2024-03-28T22:23:44Z
https://eprints.maths.manchester.ac.uk/cgi/oai2
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:184
2017-10-20T14:12:05Z
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https://eprints.maths.manchester.ac.uk/184/
Symplectic integrators and optimal control
Cross, Mathew I.
49 Calculus of variations and optimal control; optimization
65 Numerical analysis
70 Mechanics of particles and systems
When selecting a numerical method to integrate an ODE system, it is intuitively clear that preservation of geometric properties is desirable. The particular subclasses of ODE systems we will consider are Lagrangian and Hamiltonian systems. The dynamical equations for these derive from variational principles, and we obtain structure preserving integrators by discretizing the principles rather than the ODEs they generate. We demonstrate some advantages that these symplectic integrators have over methods that are more rudimentary by looking at some examples from optimal control theory.
Our major motivation for considering symplectic integrators is solving an image registration problem, where, using the least effort, we associate a set of landmark points on one image to a corresponding set of points on another. A mathematical formulation of this problem is as a Hamiltonian system; this becomes apparent once we realize that we are computing the motion of particles (the landmark points) under some appropriate potential function. We
investigate the performance of symplectic methods on this, more complex, problem. We show that by formulating the problem as a system of nonlinear equations rather than one of optimal control, the explicit Euler method performs better than the symplectic integrators, especially on a set of data points generated by a real experiment. We give some
evidence that the higher-order methods in Matlab's ODE suite
may be better still, but we do not pursue this line of investigation in any detail.
2005-05-01
Thesis
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/184/1/MSc2005.pdf
Cross, Mathew I. (2005) Symplectic integrators and optimal control. Masters thesis, University of Manchester.
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:517
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https://eprints.maths.manchester.ac.uk/517/
An alternative form of the Helmholtz criterion in the inverse problem of the calculus of variations
Voronov, Theodore
37 Dynamical systems and ergodic theory
49 Calculus of variations and optimal control; optimization
58 Global analysis, analysis on manifolds
70 Mechanics of particles and systems
We give a necessary and sufficient condition for the existence of a local solution of the inverse problem of calculus of variations in terms of the identical vanishing of the variation of a functional on an extended space (with the number of independent variables increased by one), and explain its relation with the classical Helmholtz criterion using the de Rham complex on an infinite-dimensional space of fields.
2004
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/517/1/An_Alternative_Form.pdf
Voronov, Theodore (2004) An alternative form of the Helmholtz criterion in the inverse problem of the calculus of variations. Letters in Mathematical Physics, 67 (2). pp. 103-110. ISSN 0377-9017
http://www.springerlink.com/content/g4n18g0305r25r76/?p=2ad58cc20ae545fabed4ec022e7a4d6e&pi=2
10.1023/B:MATH.0000032834.91846.02
10.1023/B:MATH.0000032834.91846.02
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:813
2017-11-08T18:18:30Z
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https://eprints.maths.manchester.ac.uk/813/
Total Variation Regularization in Electrical Impedance Tomography
Borsic, Andrea
Graham, Brad M.
Adler, Andy
Lionheart, William R.B.
35 Partial differential equations
49 Calculus of variations and optimal control; optimization
This paper presents an evaluation of the use of Primal Dual Methods for efficiently regularizing the electric impedance tomography (EIT) problem with the Total Variation (TV) functional. The Total Variation functional is assuming an important role in the regularization of inverse problems thanks to its ability to preserve dis-
continuities in reconstructed profiles. This property is desirable in many fields of application of EIT imaging, such as the medical and the industrial, where inter-organ boundaries, in the first case, and inter-phase boundaries, in the latter case, present step changes in electrical
properties which are difficult to be reconstructed with traditional regularization methods, as they tend to smooth the reconstructed image. Though desirable, the TV functional leads to the formulation of the inverse problem as a minimization of a non-differentiable function whichcannot be efficiently solved with traditional optimization techniques such as the Newton Method. In this paper we demonstrate the use of Primal Dual - Interior Point Methods (PD-IPM) as a framework for TV regularized inversion.
2007-06-02
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/813/1/TVReglnEITpreprint.pdf
Borsic, Andrea and Graham, Brad M. and Adler, Andy and Lionheart, William R.B. (2007) Total Variation Regularization in Electrical Impedance Tomography. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1019
2017-10-20T14:12:34Z
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https://eprints.maths.manchester.ac.uk/1019/
Calculating the $\mathcal{H}_{\infty}$-Norm of Large Sparse Systems via Chandrasekhar Iterations and Extrapolations
Chahlaoui, Y
Gallivan, K.A
Van Dooren, P
15 Linear and multilinear algebra; matrix theory
49 Calculus of variations and optimal control; optimization
65 Numerical analysis
93 Systems theory; control
We describe an algorithm for estimating the $\mathcal{H}_{\infty}$-norm of a large linear time invariant dynamical system described by a discrete time state-space model. The algorithm uses Chandrasekhar iterations to obtain an estimate of the $\mathcal{H}_{\infty}$-norm and then uses extrapolation to improve these estimates.
