2024-03-28T21:51:44Zhttps://bibdigital.epn.edu.ec/oai/requestoai:bibdigital.epn.edu.ec:15000/173442019-04-08T01:41:39Zcom_15000_17342com_15000_7554com_15000_7553com_15000_1col_15000_17343
Regularización de variación total generalizada para el problema mal condicionado de asimilación de datos
Loayza Romero, Karen Estefanía
In this paper we study the second order total generalized variation (TGV) regularization applied to the well-known data assimilation problem using the Burgers equation as the state equation. It has been experimentally shown that the TGV regularization preserve the sharp fronts and recovers the solutions better compared to the TV regularization, mainly because it eliminates the staircase effect. We perform the derivation of the first order optimality conditions. For the numerical solution of the Burgers equation, a semi-implicit time discretization was performed and for the spatial variables the finite differences scheme with Upwinding was used for the first order spatial derivative. The solution of the problem was made using a globalized Newton method and for this method we show theoretical results that guarantee the convergence to stationary points of the problem. The numerical experiments section is devoted to show the way each regularization recover the solutions. In particular, we show the staircase effect produced by the TV regularization and the way how the TGV regularization eliminates it.
Quito, 2017.
2017-05-29
masterThesis
spa
De los Reyes Bueno, Juan Carlos, director
http://bibdigital.epn.edu.ec/handle/15000/17344
/var/dspace/bitstream/15000/17344/1/CD-7840.pdf
c424681157319a1fb35761d836e0429a
/var/dspace/bitstream/15000/17344/2/license.txt
8a4605be74aa9ea9d79846c1fba20a33
/var/dspace/bitstream/15000/17344/3/CD-7840.pdf.txt
79bf6ce927d92f55ffc325cb2abe2729
openAccess
OPTIMIZACIÓN MATEMÁTICA
MATEMÁTICA APLICADA
oai:bibdigital.epn.edu.ec:15000/173762019-04-08T01:46:43Zcom_15000_17342com_15000_7554com_15000_7553com_15000_1col_15000_17343
Solución de un problema de asimilación de datos y un problema de localización óptima mediante métodos de optimización binivel
Castro Castro, Paula Monserratte
Data assimilation problems have been widely studied in numerical weather prediction as a technique for the reconstruction of the atmosphere’s initial condition. Taking this problem as motivation, our goal in this project is finding the optimal placements of the locations of a data assimilation problem, represented by a parabolic equation. We worked in a bilevel optimization problem where the inner level solves the data assimilation problem, and the upper level solves the optimal placement problem. To solve the data assimilation problem, we used a variational approach 4DVAR. The existence and uniqueness of the data assimilation problem were demonstrated using usual techniques in optimization. By using the Lagrangian approach, we derive the optimality system of the inner problem. As result, we got an adjoint equation with measures on the right-hand side. This was due to the structure of the objective functional. We proved the existence of a unique very weak solution of the adjoint equation by using the transposition method. The numerical solution was also worked in two levels. The inner problem was solved by using the BFGS method while the upper level used the BFGS projected method through an estimation of active sets.
Quito, 2017.
