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Development of a three-dimensional all-at-once inversion approach for the magnetotelluric methodWilhelms, Wenke 27 July 2016 (has links) (PDF)
A three-dimensional inversion was implemented for magnetotellurics, which is a passive electromagnetic method in geophysics. It exploits natural electromagnetic fields of the Earth, which function as sources. Their interaction with the conductive parts of the subsurface are registered when components of the electric and the magnetic field are measured and evaluated.
The all-at-once approach is an inversion scheme that is relatively new to geophysics. In this approach, the objective function – the basis of each inversion – is called the Lagrangian. It consists of three parts: (i) the data residual norm, (ii) the regularisation part, and (iii) the forward problem. The latter is the significant difference to conventional inversion approaches that are built up of a forward calculation part and an inversion part. In the case of all-at-once, the forward problem is incorporated in the objective function and is therefore already taken into account in each inversion iteration. Thus, an explicit forward calculation is obsolete. As an objective function, the Lagrangian shall reach a minimum and therefore its first and second derivatives are evaluated. Hence, the gradient of the Lagrangian and its Hessian are constituent parts of the KKT system – the Newton-type system that is set up in the all-at-once inversion. Conventional inversion approaches avoid the Hessian because it is a large, dense, not positive definite matrix that is challenging to handle. However, it provides additional information to the inversion, which raises hope for a high quality inversion result.
As a first step, the inversion was programmed for the more straightforward one-dimensional magnetotelluric case. This was particularly suitable to become familiar with sQMR – a Krylov subspace method which is essential for the three-dimensional case to be able to work with the Hessian and the resulting KKT system. After the implementation and validation of the one-dimensional forward operator, the Lagrangian and its derivatives were set up to complete the inversion, which successfully solved the KKT system. Accordingly, the three-dimensional forward operator also needed to be implemented and validated, which was done using published data from the 3D-2 COMMEMI model. To realise the inversion, the Lagrangian was assembled and its first and second derivatives were validated with a test that exploits the Taylor expansion. Then, the inversion was initially programmed for the Gauss-Newton approximation where second order information is neglected. Since the system matrix of the Gauss-Newton approximation is positive definite, the solution of this system of equations could be carried out by the conventional solver pcg. Based on that, the complete KKT system (Newton\\\'s method) was set up and preconditioned sQMR solved this system of equations.
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Space-time Discretization Of Optimal Control Of Burgers Equation Using Both Discretize-then-optimize And Optimize-then-discretize ApproachesYilmaz, Fikriye Nuray 01 July 2011 (has links) (PDF)
Optimal control of PDEs has a crucial place in many parts of sciences and industry. Over the
last decade, there have been a great deal in, especially, control problems of elliptic problems.
Optimal control problems of Burgers equation that is as a simplifed model for turbulence
and in shock waves were recently investigated both theoretically and numerically. In this
thesis, we analyze the space-time simultaneous discretization of control problem for Burgers
equation. In literature, there have been two approaches for discretization of optimization
problems: optimize-then-discretize and discretize-then-optimize. In the first part, we follow
optimize-then-discretize appoproach. It is shown that both distributed and boundary time dependent
control problem can be transformed into an elliptic pde. Numerical results obtained
with adaptive and non-adaptive elliptic solvers of COMSOL Multiphysics are presented for
both the unconstrained and the control constrained cases. As for second part, we consider
discretize-then-optimize approach. Discrete adjoint concept is covered. Optimality conditions,
KKT-system, lead to a saadle point problem. We investigate the numerical treatment
for the obtained saddle point system. Both direct solvers and iterative methods are considered. For iterative mehods, preconditioners are needed. The structures of preconditioners for
both distributed and boundary control problems are covered. Additionally, an a priori error
analysis for the distributed control problem is given. We present the numerical results at the
end of each chapter.
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Development of a three-dimensional all-at-once inversion approach for the magnetotelluric methodWilhelms, Wenke 21 June 2016 (has links)
A three-dimensional inversion was implemented for magnetotellurics, which is a passive electromagnetic method in geophysics. It exploits natural electromagnetic fields of the Earth, which function as sources. Their interaction with the conductive parts of the subsurface are registered when components of the electric and the magnetic field are measured and evaluated.