EDP Sciences, ESAIM
Benbourhim, Mohammed-Najib
Chenin, Patrick
Hassouni, Abdelhak
Hiriart-Urruty, Jean-Baptiste
2007-10-13
Book Section
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1019/1/proc072008.pdf
Chahlaoui, Y and Gallivan, K.A and Van Dooren, P (2007) Calculating the $\mathcal{H}_{\infty}$-Norm of Large Sparse Systems via Chandrasekhar Iterations and Extrapolations. In: RFMAO 05 - Rencontres Franco-Marocaines en Approximation et Optimisation 2005. ESAIM: Proceedings, 20 . EDP Sciences, ESAIM, France, pp. 83-92.
http://www.edpsciences.org/articles/proc/abs/2007/04/contents/contents.html
10.1051/proc:072008
10.1051/proc:072008
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1029
2017-11-07T22:38:45Z
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1042
2017-10-20T14:12:35Z
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https://eprints.maths.manchester.ac.uk/1042/
Low-rank approximation and model reduction.
Chahlaoui, Younes
15 Linear and multilinear algebra; matrix theory
34 Ordinary differential equations
35 Partial differential equations
37 Dynamical systems and ergodic theory
41 Approximations and expansions
49 Calculus of variations and optimal control; optimization
65 Numerical analysis
68 Computer science
93 Systems theory; control
The basic idea of model reduction is to represent a complex linear dynamical system by a much simpler one. This may refer to many different techniques, but in this dissertation
we focus on projection-based model reduction of linear systems. The projection is based on the dominant eigen-spaces of energy functions for ingoing and outgoing signals of the system.
These energy functions are called Gramians of the system and can be obtained as the solutions of Stein equations. When the system matrices are large and sparse, it is not
obvious how to compute efficiently these solutions or their dominant eigen-spaces. In fact, direct methods ignore sparsity in the Stein equations and are not very attractive
for parallelization. Their use is then limited if the state dimension N of the system is large.
The complexity of these methods is roughly O(N3) floating point operations and they require about O(N2) words of memory.
This thesis provides some new ideas of recursive projection-based model reduction for time-varying systems as well as time-invariant systems. We present three algorithms for the recursive computation of the projection. These algorithms combine ideas of two classical methods — namely Balanced Truncation and Krylov subspaces — to produce a
low-rank approximation of the Gramians or the input/output map of the system. We show the practical relevance of our results with real world benchmark examples. We also present some new ideas for second order systems. Such systems have a special structure which one wants to preserve in the reduced order model. We show how to adapt our projection based method to such systems.
2003-12-22
Thesis
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1042/1/Chahlaoui_thesis_A4.pdf
Chahlaoui, Younes (2003) Low-rank approximation and model reduction. Doctoral thesis, University Catholic of Louvain, Louvain-La-Neuve, Belgium.
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1223
2017-10-20T14:12:41Z
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7375626A656374733D4D5343:4D53435F3933
74797065733D626F6F6B5F73656374696F6E
https://eprints.maths.manchester.ac.uk/1223/
Calculating the $\mathcal{H}_{\infty}$-Norm of Large Sparse Systems via Chandrasekhar Iterations and Extrapolations
Chahlaoui, Y
Gallivan, K.A
Van Dooren, P
15 Linear and multilinear algebra; matrix theory
49 Calculus of variations and optimal control; optimization
65 Numerical analysis
93 Systems theory; control
We describe an algorithm for estimating the $\mathcal{H}_{\infty}$-norm of a large linear time invariant dynamical system described by a discrete time state-space model. The algorithm uses Chandrasekhar iterations to obtain an estimate of the $\mathcal{H}_{\infty}$-norm and then uses extrapolation to improve these estimates.
EDP Sciences, ESAIM
Benbourhim, Mohammed-Najib
Chenin, Patrick
Hassouni, Abdelhak
Hiriart-Urruty, Jean-Baptiste
2007-10-13
Book Section
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1223/1/MIMS_ep2008_5.pdf
Chahlaoui, Y and Gallivan, K.A and Van Dooren, P (2007) Calculating the $\mathcal{H}_{\infty}$-Norm of Large Sparse Systems via Chandrasekhar Iterations and Extrapolations. In: RFMAO 05 - Rencontres Franco-Marocaines en Approximation et Optimisation 2005. ESAIM: Proceedings, 20 . EDP Sciences, ESAIM, France, pp. 83-92.
http://www.edpsciences.org/articles/proc/abs/2007/04/contents/contents.html
10.1051/proc:072008
10.1051/proc:072008
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1480
2017-10-20T14:12:50Z
7374617475733D707562
7375626A656374733D4D5343:4D53435F3439
7375626A656374733D4D5343:4D53435F3630
7375626A656374733D4D5343:4D53435F3933
74797065733D626F6F6B5F73656374696F6E
https://eprints.maths.manchester.ac.uk/1480/
Dealing with stochastic reachability
Bujorianu, M.L.
