An Early History of Optimization Technology for Automated Design of Microwave Circuits

This paper outlines the early history of optimization technology for the design of microwave circuits—a personal journey filled with aspirations, academic contributions, and commercial innovations. Microwave engineers have evolved from being consumers of mathematical optimization algorithms to originators of exciting concepts and technologies that have spread far beyond the boundaries of microwaves. From the early days of simple direct search algorithms based on heuristic methods through gradient-based electromagnetic optimization to space mapping technology we arrive at today’s surrogate methodologies. Our path finally connects to today’s multi-physics, system-level, and measurement-based optimization challenges exploiting confined and feature-based surrogates, cognition-driven space mapping, Bayesian approaches, and more. Our story recognizes visionaries such as William J. Getsinger of the 1960s and Robert Pucel of the 1980s, and highlights a seminal decades-long collaboration with mathematician Kaj Madsen. We address not only academic contributions that provide proof of concept, but also indicate early formative milestones in the development of commercially competitive software specifically featuring optimization technology.ITESO, A.C

Optimal control and robust estimation for ocean wave energy converters

This thesis deals with the optimal control of wave energy converters and some associated observer design problems. The first part of the thesis will investigate model predictive control of an ocean wave energy converter to maximize extracted power. A generic heaving converter that can have both linear dampers and active elements as a power take-off system is considered and an efficient optimal control algorithm is developed for use within a receding horizon control framework. The optimal control is also characterized analytically. A direct transcription of the optimal control problem is also considered as a general nonlinear program. A variation of the projected gradient optimization scheme is formulated and shown to be feasible and computationally inexpensive compared to a standard nonlinear program solver. Since the system model is bilinear and the cost function is not convex quadratic, the resulting optimization problem is shown not to be a quadratic program. Results are compared with other methods like optimal latching to demonstrate the improvement in absorbed power under irregular sea condition simulations. In the second part, robust estimation of the radiation forces and states inherent in the optimal control of wave energy converters is considered. Motivated by this, low order H∞ observer design for bilinear systems with input constraints is investigated and numerically tractable methods for design are developed. A bilinear Luenberger type observer is formulated and the resulting synthesis problem reformulated as that for a linear parameter varying system. A bilinear matrix inequality problem is then solved to find nominal and robust quadratically stable observers. The performance of these observers is compared with that of an extended Kalman filter. The robustness of the observers to parameter uncertainty and to variation in the radiation subsystem model order is also investigated. This thesis also explores the numerical integration of bilinear control systems with zero-order hold on the control inputs. Making use of exponential integrators, exact to high accuracy integration is proposed for such systems. New a priori bounds are derived on the computational complexity of integrating bilinear systems with a given error tolerance. Employing our new bounds on computational complexity, we propose a direct exponential integrator to solve bilinear ODEs via the solution of sparse linear systems of equations. Based on this, a novel sparse direct collocation of bilinear systems for optimal control is proposed. These integration schemes are also used within the indirect optimal control method discussed in the first part.Open Acces

Two-dimensional modeling and inversion of the controlled-source electromagnetic and magnetotelluric methods using finite elements and full-space PDE-constrained optimization strategies