2017-05-29
bachelorThesis
spa
De los Reyes Bueno, Juan Carlos, director
http://bibdigital.epn.edu.ec/handle/15000/17376
/var/dspace/bitstream/15000/17376/1/CD-7874.pdf
a3028163bcaad5285114fcb780156a4e
/var/dspace/bitstream/15000/17376/2/license.txt
8a4605be74aa9ea9d79846c1fba20a33
/var/dspace/bitstream/15000/17376/3/CD-7874.pdf.txt
3394208f7da5ec99e7db3a28c606b9d5
openAccess
OPTIMIZACIÓN MATEMÁTICA
MATEMÁTICA APLICADA
oai:bibdigital.epn.edu.ec:15000/173772019-04-07T13:29:11Zcom_15000_17342com_15000_7554com_15000_7553com_15000_1col_15000_17343
Optimización binivel del parámetro de regularización con dependencia espacial del modelo de variación total generalizada para el filtrado de ruido en imágenes
Herrera Terán, Maribel Kateryn
In this thesis, we study and solve a nonlinear bilevel optimization problem in function spaces. The goal is to determine the optimal spatially dependent regularization parameters in total variation (TV) and generalized total variation (TGV) image denoising models. Considering the spatial dependence of the parameters allows us to filter non-uniform noise in an image, which brings us closer to real situations where the type and distribution of noise are not known. We present some analytical results like the existence of solutions of the problem of parameter optimization, the Fréchet differentiability of the solution operator, which allows to prove the existence of Lagrange multipliers. The multipliers associated with the positivity constraints are regular Borel measured which are very difficult to compute. In order to overcome this issue, we proposed a Moreau-Yosida regularization, where the optimality system associated with the regularized problem was established and we prove that the solutions of regularized problems converge to the solution of the original one.
Quito, 2017.
2017-06-07
masterThesis
spa
De Los Reyes Bueno, Juan Carlos, director
http://bibdigital.epn.edu.ec/handle/15000/17377
/var/dspace/bitstream/15000/17377/2/license.txt
8a4605be74aa9ea9d79846c1fba20a33
/var/dspace/bitstream/15000/17377/1/CD-7875.pdf
5919e9d4713cc5c55410934639ddac6e
/var/dspace/bitstream/15000/17377/3/CD-7875.pdf.txt
221dfd56f145c543884b295b3dd3af49
openAccess
OPTIMIZACIÓN MATEMATICA
METODOS VARIACIONALES
oai:bibdigital.epn.edu.ec:15000/173782019-04-08T01:46:58Zcom_15000_17342com_15000_7554com_15000_7553com_15000_1col_15000_17343
A multigrid optimization approach for the numerical solution of a class of variational inequalities of the second kind
López Ordóñez, Sofía Alejandra
In this thesis we introduce a Multigrid Optimization Algorithm (MG/OPT) for the numerical solution of a class of quasilinear variational inequalities of the second kind, which involve the p-Laplacian operator and the L1-norm of the gradient. This approach follows from the fact that the solution of the variational inequality is given by the minimizer of a nonsmooth energy functional, Therefore, we proposed a Huber
regularization of the functional and a finite element discretization for the problem. Further, we analyze the regularity of the discretized energy functional, and we are able to prove that its Jacobian is slantly differentiable. This regularity property is useful to analyze the convergence of the MG/OPT algorithm. In fact, we demostrate that the algorithm is global convergent by using the mean value theorem for slantly differentiable functions. Finally, we analyze the performance of the MG/OPT algorithm when used to simulate the visco-plastic flow of Bingham, Casson and Herschel-Bulkley fluids in a pipe. Several numerical experiments are carried out to show the main features of the proposed method.
Quito, 2017.
2017-05-31
masterThesis
spa
González Andrade, Sergio Alejandro, director
http://bibdigital.epn.edu.ec/handle/15000/17378
/var/dspace/bitstream/15000/17378/1/CD-7876.pdf
fb23136d0535fd4590f7a9cfd1f5f82c
/var/dspace/bitstream/15000/17378/2/license.txt
8a4605be74aa9ea9d79846c1fba20a33
/var/dspace/bitstream/15000/17378/3/CD-7876.pdf.txt
f4ac1a7f9b9489ae3a25b9804878574c
openAccess
OPTIMIZACIÓN MATEMATICA
DESIGUALDADES VARIACIONALES
oai:bibdigital.epn.edu.ec:15000/193752019-04-07T12:18:12Zcom_15000_17342com_15000_7554com_15000_7553com_15000_1col_15000_17343
Un método combinado de Newton Semi-smooth y conjuntos de nivel para la simulación numérica de fluidos tipo Bingham con frontera libre
Núñez Ramos, Cristhian Alexander
In this thesis we consider a quasi-static formulation of a time-dependent free boundary Bingham flow, where the static subproblem is formulated as a primal--dual optimality system of a regularized version of the original Bingham model. We propose a numerical scheme for its numerical simulation that combines a Semismooth-Newton method for solving the associated Bingham static subproblem, together with the level-set method which describes the motion of the free boundary of the Bingham fluid. We analyze the static subproblem and present numerical simulations of the proposed methodology. In order to validate our methodology, we will compare this one with known numerical results for the Newtonian case.