49 Calculus of variations and optimal control; optimization
60 Probability theory and stochastic processes
93 Systems theory; control
For stochastic hybrid systems, stochastic reachability is very little supported mainly because of complexity and difficulty of the associated mathematical problems. In this paper, we develop two main directions of studying stochastic reachability as an optimal stopping problem. The first approach studies the hypotheses for the dynamic programming corresponding with the optimal stopping problem for stochastic hybrid systems. In the second approach, we investigate the reachability problem considering approximations of stochastic hybrid systems. The main difficulty arises when we have to prove the convergence of the value functions of the approximating processes to the value function of the initial process. An original proof is provided.
IEEE Control Systems Society
2009-12
Book Section
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1480/1/cdc09_p1_finfin.tex
Bujorianu, M.L. (2009) Dealing with stochastic reachability. In: Proceedings of the 48th IEEE Conference on Decision and Control, CDC 2009. IEEE Control Systems Society, Shanghai, China, pp. 2935-2940.
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1481
2017-10-20T14:12:50Z
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https://eprints.maths.manchester.ac.uk/1481/
Large deviation methods for stochastic reachability
Bujorianu, M.L.
Wang, Hong
49 Calculus of variations and optimal control; optimization
60 Probability theory and stochastic processes
In this paper, we propose to find upper/lower bounds for different measures that characterize the reachability problem defined in the context of stochastic hybrid systems, using the theory of large deviations. For stochastic hybrid processes, criteria for large deviation results are given using properties of their infinitesimal generators. This represents just the first step towards applying large deviation methods for stochastic hybrid systems for treating new topics like robust control, metastability, performance analysis.
IEEE Control Systems Society
2009-12
Book Section
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1481/1/cdc09_p2fin.pdf
Bujorianu, M.L. and Wang, Hong (2009) Large deviation methods for stochastic reachability. In: Proceedings of the 48th IEEE Conference on Decision and Control, CDC 2009. IEEE Control Systems Society, Shanghai, China, pp. 3932-3937.
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1483
2017-10-20T14:12:50Z
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7375626A656374733D4D5343:4D53435F3630
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https://eprints.maths.manchester.ac.uk/1483/
State constrained reachability for stochastic hybrid systems
Bujorianu, M.L.
Bujorianu, M.C.
49 Calculus of variations and optimal control; optimization
60 Probability theory and stochastic processes
The stochastic hybrid systems constitute well established classes of realistic models of hybrid discrete/continuous dynamics subject to random perturbations, autonomous uncontrollable transitions, nondeterminism or uncertainty. Stochastic reachability analysis is a key factor in the verification and deployment of stochastic hybrid systems. The encouraging recent progress prompts us to rene the problem to cover more realistic situations. We extend the so called constrained reachability problem from the probabilistic discrete case to stochastic hybrid systems. Then we dene mathematically this problem, and we obtain the reach probabilities as solutions of a boundary value problem. The last problem is well studied and numerical, even symbolic solutions exist. This characterization is useful in stochastic control, in probabilistic path planning and for nano-systems.
IFAC, Elsevier
2009-08
Book Section
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1483/1/adhs09final.pdf
Bujorianu, M.L. and Bujorianu, M.C. (2009) State constrained reachability for stochastic hybrid systems. In: Proceedings of Analysis and Design of Hybrid Systems. IFAC, Elsevier, Zaragoza, Spain.
10.3182/20090916-3-US-00030
10.3182/20090916-3-US-00030
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1604
2017-10-20T14:12:54Z
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https://eprints.maths.manchester.ac.uk/1604/
Multi-Objective Hybrid Intelligent Optimization of Operational Indices for Industrial Processes and Application
Chai, Tianyou
Ding, Jinliang
Wang, Hong
49 Calculus of variations and optimal control; optimization
93 Systems theory; control
To pursuit the plant-wide optimization of multiple units industrial process, a hybrid
intelligent optimization approach under dynamic environment is proposed. The objective of
optimization is that the production indices de�ned as the performance related to the �nal
product quality, yield, energy and material consumption fall into their target ranges; whilst the
decision variables are operational indices of each unit, which is related to units� intermediate
product quality, e�ciency and consumption. In this context, the domain knowledge of process
engineers are mimicked and combined with the framework in terms of feedback, prediction and
feed-forward schemes so as to realize the required optimization. The e�ectiveness of the proposed
approach has been demonstrated by the practical application results.
