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New interpolants for asymptotically correct defect control of BVODEs
, 2008
"... The defect of a continuous approximate solution to an ODE is the amount by which that approximation fails to satisfy the ODE. A number of studies have explored the use of asymptotically correct defect estimates in the numerical solution of initial value ODEs (IVODEs). By employing an appropriately c ..."
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The defect of a continuous approximate solution to an ODE is the amount by which that approximation fails to satisfy the ODE. A number of studies have explored the use of asymptotically correct defect estimates in the numerical solution of initial value ODEs (IVODEs). By employing an appropriately constructed interpolant to an approximate discrete solution to the ODE, various researchers have shown that it is possible to obtain estimates of the local error and/or the maximum defect that are asymptotically correct on each step, as the stepsize h → 0. In this paper, we investigate the usefulness of asymptotically correct defect estimates for defect control in boundary value ODE (BVODE) codes. In the BVODE context, for a sequence of meshes which partition the problem interval, one computes a discrete numerical solution, constructs an interplant, and estimates the maximum defect. The estimates (typically obtained by sampling the defect at a small number of points on each subinterval of the mesh) are used in a redistribution process to determine the next mesh and thus the availability of these more reliable maximum defect estimates can lead to improved meshes. As well, when such estimates are available, the code can terminate with more confidence that the defect is bounded throughout the problem domain by the userprescribed tolerance. In this paper we employ a bootstrapping approach to derive interpolants that allow asymptotically correct defect estimates. Numerical results are included to demonstrate the validity of this approach. Subject Classification: 65L05, 65L10. Keywords: RungeKutta schemes, boundary value ODEs, defect control.
Stationary solutions of driven fourth and sixthorder CahnHilliard type equations
, 2008
"... New types of stationary solutions of a onedimensional driven sixthorder CahnHilliard type equation that arises as a model for epitaxially growing nanostructures such as quantum dots, are derived by an extension of the method of matched asymptotic expansions that retains exponentially small terms. ..."
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New types of stationary solutions of a onedimensional driven sixthorder CahnHilliard type equation that arises as a model for epitaxially growing nanostructures such as quantum dots, are derived by an extension of the method of matched asymptotic expansions that retains exponentially small terms. This method yields analytical expressions for farfield behavior as well as the widths of the humps of these spatially nonmonotone solutions in the limit of small driving force strength which is the deposition rate in case of epitaxial growth. These solutions extend the family of the monotone kink and antikink solutions. The hump spacing is related to solutions of the Lambert W function. Using phase space analysis for the corresponding fifthorder dynamical system, we use a numerical technique that enables the efficient and accurate tracking of the solution branches, where the asymptotic solutions are used as initial input. Additionally, our approach is first demonstrated for the related but simpler driven fourthorder CahnHilliard equation, also known as the convective CahnHilliard equation. Keywords: convective CahnHilliard, quantum dots, exponential asymptotics, matching, dynamical systems 1
Singular Perturbation Problems
, 2009
"... Abstract: We propose a simple and quite efficient code to solve singular perturbation problems when the perturbation parameter ǫ is very small. The code is based on generalized upwind methods of order ranging from 4 to 10 and uses highly variable stepsize to fit the boundary regions with relatively ..."
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Abstract: We propose a simple and quite efficient code to solve singular perturbation problems when the perturbation parameter ǫ is very small. The code is based on generalized upwind methods of order ranging from 4 to 10 and uses highly variable stepsize to fit the boundary regions with relatively few points. An extensive numerical test section shows the effectiveness of the proposed technique on linear problems.
MPRA Paper No. 50203, posted 25. September 2013 22:44 UTCSwitching from Patents to an Intertemporal Bounty in a NonScale Growth Model: Transitional Dynamics and Welfare Evaluation ∗
, 2013
"... The prize system for innovation has been criticized as impractical due to the lack of a workable formula or algorithm to determine the size of prizes. In this paper, a decentralized market mechanism via the intertemporal bounty (IB) system can function to duplicate Pareto optimality. Under this syst ..."
