Adaptive Mesh Selection Strategies for Solving Boundary Value Problems

Russell, Robert D.; Christiansen, J.

doi:10.1137/0715004

Cited by 190 publications

(64 citation statements)

References 17 publications

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“…In addition, in order to estimate and bound the discretized and roundoff errors, a wide spectrum of element placement constraints can be used, ranging from highly nonlinear constraints, based on the residual at noncollocation points, to simple ad hoc bounds on elements. An extensive review of these constraints can be found in Russell and Christiansen (1978). In this study, we incorporate the following relation, (67) where hi is step size, k is the order of the method, ||e(t)|| is approximated local error, and j|€(inc)l!…”

Section: Element Placement and Error Controlmentioning

confidence: 99%

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Stable Decomposition for Dynamic Optimization

Tanartkit

Biegler

1995

Ind. Eng. Chem. Res.

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Abstract, Dynamic optimisation problems axe usually solved by transforming them to nonlinear programming (NLP) problems with either sequential or simultaneous approaches. However, both approaches can still be inefficient to tackle complex problems. In addition, many problems in chemical engineering have unstable components which lead to unstable intermediate profiles during the solution procedure. If the numerical algorithm chosen utilises an initial value formulation, the error from decomposition or integration can accumulate and the Newton iteration* then faiL On the other hand, by njiiigsmUhkŜ hooting or collocation, our algorithm has favorable numerical characteristics for both stable and unstable problems; by exploiting the structure of the resulting system, a stable and efficient decomposition algorithm results.'Here solution of this NLP formulation is considered through a reduced Hessian Successive Quadratk Programming (SQP) approach. The routine chosen for the decomposition of the system equations is CX)LDAE, in which the stable multipk shooting scheme is implemented. To address the mesh selection, we will introduce a new bilevd framework that wiD decouple the element placement from the optimal control procedure. We will also provide a proof for the connection of our algorithm and the calculus of variations.

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Section: Element Placement and Error Controlmentioning

confidence: 99%

“…as proposed in Russell and Christiansen (1978). Next we enforce constraint (67) without the O(/** +1 ) term by bounding the residual for each equation, including algebraic equations, at noncollocation points as, 6 = F( z » Zih VH, u ih P> f nc) (68) where (&)/ is an interpolated residual for equation / at element i.…”

Section: Element Placement and Error Controlmentioning

confidence: 99%

Stable Decomposition for Dynamic Optimization

Tanartkit

Biegler

1995

Ind. Eng. Chem. Res.

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“…We note that other reparameterisations are possible with the most common choices being energy-weighted arc-length [13] or equi-distribution of error [28]. The reparameterisation of the path has strong links with moving meshes for the solution of parabolic partial differential equations [4].…”

Section: The String Methodsmentioning

confidence: 99%

Computation of Minimum Energy Paths for Quasi-Linear Problems

2011

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We investigate minimum energy paths of the quasi-linear problem with the p-Laplacian operator and a double-well potential. We adapt the String method of E., Ren, and Vanden-Eijnden (J. Chem. Phys., vol. 126 2007) to locate saddle-type solutions. In one-dimension, the String method is shown to find a minimum energy path that can align along one-dimensional "ridges" of saddle-continua. We then apply the same method to locate saddle solutions and transition paths of the two-dimensional quasi-linear problem. The method developed is applicable to a general class of quasi-linear PDEs.

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“…Let X coll denote a grid consisting of the mesh points x j from the coarser grid X N and its collocation points t j,l . Then we compute G X coll := max x |g(x)| ∞ for x ∈ X coll , see (30). The number of points for the next iteration is predicted according tô…”

Section: Description Of Bvpsuitenga Adaptivitymentioning

confidence: 99%

Automatic grid control in adaptive BVP solvers

2010

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Grid adaptation in two-point boundary value problems is usually based on mapping a uniform auxiliary grid to the desired nonuniform grid. Here we combine this approach with a new control system for constructing a grid density function φ(x). The local mesh width ∆x j+1/2 = x j+1 − x j with 0 = x 0 < x 1 < · · · < x N = 1 is computed as ∆x j+1/2 = ǫ N /ϕ j+1/2 , where {ϕ j+1/2 } N −1 0 is a discrete approximation to the continuous density function φ(x), representing mesh width variation. The parameter ǫ N = 1/N controls accuracy via the choice of N . For any given grid, a solver provides an error estimate. Taking this as its input, the feedback control law then adjusts the grid, and the interaction continues until the error has been equidistributed. Digital filters may be employed to process the error estimate as well as the density to ensure the regularity of the grid. Once φ(x) is determined, another control law determines N based on the prescribed tolerance tol. The paper focuses on the interaction between control system and solver, and the controller's ability to produce a nearoptimal grid in a stable manner as well as correctly predict how many grid points are needed. Numerical tests demonstrate the advantages of the new control system within the bvpsuite solver, ceteris paribus, for a selection of problems and over a wide range of tolerances. The control system is modular and can be adapted to other solvers and error criteria.

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Adaptive Mesh Selection Strategies for Solving Boundary Value Problems

Cited by 190 publications

References 17 publications

Stable Decomposition for Dynamic Optimization

Stable Decomposition for Dynamic Optimization

Computation of Minimum Energy Paths for Quasi-Linear Problems

Automatic grid control in adaptive BVP solvers

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