2011-09-01
Conference or Workshop Item
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1604/1/IFAC11_1753_FI.pdf
Chai, Tianyou and Ding, Jinliang and Wang, Hong (2011) Multi-Objective Hybrid Intelligent Optimization of Operational Indices for Industrial Processes and Application. In: 18th IFAC World Congress, August 28 - September 2, 2011, Milano, Italy. (Submitted)
https://ifac.papercept.net/conferences/scripts/abstract.pl?ConfID=30&Number=1753
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1605
2017-10-20T14:12:54Z
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https://eprints.maths.manchester.ac.uk/1605/
Product Scheduling for Thermal Energy Reduction in Papermakig Industries
Ghasemi Afshar, Puya
Brown, Martin
Wang, Hong
Maciejowski, Jan
49 Calculus of variations and optimal control; optimization
93 Systems theory; control
Papermaking is considered as an energy-intensive industry partly due to the fact
that the machinery and procedures have been designed at the time when energy was both cheap
and plentiful. A typical paper machine manufactures a variety of di�erent products (grades)
which impose variable per-unit raw material and energy costs to the mill. It is known that
during a grade change operation the products are not market-worthy. Therefore, two di�erent
production regimes, i.e. steady state and grade transition can be recognised in papermaking
practice. Among the costs associated with paper manufacture, the energy cost is �more variable�
due to (usually) day-to-day variations of the energy prices. Moreover, the production of a grade
is often constrained by customer delivery time requirements. Given the above constraints and
production modes, the product scheduling technique proposed in this paper aims at optimising
the sequence of orders in a single machine so that the cost of production (mainly determined
by the energy) is minimised. Simulation results obtained from a commercial board machine in
the UK con�rm the e�ectiveness of the proposed method.
2011-08-31
Conference or Workshop Item
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1605/1/IFAC11_1566_FI.pdf
Ghasemi Afshar, Puya and Brown, Martin and Wang, Hong and Maciejowski, Jan (2011) Product Scheduling for Thermal Energy Reduction in Papermakig Industries. In: 18th IFAC World Congress, August 28 - September 2, 2011, Milano, Italy. (Submitted)
https://ifac.papercept.net/conferences/scripts/abstract.pl?ConfID=30&Number=1566
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1607
2017-10-20T14:12:54Z
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74797065733D636F6E666572656E63655F6974656D
https://eprints.maths.manchester.ac.uk/1607/
Dynamic Optimal Scheduling Method and Its Application for Converter Fault in Steelmaking and Continuous Casting Production Process
Yu, Shengping
Chai, Tianyou
Wang, Hong
Pang, Xinfu
Zheng, Binglin
49 Calculus of variations and optimal control; optimization
65 Numerical analysis
90 Operations research, mathematical programming
In steelmaking and continuous casting (SMCC) production process, converter fault can lead to unexpected changes to the pre-specified converter-continuous caster production mode so that the original scheduling plan becomes unrealizable. In this paper, the dynamic scheduling problem in response to converter fault is firstly analyzed. This is then followed by the establishment of a novel multi-objective nonlinear programming model (MONPM) by introducing the production mode parameter α, production schedule parameter β and �. The proposed method considers changes in production mode, production schedule of charge, the interval characteristics of processing time. In specific, a two-stage dynamic optimal scheduling method is proposed including the production path planning of charges (PPP) and the production time scheduling (PTS). As a result, a dynamic optimal scheduling software system (DOSSS)
is developed and is successfully applied to the scheduling of the largest iron and steel company (BaoSteel)
in China. The real-time application shows that the proposed method can efficiently reduce scheduling time, significantly increase the outputs of converters and dramatically shorten the redundant waiting time for molten steel.
2011-08-31
Conference or Workshop Item
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1607/1/IFAC11_1793_FI.pdf
Yu, Shengping and Chai, Tianyou and Wang, Hong and Pang, Xinfu and Zheng, Binglin (2011) Dynamic Optimal Scheduling Method and Its Application for Converter Fault in Steelmaking and Continuous Casting Production Process. In: 18th IFAC World Congress, August 28 - September 2, 2011, Milano, Italy. (Submitted)
https://ifac.papercept.net/conferences/scripts/abstract.pl?ConfID=30&Number=1793
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1631
2017-10-20T14:12:55Z
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https://eprints.maths.manchester.ac.uk/1631/
State constrained reachability for stochastic hybrid systems
Bujorianu, L.M.
Bujorianu, M.C.
49 Calculus of variations and optimal control; optimization
60 Probability theory and stochastic processes
Many control problems can be formulated as driving a system to reach some target states while avoiding some unwanted states. We study this problem for systems with regime change operating in uncertain environments. Nowadays, it is a common practice to model such systems in the framework of stochastic hybrid system models. In this casting, the problem is formalized as a mathematical problem named state constrained stochastic reachability analysis. In the state constrained stochastic reachability analysis, this probability is computed by imposing a constraint on the system to avoid the unwanted states. The scope of this paper is twofold. First we define and investigate the state constrained reachability analysis in an abstract mathematical setting. We define the problem for a general model of stochastic hybrid systems, and we show that the reach probabilities can be computed as solutions of an elliptic integro-differential equation. Moreover, we extend the problem by considering randomized targets. We approach this extension using stochastic dynamic programming. The second scope is to define a developmental setting in which the state constrained reachability analysis becomes more tractable. This framework is based on multilayer modelling of a stochastic system using hierarchical viewpoints. Viewpoints represent a method originated from software engineering, where a system is described by multiple models created from different perspectives. Using viewpoints, the reach probabilities can be easily computed, or even symbolically calculated. The reach probabilities computed in one viewpoint can be used in another viewpoint for improving the system control. We illustrate this technique for trajectory design.