Quito, 2018.
2018-04-26
bachelorThesis
spa
Merino Rosero, Pedro Martín, director
http://bibdigital.epn.edu.ec/handle/15000/19375
/var/dspace/bitstream/15000/19375/1/CD-8759.pdf
ebf9e693ce0e45d57c2b59634ca23f2e
/var/dspace/bitstream/15000/19375/2/license.txt
8a4605be74aa9ea9d79846c1fba20a33
/var/dspace/bitstream/15000/19375/3/CD-8759.pdf.txt
74c396d698721cb9a75f0673082ea4e2
/var/dspace/bitstream/15000/19375/4/CD-8759.pdf.txt
74c396d698721cb9a75f0673082ea4e2
openAccess
SIMULACIÓN NUMÉRICA
METODOS NUMERICOS
oai:bibdigital.epn.edu.ec:15000/217292021-07-13T15:29:54Zcom_15000_17342com_15000_7554com_15000_7553com_15000_1col_15000_17343
Aproximación por elementos finitos de una clase problemas de control óptimo no convexos gobernados por ecuaciones diferenciales parciales elíptica
Nenjer Morillo, Hernán Alexander
The purpose of this study is deriving an estimation of the error generated by the approximation, by the
finite element method, an optimal control problem of the elliptical type, whose cost functional has a nonconvex term that regularizes locally the quasi-norm L^q (with q\in (0.1)). Our study is based on
obtaining the optimality system by formulating the cost functional as a difference of convex functions
(DC), which involves a L^1 penalty optimal control problem.
We consider the space of piece-wise linear continuous functions for the discretization of the state
equation and the space of piece-wise constant functions for the approximation of the controls. Under
certain conditions established for the parameters of regularization of the problem we obtain that the
convergence of our approximation can reach a linear order of convergence as the size of the mesh
decreases. Finally, numerical experiments of the obtained theoretical results will be illustrated.
Quito, 2021
2021-07-05
masterThesis
spa
Merino Rosero, Pedro Martin, director
http://bibdigital.epn.edu.ec/handle/15000/21729
/var/dspace/bitstream/15000/21729/2/license.txt
8a4605be74aa9ea9d79846c1fba20a33
/var/dspace/bitstream/15000/21729/1/CD%2011209.pdf
bd3d0649c11dfd31c923544581314739
openAccess
CONTROL ELÍPTICO
NO CONVEXIDAD
oai:bibdigital.epn.edu.ec:15000/217512021-07-28T17:33:25Zcom_15000_17342com_15000_7554com_15000_7553com_15000_1col_15000_17343
Particionamiento de un Grafo General en k Componentes Conexas usando Generación de Columnas.
Cordero Cárdenas, Mishelle Briggitte
In this work we study the problem of partitioning general graphs into a fixed
number of connected components. The problem takes as input an undirected general
graph G := (V, E) with costs on the edges. The partitioning problem consists of
partitioning the set of nodes into a fixed number of subsets such that each subset
induces a connected subgraph and the total cost of the edges in each subgraph must
be minimized. Three Integer Programming formulations are provided and a heuristic
method for column generation is proposed. Finally, computational results based
on simulated instances are reported.
