel implementation of our multisplitting methods. In this context, we carry out parallel computations on GPU with CUDA.     

On the Modeling of CO2 EUA and CER Prices of EU‐ETS for the 2008–2012 Period

Increased consumption of fossil fuels in industrial production has led to a significant elevation in the emission of greenhouse gases and to global warming. The most effective international action against global warming is the Kyoto Protocol, which aims to reduce carbon emissions to desired levels in a certain time span. Carbon trading is one of the mechanisms used to achieve the desired reductions. One of the most important implications of carbon trading for industrial systems is the risk of uncertainty about the prices of carbon allowance permits traded in the carbon markets. In this paper, we consider stochastic and time series modeling of carbon market prices and provide estimates of the model parameters involved, based on the European Union emissions trading scheme carbon allowances data obtained for 2008–2012 period. In particular, we consider fractional Brownian motion and autoregressive moving average–generalized autoregressive conditional heteroskedastic modeling of the European Union emissions trading scheme data and provide comparisons with benchmark models. Our analysis reveals evidence for structural changes in the underlying models in the span of the years 2008–2012. Data‐driven methods for identifying possible change‐points in the underlying models are employed, and a detailed analysis is provided. Our analysis indicated change‐points in the European Union Allowance (EUA) prices in the first half of 2009 and in the second half of 2011, whereas in the Certified Emissions Reduction (CER) prices three change‐points have appeared, in the first half of 2009, the middle of 2011, and in the second half of 2012. These change‐points seem to parallel the global economic indicators as well. Copyright © 2016 John Wiley & Sons, Ltd.     

An extended TODIM method for multiple attribute group decision‐making based on 2‐dimension uncertain linguistic Variable

The significant characteristic of the TODIM (an acronym in Portuguese of Interactive and Multiple Attribute Decision Making) method is that it can consider the bounded rationality of the decision makers. However, in the classical TODIM method, the rating of the attributes only can be used in the form of crisp numbers. Because 2‐dimension uncertain linguistic variables can easily express the fuzzy information, in this article, we extend the TODIM method to 2‐dimension uncertain linguistic information. First of all, the definition, characteristics, expectation, comparative method and distance of 2‐dimension uncertain linguistic information are introduced, and the steps of the classical TODIM method for Multiple attribute decision making (MADM) problems are presented. Second, on the basis of the classical TODIM method, the extended TODIM method is proposed to deal with MADM problems in which the attribute values are in the form of 2‐dimension uncertain linguistic variables, and detailed decision steps are given. Its significant characteristic is that it can fully consider the bounded rationality of the decision makers, which is a real action in real decision making. Finally, a numerical example is provided to verify the developed approach and its practicality and effectiveness. © 2014 Wiley Periodicals, Inc. Complexity 21: 20–30, 2016     

The upwind finite difference fractional steps methods for two‐phase compressible flow in porous media

The upwind finite difference fractional steps methods are put forward for the two‐phase compressible displacement problem. Some techniques, such as calculus of variations, multiplicative commutation rule of difference operators, decomposition of high‐order difference operators, and prior estimates, are adopted. Optimal order estimates in L² norm are derived to determine the error in the approximate solution. This method has already been applied to the numerical simulation of seawater intrusion and migration‐accumulation of oil resources. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 67–88, 2003     

Some new properties of asynchronous algorithms of theta scheme combined with finite elements methods for an evolutionary implicit 2‐sided obstacle problem

In this paper, we give some new properties of the presented asynchronous algorithms of theta scheme combined with finite elements methods (App. Math. Comp., 217 (2011), 6443‐6450) for an evolutionary implicit 2‐sided obstacle problem to prove the existence and uniqueness of the discrete solution. Furthermore, an error estimate on the uniform norm is given.