2011-05
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1631/1/ifacJournalrevised20.pdf
Bujorianu, L.M. and Bujorianu, M.C. (2011) State constrained reachability for stochastic hybrid systems. Nonlinear Analysis: Hybrid Systems, 5 (2). pp. 320-342. ISSN 1751-570X
http://www.sciencedirect.com/science/article/pii/S1751570X10000786
doi:10.1016/j.nahs.2010.10.008
doi:10.1016/j.nahs.2010.10.008
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1651
2017-11-07T22:38:46Z
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1673
2017-11-07T22:38:46Z
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1705
2017-10-20T14:12:58Z
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https://eprints.maths.manchester.ac.uk/1705/
The Determination of a Dynamic Cut-Off
Grade for the Mining Industry
Johnson, P.V.
Evatt, G.W.
Duck, P.W.
Howell, S
35 Partial differential equations
49 Calculus of variations and optimal control; optimization
Prior to extraction from a mine, a pit is usually divided up into 3-D `blocks' which contain varying levels of estimated ore-grades. From these, the order (or `pathway') of extraction is decided, and this order of extraction can remain unchanged for several years. However, because commodity prices are uncertain, once each block is extracted from the mine, the company must decide in real-time whether the ore grade is high enough to warrant processing the block further in readiness for sale, or simply to waste the block. This paper first shows how the optimal cut-off ore grade—the level below which a block should be wasted—is not simply a function of the current commodity price and the ore grade, but also a function of the ore-grades of subsequent blocks, the costs of processing, and the bounds on the rates of processing and extraction. Secondly, the paper applies a stochastic price uncertainty, and shows how to derive an efficient mathematical algorithm to calculate and operate a dynamic optimal cut-off grade criterion throughout the extraction process, allowing the mine operator to respond to future market movements. The model is applied to a real mine composed of some 60,000 blocks, and shows that an extra 10% of value can be created by implementing such an optimal regime.
Springer
Ao, S.I.
Gelman, L.
2011
Book Section
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1705/1/johnson_evatt_book_cutoff.pdf
Johnson, P.V. and Evatt, G.W. and Duck, P.W. and Howell, S (2011) The Determination of a Dynamic Cut-Off Grade for the Mining Industry. In: Electrical Engineering and Applied Computing. Lecture Notes in Electrical Engineering, 90 . Springer, pp. 391-403. ISBN 978-94-007-1192-1
http://www.springerlink.com/content/x6875618401664l3/
10.1007/978-94-007-1192-1_32
10.1007/978-94-007-1192-1_32
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1806
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https://eprints.maths.manchester.ac.uk/1806/
Continuous-Time Revenue Management in Carparks
Papayiannis, A.
Johnson, P.
Yumashev, D.
Howell, S.
Proudlove, N.
Duck, P.
49 Calculus of variations and optimal control; optimization
90 Operations research, mathematical programming
In this paper, we study optimal revenue management applied to carparks, with primary objective to maximize
revenues under a continuous-time framework. We develop a stochastic discrete-time model and propose a
rejection algorithm that makes optimal decisions (accept/reject) according to the future expected revenues
generated and on the opportunity cost that arises before each sale. For this aspect of the problem, a Monte
Carlo approach is used to derive optimal rejection policies. We then extend this approach to show that there
exists an equivalent continuous-time methodology that yields to a partial differential equation (PDE). The
nature of the PDE, as opposed to theMonte Carlo approach, generates the rejection policies quicker and causes
the optimal surfaces to be significantly smoother. However, because the solution to the PDE is considered not
to solve the `full' problem, we propose an approach to generate optimal revenues using the discrete-time
model by exploiting the information coming from the PDE. We give a worked example of how to generate
near-optimal revenues with an order of magnitude decrease in computation speed.
2012-02-06
Conference or Workshop Item
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1806/1/con_time_rev_man_carparks.pdf
Papayiannis, A. and Johnson, P. and Yumashev, D. and Howell, S. and Proudlove, N. and Duck, P. (2012) Continuous-Time Revenue Management in Carparks. In: 1st International Conference on Operations Research and Enterprise Systems, 04-06 February 2012, Vilamoura, Portugal.
http://www.icores.org/
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1808
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https://eprints.maths.manchester.ac.uk/1808/
Stochastic Reachability Analysis of Hybrid Systems
Bujorianu, L. M.
49 Calculus of variations and optimal control; optimization
60 Probability theory and stochastic processes
93 Systems theory; control
Stochastic reachability analysis (SRA) is a method of analyzing the behavior of control systems which mix discrete and continuous dynamics. For probabilistic discrete systems it has been shown to be a practical verification method but for stochastic hybrid systems it can be rather more. As a verification technique SRA can assess the safety and performance of, for example, autonomous systems, robot and aircraft path planning and multi-agent coordination but it can also be used for the adaptive control of such systems.