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The prize system for innovation has been criticized as impractical due to the lack of a workable formula or algorithm to determine the size of prizes. In this paper, a decentralized market mechanism via the intertemporal bounty (IB) system can function to duplicate Pareto optimality. Under this system, any bountiable innovation is placed in the public domain, and the prize of innovation is dynamically amortized in an infinitely time domain as periodic bounties paid to holders of bounty claims. Periodic bounties are calculated using a governmentdetermined bounty rate times observed market sales. Two formulas are derived to calculate “longrun Pareto optimal bounty rate ” and “longrun suboptimal bounty rate. ” The former can correct monopoly distortions and externalities, while the latter can only address monopoly distortions. They are empirically computable and can serve as an upper bound and the lower bound of the bounty rate. This paper provides a dynamic generalequilibrium analysis of replacing finitelylived patents with the IB system using either of these two bounty rates. Based on a nonscale growth model calibrated to the US economy, transition paths are worked out to compute welfare gains.
A PETSc Parallel BVP Code
"... Solutions to ordinary differential equations with boundary conditions (BVPs) are often approximated by a numerical method. Typically, this method involves setting up a large system of nonlinear equations for which the unknowns are discrete numerical approximations of the solution to the BVP. This no ..."
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Solutions to ordinary differential equations with boundary conditions (BVPs) are often approximated by a numerical method. Typically, this method involves setting up a large system of nonlinear equations for which the unknowns are discrete numerical approximations of the solution to the BVP. This nonlinear system has the potential to be large. Therefore, routines that parallelize both the generation and solution of the nonlinear system could reduce the overall computational time. This report describes a BVP code called PETSc BVP that contains parallel implementations of both of these routines. The parallel scientific toolbox PETSc is used to create the code. The report also describes the results of several numerical experiments used to show that for certain problems, a reduction of overall computational time is achieved when multiple processors are used to solve the problem. 1
Switching from Patents to an Intertemporal Bounty in a NonScale Growth Model
, 2013
"... The prize system for innovation has been criticized as impractical due to the lack of a workable formula or algorithm to determine the size of prizes. In this paper, a decentralized market mechanism via the intertemporal bounty (IB) system can function to duplicate Pareto optimality. Under this syst ..."
Abstract
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The prize system for innovation has been criticized as impractical due to the lack of a workable formula or algorithm to determine the size of prizes. In this paper, a decentralized market mechanism via the intertemporal bounty (IB) system can function to duplicate Pareto optimality. Under this system, any bountiable innovation is placed in the public domain, and the prize of innovation is dynamically amortized in an infinitely time domain as periodic bounties paid to holders of bounty claims. Periodic bounties are calculated using a governmentdetermined bounty rate times observed market sales. Two formulas are derived to calculate “longrun Pareto optimal bounty rate ” and “longrun suboptimal bounty rate. ” The former can correct monopoly distortions and externalities, while the latter can only address monopoly distortions. They are empirically computable and can serve as an upper bound and the lower bound of the bounty rate. This paper provides a dynamic generalequilibrium analysis of replacing finitelylived patents with the IB system using either of these two bounty rates. Based on a nonscale growth model calibrated to the US economy, transition paths are worked out to compute welfare gains. [JEL Classification: C63, O31, O34]
A RungeKutta BVODE Solver . . .
, 2013
"... Boundary value ordinary differential equations (BVODEs) are systems of ODEs with boundary conditions imposed at two or more distinctpoints. The global error(GE) of a numerical solution to a BVODE is the amount by which the numerical solution differs from the exact solution. Thedefect is the amount b ..."
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Boundary value ordinary differential equations (BVODEs) are systems of ODEs with boundary conditions imposed at two or more distinctpoints. The global error(GE) of a numerical solution to a BVODE is the amount by which the numerical solution differs from the exact solution. Thedefect is the amount by which the numerical solution fails to satisfy the ODEs and boundary conditions. Although GE control is often familiar to users, the defect controlled numerical solution can be interpreted as the exact solution to a perturbation of the original BVODE. Software packages based on GE control and on defect control are in wide use. The defect control solver, BVP SOLVER, can provide an a posteriori estimate of the GE using Richardson extrapolation. In this paper, we consider three more strategies for GE estimation based on (i) the direct use of a higher order discretization formula (HO), (ii) the use of a higher order discretization formula within a deferred correction (DC) framework, and (iii) the product of an estimate of the maximum defect and an estimate of the BVODE conditioning constant, and demonstrate that the HO and DC approaches have superior performance. We also modify BVP SOLVER to introduce GE control.