[eng] The controlled-source electromagnetics (CSEM) and magnetotellurics (MT) methods are common geophysical tools for imaging the Earth's electrical interior. To appreciate measured data, both methods require forward and inverse modeling of the subsurface with the ultimate goal of finding a feasible model for which the simulated data reasonably fits the observations. The goodness of this fit depends on the error in the measured data, on the numerical error and on the degree of approximation inferred by numerical modeling. Therefore, active research focuses on new methods for modeling and inversion to improve accuracy and reliability for increasingly complex scenarios. In a first step, physical factors such as anisotropy, topography and realistic sources must be taking into account. Second, numerical methods need to be assessed in terms of solution accuracy, time efficiency and memory demand. The finite elements (FE) methods offer much flexibility in model geometry and contain quality control mechanisms for the solution, as shape function order and adaptive mesh refinement. Most emerging modeling programs are based on FE, however, inversion programs are generally based on finite differences (FD) or integral equation (IE) methods. On the other hand, inverse modeling is usually based on gradient methods and formulated in the reduced-space, where the electrical conductivity is the only optimization variable. Originally, the inverse problem is stated for the EM fields and the conductivity parameter (in the full-space), and constrained by governing partial differential equations (PDEs). The reduced-space strategy eliminates the field variables by applying equality constraints and solves then, the unconstrained problem. A common drawback of this is the repeated costly computation of the forward solution. Solving the PDE-constrained optimization problem directly, in the full-space, has the advantage that it is only necessary to exactly solve the PDEs at the end of the optimization, but it comes at the cost of a larger number of variables. This thesis develops a robust and versatile adaptive unstructured mesh FE program to model the total field for two-dimensional (2-D) anisotropic CSEM and MT data, allowing for arbitrarily oriented, three-dimensional (3-D) sources, for which a two-and-a-half-dimensional (2.5-D) approximation is employed. The formulations of the problems in a FE framework are derived for isotropic and anisotropic subsurface conductivity structures. The accuracy of the solution is controlled and improved by a goal-oriented adaptive mesh refinement algorithm. Exhaustive numerical experiments validate the adaptive FE program for both CSEM and MT methods and on land and marine environments. The influence of the model dimensions, mesh design and order of shape functions on the solution accuracy is studied and notably, an outperformance of quadratic shape functions is found (compared to linear and cubic). Several examples demonstrate the effect of complex scenarios on EM data. In particular, we study the distortion caused by: the bathymetry, the orientation and geometry of the sources and the anisotropy, considering vertical and dipping cases. All examples showcase the importance of adequate consideration of these very common physical features of real world data. Further, a formulation for the 2.5-D CSEM inversion as a PDE-constrained optimization in full-space is derived within a FE framework following two strategies: discretize-optimize and optimize-discretize. The discretize-optimize formulation is implemented using a general purpose optimization algorithm. Two examples, a canonical reservoir model and a more realistic marine model with topography, demonstrate the performance of this inversion scheme, recovering in both cases the model’s main structures within an acceptable data misfit. Finally, the optimize-discretize formulation is derived in a FE framework, as a first step towards a development of an inversion scheme using adaptive FE meshes.[cat] El mètode de font electromagnètica controlada (CSEM) i el mètode magnetotel.lúric (MT) són tècniques geofísiques usades habitualment per obtenir una imatge de les propietats elèctriques del subsòl terrestre i s'utilitzen independentment, conjuntament i en combinació amb altres tècniques geofísiques. Per poder interpretar les dades, ambdós mètodes necessiten la modelització directa i inversa de la conductivitat elèctrica del subsòl amb l'objectiu final d'obtenir un model coherent per al qual les dades simulades s'ajustin de forma raonable a les observacions. Naturalment, la qualitat d'aquest ajust no només depèn de l'error en les dades mesurades i de l'error numèric, sinó també del grau en l'aproximació física inferit per la modelització numèrica. D'aquesta manera, les recerques actuals se centren a investigar noves metodologies per a la modelització i inversió, per tal d'obtenir models acurats i fiables de les estructures de la Terra en escenaris cada cop més complexos. Un primer pas és millorar les aproximacions en la modelització tenint en compte factors físics com ara l'anisotropia, la topografia o fonts més realistes. En segon lloc, per tal d'acomodar aquests factors en un programa de modelització i inversió i per poder tractar els conjunts de dades típicament llargs, els mètodes numèrics han de ser avaluats en termes de la precisió de la solució, l'eficiència