Stochastic Reachability Analysis of Hybrid Systems is a self-contained and accessible introduction to this novel topic in the analysis and development of stochastic hybrid systems. Beginning with the relevant aspects of Markov models and introducing stochastic hybrid systems, the book then moves on to coverage of reachability analysis for stochastic hybrid systems. Following this build up, the core of the text first formally defines the concept of reachability in the stochastic framework and then treats issues representing the different faces of SRA:
· stochastic reachability based on Markov process theory;
· martingale methods;
· stochastic reachability as an optimal stopping problem; and
· dynamic programming.
The book is rounded off by an appendix providing mathematical underpinning on subjects such as ordinary differential equations, probabilistic measure theory and stochastic modeling, which will help the non-expert-mathematician to appreciate the text.
Stochastic Reachability Analysis of Hybrid Systems characterizes a highly interdisciplinary area of research and is consequently of significant interest to academic researchers and graduate students from a variety of backgrounds in control engineering, applied mathematics and computer science.
Springer
2012-03
Book
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1808/1/9781447127949-t1%281%29.pdf
Bujorianu, L. M. (2012) Stochastic Reachability Analysis of Hybrid Systems. Communications and Control Engineering . Springer, London, UK. ISBN 978-1-4471-2794-9
http://www.springer.com/mathematics/applications/book/978-1-4471-2794-9
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1989
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https://eprints.maths.manchester.ac.uk/1989/
Continuous-Time Revenue Managment in Carparks - Part two: Refining the PDE
Andreas, Papayiannis
Paul, Johnson
Dmitry, Yumashev
Peter, Duck
35 Partial differential equations
49 Calculus of variations and optimal control; optimization
In this paper, we study optimal revenue management applied to carparks, with the primary objective
to maximize revenues under a continuous-time framework. This work is an extension to Papayiannis et al (2012) where the authors developed a Partial Differential Equation (PDE) model that could solve for the rate at which cash is generated through an infinitesimal time. However, in practice, carpark managers charge customers per day or per hour which is a finite period of time. Unfortunately, this situation was currently not captured by this previous work. Therefore, our current work attempts to reformulate the existing PDE in a way that it does capture the revenue that is generated within any finite time interval of length $\Delta T$. The new model is compared against the Monte Carlo (MC) approach for several choices of $\Delta T$; the results are remarkable as the improvement in computation speed and efficiency are significant. Since, the algorithm in the PDE still does not solve the `exact' problem, a method is proposed to marry the benefits of the PDE with those of the MC approach. Our results are prominent as the optimal values generated in this case have shown to be extremely close to the MC ones while the computation times are kept to a minimum.
2013-06-18
Conference or Workshop Item
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/1989/1/paper_camera_ready_version.pdf
Andreas, Papayiannis and Paul, Johnson and Dmitry, Yumashev and Peter, Duck (2013) Continuous-Time Revenue Managment in Carparks - Part two: Refining the PDE. In: 2nd International Conference on Operations Research and Enterprise Systems, 16-18 February, Barcelona, Spain.
http://www.scitepress.org/DigitalLibrary/
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:2002
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https://eprints.maths.manchester.ac.uk/2002/
Optimal Staffing Policy:
A Service System with Stochastic Travel Times
Al-Foraih, Mishari
Johnson, Paul
Evatt, Geoff
Duck, Peter
35 Partial differential equations
49 Calculus of variations and optimal control; optimization
90 Operations research, mathematical programming
Private sector operators of response services such as ambulance, fire or police etc. are often regulated with
targets on the distribution of response times, which may result in inefficient over staffing to ensure those targets
are met. In this paper, we use a network chain of M=M=K queues to model the arrival and completion of jobs
on the system so that quantities such as the expected total time waiting for all jobs can be calculated. The
Markov nature enables us to evoke the Hamilton Jacobi Bellman equation (HJB) principle to optimize the
required number of staff whilst still meeting targets.
2012-11-22
Conference or Workshop Item
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/2002/1/mishari-paper70.pdf
Al-Foraih, Mishari and Johnson, Paul and Evatt, Geoff and Duck, Peter (2012) Optimal Staffing Policy: A Service System with Stochastic Travel Times. In: 2nd International Conference on Operation Research and Enterprise Systems, 16-18 February 2013, Barcelona, Spain.
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:2137
2017-11-07T22:38:47Z
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:2144
2017-10-20T14:13:13Z
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https://eprints.maths.manchester.ac.uk/2144/
Investment Lags:
A Numerical Approach
Al-Foraih, Mishari
Johnson, Paul
Duck, Peter
35 Partial differential equations
49 Calculus of variations and optimal control; optimization
In this paper we use a mixture of numerical methods including finite difference and body fitted co-ordinates
to form a robust stable numerical scheme to solve the investment lag model presented in the paper by Bar-Ilan
and Strange (1996). This allows us to apply our methodology to models with different stochastic processes
that does not have analytic solutions.
2014-03-06
Conference or Workshop Item
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/2144/1/ICORESp.pdf
Al-Foraih, Mishari and Johnson, Paul and Duck, Peter (2014) Investment Lags: A Numerical Approach. In: 3rd International Conference on Operations Research and Enterprise Systems, 6-8 March 2014, Angers, France.