The all-at-once approach is an inversion scheme that is relatively new to geophysics. In this approach, the objective function – the basis of each inversion – is called the Lagrangian. It consists of three parts: (i) the data residual norm, (ii) the regularisation part, and (iii) the forward problem. The latter is the significant difference to conventional inversion approaches that are built up of a forward calculation part and an inversion part. In the case of all-at-once, the forward problem is incorporated in the objective function and is therefore already taken into account in each inversion iteration. Thus, an explicit forward calculation is obsolete. As an objective function, the Lagrangian shall reach a minimum and therefore its first and second derivatives are evaluated. Hence, the gradient of the Lagrangian and its Hessian are constituent parts of the KKT system – the Newton-type system that is set up in the all-at-once inversion. Conventional inversion approaches avoid the Hessian because it is a large, dense, not positive definite matrix that is challenging to handle. However, it provides additional information to the inversion, which raises hope for a high quality inversion result.
As a first step, the inversion was programmed for the more straightforward one-dimensional magnetotelluric case. This was particularly suitable to become familiar with sQMR – a Krylov subspace method which is essential for the three-dimensional case to be able to work with the Hessian and the resulting KKT system. After the implementation and validation of the one-dimensional forward operator, the Lagrangian and its derivatives were set up to complete the inversion, which successfully solved the KKT system. Accordingly, the three-dimensional forward operator also needed to be implemented and validated, which was done using published data from the 3D-2 COMMEMI model. To realise the inversion, the Lagrangian was assembled and its first and second derivatives were validated with a test that exploits the Taylor expansion. Then, the inversion was initially programmed for the Gauss-Newton approximation where second order information is neglected. Since the system matrix of the Gauss-Newton approximation is positive definite, the solution of this system of equations could be carried out by the conventional solver pcg. Based on that, the complete KKT system (Newton\\\'s method) was set up and preconditioned sQMR solved this system of equations.
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Solving Optimal Control Time-dependent Diffusion-convection-reaction Equations By Space Time DiscretizationsSeymen, Zahire 01 February 2013 (has links) (PDF)
Optimal control problems (OCPs) governed by convection dominated diffusion-convection-reaction
equations arise in many science and engineering applications such as shape optimization of the technological
devices, identification of parameters in environmental processes and flow control problems.
A characteristic feature of convection dominated optimization problems is the presence of sharp layers.
In this case, the Galerkin finite element method performs poorly and leads to oscillatory solutions.
Hence, these problems require stabilization techniques to resolve boundary and interior layers accurately.
The Streamline Upwind Petrov-Galerkin (SUPG) method is one of the most popular stabilization
technique for solving convection dominated OCPs.
The focus of this thesis is the application and analysis of the SUPG method for distributed and
boundary OCPs governed by evolutionary diffusion-convection-reaction equations. There are two approaches
for solving these problems: optimize-then-discretize and discretize-then-optimize. For the
optimize-then-discretize method, the time-dependent OCPs is transformed to a biharmonic equation,
where space and time are treated equally. The resulting optimality system is solved by the finite
element package COMSOL. For the discretize-then-optimize approach, we have used the so called allv
at-once method, where the fully discrete optimality system is solved as a saddle point problem at once
for all time steps. A priori error bounds are derived for the state, adjoint, and controls by applying
linear finite element discretization with SUPG method in space and using backward Euler, Crank-
Nicolson and semi-implicit methods in time. The stabilization parameter is chosen for the convection
dominated problem so that the error bounds are balanced to obtain L2 error estimates. Numerical examples
with and without control constraints for distributed and boundary control problems confirm the
effectiveness of both approaches and confirm a priori error estimates for the discretize-then-optimize
approach.
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Simultaneous Plant/Controller Optimization of Traction Control for Electric VehicleTong, Kuo-Feng January 2007 (has links)
Development of electric vehicles is motivated by global concerns over the need
for environmental protection. In addition to its zero-emission characteristics, an
electric propulsion system enables high performance torque control that may be
used to maximize vehicle performance obtained from energy-efficient, low rolling
resistance tires typically associated with degraded road-holding ability.
A simultaneous plant/controller optimization is performed on an electric vehicle
traction control system with respect to conflicting energy use and performance
objectives. Due to system nonlinearities, an iterative simulation-based optimization
approach is proposed using a system model and a genetic algorithm (GA) to guide
search space exploration.
The system model consists of: a drive cycle with a constant driver torque request
and a step change in coefficient of friction, a single-wheel longitudinal vehicle model,
a tire model described using the Magic Formula and a constant rolling resistance,
and an adhesion gradient fuzzy logic traction controller.
