A globally convergent method for finding all steady‐state solutions of distillation columns

Baharev, Ali; Neumaier, Arnold

doi:10.1002/aic.14305

Cited by 6 publications

(6 citation statements)

References 11 publications

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“…4. The algorithm of the present paper is a significant improvement over older algorithms discussed in [3, 4], both algorithmically and on the implementation level. The entire algorithm has been redesigned and rewritten from scratch, and in particular, the backsolve step is radically different.…”

Section: Overview Of the Proposed Algorithmmentioning

confidence: 97%

A manifold-based approach to sparse global constraint satisfaction problems

2019

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We consider square, sparse nonlinear systems of equations whose Jacobian is structurally nonsingular, with reasonable bound constraints on all variables. We propose an algorithm for finding good approximations to all well-separated solutions of such systems. We assume that the input system is ordered such that its Jacobian is in bordered block lower triangular form with small diagonal blocks and with small border width; this can be performed fully automatically with off-the-shelf decomposition methods. Five decades of numerical experience show that models of technical systems tend to decompose favorably in practice. Once the block decomposition is available, we reduce the task of solving the large nonlinear system of equations to that of solving a sequence of low-dimensional ones. The most serious weakness of this approach is well-known: It may suffer from severe numerical instability. The proposed method resolves this issue with the novel backsolve step. We study the effect of the decomposition on a sequence of challenging problems. Beyond a certain problem size, the computational effort of multistart (no decomposition) grows exponentially. In contrast, thanks to the decomposition, for the proposed method the computational effort grows only linearly with the problem size. It depends on the problem size and on the hyperparameter settings whether the decomposition and the more sophisticated algorithm pay off. Although there is no theoretical guarantee that all solutions will be found in the general case, increasing the so-called sample size hyperparameter improves the robustness of the proposed method.

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Section: Overview Of the Proposed Algorithmmentioning

confidence: 97%

A manifold-based approach to sparse global constraint satisfaction problems

2019

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“…Staircase sampling was inspired by (Baharev and Neumaier, 2014), and proposed in (Baharev et al, 2016) to mitigate all of the issues listed in Sections 3 and 4. A detailed presentation of this method is outside the scope of this paper; here we only sketch the basic idea.…”

Section: A Novel Robust Approachmentioning

confidence: 99%

Failure Modes of Tearing and a Novel Robust Approach

Baharev

Neumaier

Schichl

2017

Linköping Electronic Conference Proceedings

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State-of-the-art Modelica implementations may fail in various ways when tearing is turned on: Completely incorrect results are returned without a warning, or the software fails with an obscure error message, or it hangs for several minutes although the problem is solvable in milliseconds without tearing. We give three detailed examples and an in-depth discussion why such failures are inherent in tearing and cannot be fixed within the traditional approach.Without compromising the advantages of tearing, these issues are resolved for the first time with staircase sampling. This is a non-tearing method capable of robustly finding all well-separated solutions of sparse systems of nonlinear equations without any initial guesses. Its robustness is demonstrated on the steady-state simulation of a particularly challenging distillation column. This column has three solutions, one of which is missed by most methods, including problem-specific tearing methods. All three solutions are found with staircase sampling.

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“…But working with a point cloud allows us to counteract conditioning problems. Inspired by our earlier results for the univariate case [5], this is achieved by redistributing the sample points after each block. This redistribution step strives to ensure in each iteration that the sample of the solution set of (8) remains representative within the bound constraints, in the sense that (a) no point of the solution set is too far from the sample, and (b) the points in the sample are well-separated.…”

Section: The Idea In a Nutshellmentioning

confidence: 99%

A robust approach for finding all well-separated solutions of sparse systems of nonlinear equations

2016

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Tearing is a long-established decomposition technique, widely adapted across many engineering fields. It reduces the task of solving a large and sparse nonlinear system of equations to that of solving a sequence of low-dimensional ones. The most serious weakness of this approach is well-known: It may suffer from severe numerical instability. The present paper resolves this flaw for the first time. The new approach requires reasonable bound constraints on the variables. The worst-case time complexity of the algorithm is exponential in the size of the largest subproblem of the decomposed system. Although there is no theoretical guarantee that all solutions will be found in the general case, increasing the so-called sample size parameter of the method improves robustness. This is demonstrated on two particularly challenging problems. Our first example is the steady-state simulation a challenging distillation column, belonging to an infamous class of problems where tearing often fails due to numerical instability. This column has three solutions, one of which is missed using tearing, but even with problem-specific methods that are not based on tearing. The other example is the Stewart-Gough platform with 40 real solutions, an extensively studied benchmark in the field of numerical algebraic geometry. For both examples, all solutions are found with a fairly small amount of sampling.

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A globally convergent method for finding all steady‐state solutions of distillation columns

Cited by 6 publications

References 11 publications

A manifold-based approach to sparse global constraint satisfaction problems

A manifold-based approach to sparse global constraint satisfaction problems

Failure Modes of Tearing and a Novel Robust Approach

A robust approach for finding all well-separated solutions of sparse systems of nonlinear equations

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