10.5220/0004920702820287
10.5220/0004920702820287
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:2173
2017-11-07T22:38:47Z
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:2299
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https://eprints.maths.manchester.ac.uk/2299/
Analysis of optimal liquidation in limit order books
Blair, James W.
Johnson, Paul V.
Duck, Peter W.
35 Partial differential equations
49 Calculus of variations and optimal control; optimization
91 Game theory, economics, social and behavioral sciences
In this paper we study the optimal trading strategy of a passive trader who is trading in the limit order book. Using a combined approach of accurate numerical methods and asymptotical analysis we examine the problem using different stochastic processes to model the asset price, as well as introducing a proportional resilience for the limit order book.
This results in more complex equations to solve than when examined under the case of standard Brownian motion, allowing us to perform interesting analytical (asymptotic) analysis which adds insight into the solution space.
Under Geometric Brownian Motion, we reduce the resulting four-dimensional Hamilton-Jacobi-Bellman partial differential equation (PDE) to a novel three-dimensional non-linear PDE, as well as rescaling the variables to reduce the number of input parameters by two. We use numerical methods to solve the PDE before asymptotically examining it in several limits, with each approach informing and confirming the other. We find the transition from a time-varying solution to a perpetual-type solution results in the development of singular behaviour, and this transition is examined in some detail. Finally we emphasise the adaptability of our proposed methodologies by implementing the same methods on a mean-reverting process for the asset price.
Throughout the paper we also analyse the resulting trading strategies from a financial perspective. The trading strategies we develop are asset-price dependent, which to our knowledge is a unique concept in the passive optimal trading literature, and is arguably more realistic.
2015-05-20
Article
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/2299/1/analysis_of_optimal_liquidation.pdf
Blair, James W. and Johnson, Paul V. and Duck, Peter W. (2015) Analysis of optimal liquidation in limit order books. Preprint. (Submitted)
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:2325
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https://eprints.maths.manchester.ac.uk/2325/
Analysis of optimal liquidation in limit order books for portfolios of correlated assets with stochastic volatility
Blair, James W.
Johnson, Paul V.
Duck, Peter W.
35 Partial differential equations
49 Calculus of variations and optimal control; optimization
91 Game theory, economics, social and behavioral sciences
In this paper we study optimal liquidation under two settings: the first being for a basket of correlated assets, the second being for a portfolio of a single asset but under a stochastic volatility model. Under both frameworks we use a combined approach of accurate numerical methods and asymptotic analysis to investigate and gain insight into the solution, with each approach informing and confirming the other. We are able to make a significant improvement in efficiency in both problems, reducing the resulting Hamiliton-Jacobi-Bellman (HJB) partial differential equations (PDEs) to classical non-linear PDEs, as well as reducing the number of variables and input parameters, the latter through non-dimensionalisation. We present numerical solutions to both problems, before further investigating the solution topology through the use of asymptotic analysis in various limits. In some cases we are able to find analytic approximations for both the value function and indeed the optimal liquidation strategies. Furthermore, the solutions we present are comparable with those of Markowitz Portfolio Theory (MPT) for the multiple-asset case, and to those of option pricing theory under stochastic volatility for the stochastic volatility model. For the former we find the trader trades in a way that results in a diversified portfolio, while for the latter we find that more noise in the volatility can be beneficial for the trader in certain regimes, despite being risk-averse.
2015-06
Article
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/2325/1/multiple_assets_article_d3.pdf
Blair, James W. and Johnson, Paul V. and Duck, Peter W. (2015) Analysis of optimal liquidation in limit order books for portfolios of correlated assets with stochastic volatility. Preprint. (Unpublished)
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:2406
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https://eprints.maths.manchester.ac.uk/2406/
Modelling and controlling risk in energy systems
Gonzalez, J
49 Calculus of variations and optimal control; optimization
60 Probability theory and stochastic processes
62 Statistics
The Autonomic Power System (APS) grand challenge was a multi-disciplinary EPSRC-funded research project that examined novel techniques that would enable the transition between today's and 2050's highly uncertain and complex energy network. Being part of the APS, this thesis reports on the sub-project `RR2: Avoiding High-Impact Low Probability events'. The goal of RR2 is to develop new algorithms for controlling risk exposure to high-impact low probability (Hi-Lo) events through the provision of appropriate risk-sensitive control strategies. Additionally, RR2 is concerned with new techniques for identifying and modelling risk in future energy networks, in particular, the risk of Hi-Lo events.
In this context, this thesis investigates two distinct problems arising from energy risk management. On the one hand, we examine the problem of finding managerial strategies for exercising the operational flexibility of energy assets. We look at this problem from a risk perspective taking into account non-linear risk preferences of energy asset managers. Our main contribution is the development of a risk-sensitive approach to the class of optimal switching problems. By recasting the problem as an iterative optimal stopping problem, we are able to characterise the optimal risk-sensitive switching strategies. As byproduct, we obtain a multiplicative dynamic programming equation for the value function, upon which we propose a numerical algorithm based on least squares Monte Carlo regression.