Optimization is defined in terms of the all at once variable selection of: either
a performance oriented or low rolling resistance tire, the shape of a fuzzy logic
controller membership function, and a set of fuzzy logic controller rule base conclusions.
A mixed encoding, multi-chromosomal GA is implemented to represent the
variables, respectively, as a binary string, a real-valued number, and a novel rule
base encoding based on the definition of a partially ordered set (poset) by delta
inclusion.
Simultaneous optimization results indicate that, under straight-line acceleration
and unless energy concerns are completely neglected, low rolling resistance tires
should be incorporated in a traction control system design since the energy saving
benefits outweigh the associated degradation in road-holding ability. The results
also indicate that the proposed novel encoding enables the efficient representation
of a fix-sized fuzzy logic rule base within a GA.
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Simultaneous Plant/Controller Optimization of Traction Control for Electric VehicleTong, Kuo-Feng January 2007 (has links)
Development of electric vehicles is motivated by global concerns over the need
for environmental protection. In addition to its zero-emission characteristics, an
electric propulsion system enables high performance torque control that may be
used to maximize vehicle performance obtained from energy-efficient, low rolling
resistance tires typically associated with degraded road-holding ability.
A simultaneous plant/controller optimization is performed on an electric vehicle
traction control system with respect to conflicting energy use and performance
objectives. Due to system nonlinearities, an iterative simulation-based optimization
approach is proposed using a system model and a genetic algorithm (GA) to guide
search space exploration.
The system model consists of: a drive cycle with a constant driver torque request
and a step change in coefficient of friction, a single-wheel longitudinal vehicle model,
a tire model described using the Magic Formula and a constant rolling resistance,
and an adhesion gradient fuzzy logic traction controller.
Optimization is defined in terms of the all at once variable selection of: either
a performance oriented or low rolling resistance tire, the shape of a fuzzy logic
controller membership function, and a set of fuzzy logic controller rule base conclusions.
A mixed encoding, multi-chromosomal GA is implemented to represent the
variables, respectively, as a binary string, a real-valued number, and a novel rule
base encoding based on the definition of a partially ordered set (poset) by delta
inclusion.
Simultaneous optimization results indicate that, under straight-line acceleration
and unless energy concerns are completely neglected, low rolling resistance tires
should be incorporated in a traction control system design since the energy saving
benefits outweigh the associated degradation in road-holding ability. The results
also indicate that the proposed novel encoding enables the efficient representation
of a fix-sized fuzzy logic rule base within a GA.
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Development and Implementation of Rotorcraft Preliminary Design Methodology using Multidisciplinary Design OptimizationKhalid, Adeel S. 14 November 2006 (has links)
A formal framework is developed and implemented in this research for preliminary rotorcraft design using IPPD methodology. All the technical aspects of design are considered including the vehicle engineering, dynamic analysis, stability and control, aerodynamic performance, propulsion, transmission design, weight and balance, noise analysis and economic analysis. The design loop starts with a detailed analysis of requirements. A baseline is selected and upgrade targets are identified depending on the mission requirements. An Overall Evaluation Criterion (OEC) is developed that is used to measure the goodness of the design or to compare the design with competitors. The requirements analysis and baseline upgrade targets lead to the initial sizing and performance estimation of the new design. The digital information is then passed to disciplinary experts. This is where the detailed disciplinary analyses are performed. Information is transferred from one discipline to another as the design loop is iterated. To coordinate all the disciplines in the product development cycle, Multidisciplinary Design Optimization (MDO) techniques e.g. All At Once (AAO) and Collaborative Optimization (CO) are suggested. The methodology is implemented on a Light Turbine Training Helicopter (LTTH) design. Detailed disciplinary analyses are integrated through a common platform for efficient and centralized transfer of design information from one discipline to another in a collaborative manner. Several disciplinary and system level optimization problems are solved. After all the constraints of a multidisciplinary problem have been satisfied and an optimal design has been obtained, it is compared with the initial baseline, using the earlier developed OEC, to measure the level of improvement achieved. Finally a digital preliminary design is proposed. The proposed design methodology provides an automated design framework, facilitates parallel design by removing disciplinary interdependency, current and updated information is made available to all disciplines at all times of the design through a central collaborative repository, overall design time is reduced and an optimized design is achieved.
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