en temps i la demanda en memòria. Els mètodes de modelització en elements finits (FE) són coneguts per oferir una major flexibilitat en la modelització de la geometria i contenen mecanismes de control de la solució, com ara l'ordre de les funcions forma i la tècnica de refinament adaptatiu de la malla. La majoria de programes de modelització emergents estan basats en els FE, i mostren avantatges significatius, però gairebé tots els programes de modelització inversa, encara avui dia, estan basats en el mètode de les diferències finites (FD) o en el mètode de l'equació integral (IE). A més a més, la modelització inversa desenvolupada per a dades electromagnètiques (EM) es basa generalment en mètodes del gradient i es formula en un espai reduït, on les úniques variables d'optimització són els paràmetres del model, és a dir, la conductivitat elèctrica del subsòl. Originalment, el problema invers es formula per als camps EM i per al paràmetre conductivitat, i està constret per les equacions diferencials en derivades parcials (PDEs) que governen les variables camps EM. L'estratègia d'espai reduït elimina les variables camps aplicant lligams d'igualtat i soluciona, doncs, el problema no constret en l'espai reduït dels paràmetres del model. Un desavantatge general d'aquests mètodes és la costosa repetició del càlcul de la solució del problema directe i de la matriu jacobiana de sensibilitats (per mètodes basats en Newton). D'altra banda, també és possible de solucionar el problema invers en l'espai complet de les variables camps EM i del paràmetre conductivitat. Solucionar-hi el problema d'optimització constret per les PDEs té l'avantatge que només és necessari de solucionar exactament el problema directe al final del procés d'optimització, però això comporta el cost addicional de tenir moltes més variables d'optimització i de la presència de lligams d'igualtat. També, en particular, en el marc dels FE, el problema d'optimització constret per les PDEs té l'avantatge afegit d'incloure tècniques sofisticades pròpies dels FE en el procés d'inversió, com ara el refinament adaptatiu de la malla. Aquesta tesi desenvolupa un programa robust i versàtil amb FE i malles irregulars adaptatives per modelar numèricament el camp total de dades CSEM i MT bidimensionals (2D) i anisòtropes, que permet l'ús de fonts tridimensionals (3D) orientades arbitràriament. Per tal de representar fonts CSEM 3D en un model físic 2D, s'utilitza una aproximació dos i mig dimensional (2.5D). Les formulacions FE es deriven per a ambdós mètodes, per a estructures de conductivitat del subsòl isòtropes i anisòtropes. Encara que el cas anisòtrop no és general, inclou anisotropia vertical i de cabussament. La precisió en la solució es controla i millora amb un algoritme de refinament adaptatiu de la malla utilitzant mètodes d'estimació de l'error a posteriori. Una sèrie exhaustiva d'experiments numèrics valida el programa de FE adaptatius per ambdós mètodes, CSEM i MT, i en escenaris terrestres i marins. S'estudia la influència de les dimensions del model, del disseny de la malla i de l'ordre de les funcions forma en l'exactitud de la solució i es troba un comportament notablement superior de les funcions forma quadràtiques comparades amb les lineals o cúbiques. Diferents exemples mostren l'efecte d'escenaris complexos sobre les dades EM, en particular, un model amb batimetria, un model terrestre i un de marí amb fonts orientades i de dimensió finita, un medi amb anisotropia vertical amb un reservori encastat i un altre amb un reservori encastat en una estructura anticlinal. Aquests exemples demostren la importància de considerar adequadament (en termes de modelització directa) característiques físiques com la topografia, l'orientació i geometria de la font i l'anisotropia del medi, que sovint es troben en mesures reals. Juntament amb això, es deriva una formulació per al problema invers 2.5D CSEM com una optimització constreta per les PDEs en l'espai complet i en un marc de FE, seguint dues estratègies diferents: discretització-optimització i optimització-discretització. L'estratègia de discretització-optimització considera que el problema invers es troba en forma discretitzada i deriva les condicions d'optimitat de la Lagrangiana i el pas de Newton. Contràriament, l'aproximació optimització-discretització deriva primer les condicions d'optimitat i el pas de Newton o una aproximació d'aquest, i després discretitza les equacions resultants. La implementació de la formulació discretització-optimització es mostra en dos exemples, un model canònic de reservori i un model marí més realista amb topografia, utilitzant un programa d'optimització de propòsit general, que és una implementació d'un algoritme de programació quadràtica seqüencial (SQP). Encara que no s'utilitza una regularització explícita, l'ús de diferents malles per al paràmetre del model i per a les variables camps, permet recuperar les principals estructures del model i obtenir un ajust de les dades acceptable. Cal dir, però, que l'eficiència en temps i memòria del programa hauria de millorar-se. Finalment, el problema invers 2.5D CSEM es formula com un problema d'optimització constret per les PDEs en l'espai complet i en un marc de FE utilitzant una estratègia d'optimització-discretització i com un primer pas per al desenvolupament d'un esquema d'inversió que utilitzi malles adaptatives de FE