On the other hand, we develop tools to identify and model the risk factors faced by energy asset managers. For this, we consider a class of models consisting of superposition of Gaussian and non-Gaussian Ornstein-Uhlenbeck processes. Our main contribution is the development of a Bayesian methodology based on Markov chain Monte Carlo (MCMC) algorithms to make inference into this class of models. On extensive simulations, we demonstrate the robustness and efficiency of the algorithms to different data features. Furthermore, we construct a diagnostic tool based on Bayesian p-values to check goodness-of-fit of the models on a Bayesian framework. We apply this tool to MCMC results from fitting historical electricity and gas spot price datasets corresponding to the UK and German energy markets. Our analysis demonstrates that the MCMC-estimated models are able to capture not only long- and short-lived positive price spikes, but also short-lived negative price spikes which are typical of UK gas prices and German electricity prices.
Combining together the solutions to the two problems above, we strive to capture the interplay between risk, uncertainty, flexibility and performance in various applications to energy systems. In these applications, which include power stations, energy storage and district energy systems, we consistently show that our risk management methodology offers a tradeoff between maximising average performance and minimising risk, while accounting for the jump dynamics of energy prices. Moreover, the tradeoff is achieved in such way that the benefits in terms of risk reduction outweigh the loss in average performance.
2015-06
Thesis
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/2406/1/PhDthesis_Jhonny_Gonzalez.pdf
Gonzalez, J (2015) Modelling and controlling risk in energy systems. Doctoral thesis, The University of Manchester.
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:2497
2017-11-08T18:18:38Z
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https://eprints.maths.manchester.ac.uk/2497/
An algorithm for computing the eigenvalues of a max-plus matrix polynomial
James, Hook
49 Calculus of variations and optimal control; optimization
65 Numerical analysis
Max-plus matrix polynomial eigenvalues provide a useful approximation to the order of magnitude of the eigenvalues of a classical (i.e. real or complex) matrix polynomial. In this paper we review the max-plus matrix eigensolver of Gassner and Klinz [1] and present our extension of this algorithm to the max-plus matrix polynomial case. Our max-plus matrix polynomial algo- rithm computes all nd max-plus eigenvalues of a n � n degree d max-plus matrix polynomial with worst case cost O(n3d) in the dense case, which is the best that we are aware of.
2016-09-05
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/2497/1/LAA.pdf
James, Hook (2016) An algorithm for computing the eigenvalues of a max-plus matrix polynomial. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:2603
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https://eprints.maths.manchester.ac.uk/2603/
Inverse Problems and Control for Lung Dynamics
Tregidgo, Henry F.J.
34 Ordinary differential equations
35 Partial differential equations
49 Calculus of variations and optimal control; optimization
92 Biology and other natural sciences
Mechanical ventilation is vital for the treatment of patients in respiratory intensive care and can be life saving. However, the risks of regional pressure gradients and over-distension must be balanced with the need to maintain function. For these reasons mechanical ventilation can benefit from the regional information provided by bedside imaging such as electrical impedance tomography (EIT).
In this thesis we develop and test methods to retrieve clinically meaningful measures of lung function from EIT and examine the feasibility of closing the feedback loop to enable EIT-guided control of mechanical ventilation. Working towards this goal we develop a reconstruction algorithm capable of providing fast absolute values of conductivity from EIT measurements. We couple the resulting conductivity time series to a compartmental ordinary differential equation (ODE) model of lung function in order to recover regional parameters of elastance and airway resistance. We then demonstrate how these parameters may be used to generate optimised pressure controls for mechanical ventilation that expose the lungs to minimal gradients of pressure and are stable with respect to EIT measurement errors.
The EIT reconstruction algorithm we develop is capable of producing low dimensional absolute values of conductivity in real time after a limited additional setup time. We show that this algorithm retains the ability to give fast feedback on regional lung changes. We also describe methods of improving computational efficiency for general Gauss-Newton type EIT algorithms.
In order to couple reconstructed conductivity time series to our ODE model we describe and test the recovery of regional ventilation distributions through a process of regularised differentiation. We prove that the parameters of our ODE model are recoverable from these ventilation distributions apart from the degenerate case where all compartments have the same parameters. We then test this recovery process under varying levels of simulated EIT measurement and modelling errors.
Finally we examine the ODE lung model using control theory. We prove that the ODE model is controllable for a wide range of parameter values and link controllability to observable ventilation patterns in the lungs. We demonstrate the generation and optimisation of pressure controls with minimal time gradients and provide a bound on the resulting magnitudes of these pressures. We then test the control generation process using ODE parameter values recovered through EIT simulations at varying levels of measurement noise.
Through this work we have demonstrated that EIT reconstructions can be of benefit to the control of mechanical ventilation.
2017
Thesis
NonPeerReviewed
text
en
https://eprints.maths.manchester.ac.uk/2603/1/Tregidgo_LungEITThesis.pdf
Tregidgo, Henry F.J. (2017) Inverse Problems and Control for Lung Dynamics. Doctoral thesis, University of Manchester.