Structure-Preserving Model Reduction of Physical Network Systems

This paper considers physical network systems where the energy storage is naturally associated to the nodes of the graph, while the edges of the graph correspond to static couplings. The first sections deal with the linear case, covering examples such as mass-damper and hydraulic systems, which have a structure that is similar to symmetric consensus dynamics. The last section is concerned with a specific class of nonlinear physical network systems; namely detailed-balanced chemical reaction networks governed by mass action kinetics. In both cases, linear and nonlinear, the structure of the dynamics is similar, and is based on a weighted Laplacian matrix, together with an energy function capturing the energy storage at the nodes. We discuss two methods for structure-preserving model reduction. The first one is clustering; aggregating the nodes of the underlying graph to obtain a reduced graph. The second approach is based on neglecting the energy storage at some of the nodes, and subsequently eliminating those nodes (called Kron reduction).</p

Numerical methods for phase retrieval

In this work we consider the problem of reconstruction of a signal from the magnitude of its Fourier transform, also known as phase retrieval. The problem arises in many areas of astronomy, crystallography, optics, and coherent diffraction imaging (CDI). Our main goal is to develop an efficient reconstruction method based on continuous optimization techniques. Unlike current reconstruction methods, which are based on alternating projections, our approach leads to a much faster and more robust method. However, all previous attempts to employ continuous optimization methods, such as Newton-type algorithms, to the phase retrieval problem failed. In this work we provide an explanation for this failure, and based on this explanation we devise a sufficient condition that allows development of new reconstruction methods---approximately known Fourier phase. We demonstrate that a rough (up to $\pi/2$ radians) Fourier phase estimate practically guarantees successful reconstruction by any reasonable method. We also present a new reconstruction method whose reconstruction time is orders of magnitude faster than that of the current method-of-choice in phase retrieval---Hybrid Input-Output (HIO). Moreover, our method is capable of successful reconstruction even in the situations where HIO is known to fail. We also extended our method to other applications: Fourier domain holography, and interferometry. Additionally we developed a new sparsity-based method for sub-wavelength CDI. Using this method we demonstrated experimental resolution exceeding several times the physical limit imposed by the diffraction light properties (so called diffraction limit).Comment: PhD. Thesi

Large Scale Inverse Problems

This book is thesecond volume of a three volume series recording the "Radon Special Semester 2011 on Multiscale Simulation &amp Analysis in Energy and the Environment" that took placein Linz, Austria, October 3-7, 2011. This volume addresses the common ground in the mathematical and computational procedures required for large-scale inverse problems and data assimilation in forefront applications. The solution of inverse problems is fundamental to a wide variety of applications such as weather forecasting, medical tomography, and oil exploration. Regularisation techniques are needed to ensure solutions of sufficient quality to be useful, and soundly theoretically based. This book addresses the common techniques required for all the applications, and is thus truly interdisciplinary. This collection of survey articles focusses on the large inverse problems commonly arising in simulation and forecasting in the earth sciences

Full waveform inversion procedures with irregular topography

Full waveform inversion (FWI) is a form of seismic inversion that uses data residual, found as the misfit, between the whole waveform of field acquired and synthesized seismic data, to iteratively update a model estimate until such misfit is sufficiently reduced, indicating synthetic data is generated from a relatively accurate model. The aim of the thesis is to review FWI and provide a simplified explanation of the techniques involved to those who are not familiar with FWI. In FWI the local minima problem causes the misfit to decrease to its nearest minimum and not the global minimum, meaning the model cannot be accurately updated. Numerous objective functions were proposed to tackle different sources of local minima. The ‘joint deconvoluted envelope and phase residual’ misfit function proposed in this thesis aims to combine features of these objective functions for a comprehensive inversion. The adjoint state method is used to generate an updated gradient for the search direction and is followed by a step-length estimation to produce a scalar value that could be applied to the search direction to reduce the misfit more efficiently. Synthetic data are derived from forward modelling involving simulating and recording propagating waves influenced by the mediums’ properties. The ‘generalised viscoelastic wave equation in porous media’ was proposed by the author in sub-chapter 3.2.5 to consider these properties. Boundary layers and conditions are employed to mitigate artificial reflections arising from computational simulations. Linear algebra solvers are an efficient tool that produces wavefield vectors for frequency domain synthetic data. Regions with topography require a grid generation scheme to adjust a mesh of nodes to fit into its non-quadrilateral shaped body. Computational co-ordinate terms are implemented within wave equations throughout topographic models where a single point in the model in physical domain are represented by cartesian nodes in the computational domains. This helps to generate an accurate and appropriate synthetic data, without complex modelling computations. Advanced FWI takes a different approach to conventional FWI, where they relax upon the use of misfit function, however none of their proponents claims the former can supplant the latter but suggest that they can be implemented together to recover the true model.Open Acces