Quito, 2021
2021-07-27
masterThesis
spa
Recalde Calahorrano, Diego Fernando, director
http://bibdigital.epn.edu.ec/handle/15000/21751
/var/dspace/bitstream/15000/21751/2/license.txt
8a4605be74aa9ea9d79846c1fba20a33
/var/dspace/bitstream/15000/21751/1/CD%2011233.pdf
15b5502ac225322aa97ac29776618905
openAccess
BRANCH & BOUND
GENERACIÓN DE COLUMNAS
oai:bibdigital.epn.edu.ec:15000/218072021-09-06T17:49:27Zcom_15000_17342com_15000_7554com_15000_7553com_15000_1col_15000_17343
Análisis de un problema de control óptimo no-suave asociado a un fluido dilatante.
Quiloango Chimarro, Paola Nathaly
This thesis focuses on the analysis of an optimal control problem governed by a nonsmooth equation that
models a shear-thickening fluid. First, we study the directional differentiability of the non-smooth nonlinearity
in the state equation. Next, we analyze the directional differentiability of the solution operator. We establish a
necessary first order optimality condition, which is deduced from the directional differentiability of the solution
operator. This condition is related to the concept of Bouligand stationarity. Also, we derive another necessary
optimality condition by using a regularization and passage to the limit technique. The optimality system
derived in this way is considered a weak stationarity condition. Finally, we combine both optimality conditions
to obtain an optimality system, which is related to a stronger stationarity concept. Furthermore, we show that
this stronger stationarity concept is equivalent to the Bouligand stationarity concept, in the case of our problem.
Quito, 2021
2021-09-01
masterThesis
spa
De los Reyes Bueno, Juan Carlos, director
http://bibdigital.epn.edu.ec/handle/15000/21807
/var/dspace/bitstream/15000/21807/3/license.txt
8a4605be74aa9ea9d79846c1fba20a33
/var/dspace/bitstream/15000/21807/2/CD%2011289.pdf
36d58be59290175dfd57c77e0e2ba411
openAccess
OPTIMIZACIÓN
CONTROL ÓPTIMO EDP
oai:bibdigital.epn.edu.ec:15000/222722022-03-21T14:27:05Zcom_15000_17342com_15000_7554com_15000_7553com_15000_1col_15000_17343
Control óptimo no convexo de ecuaciones diferenciales elípticas con restricciones puntuales de estado.
Vargas Jaramillo, Diego Mauricio
We consider an optimal control problem governed by an elliptic partial differential equation with homogeneous Neumann conditions, whose cost function involves a non-convex penalty term and pointwise state constraints are imposed. The usual tools from convex analysis cannot be applied directly for deriving an optimality system therefore, it is necessary to introduce techniques from nonsmooth analysis. In addition, the presence of pointwise state constraints leads to poor regularity Lagrange multipliers, making it difficult to compute solutions numerically. Due to these difficulties, we introduce two regularizations to deal with the lack of convexity of the cost function and the low regularity of the Lagrange multipliers associated with the pointwise state constraints. Specifically, we use a Huber-like regularization to cope with the quasinorm, whereas a Lavrentiev regularization is considered for the state constraints. Thanks to these regularizations, it is possible to formulate a family of optimal control problems (depending on the regularization parameters) for which we derive an optimality system by using the Difference of Convex Functions (DC) theory involving more regular Lagrange multipliers.
Later, we solve the regularized problem numerically by applying the semi-smooth Newton method. We analyze the qualitative properties of the solution of the regularized problem, and we numerically asses the variation of the parameters associated to the regularized problem. Finally, we study the convergence rate of the proposed algorithm numerically, and we draw conclusions and discuss open questions for future research.
Quito, 2022
2022-03-21
bachelorThesis
spa
Merino Rosero, Pedro Martín, director
http://bibdigital.epn.edu.ec/handle/15000/22272
/var/dspace/bitstream/15000/22272/2/license.txt
8a4605be74aa9ea9d79846c1fba20a33
/var/dspace/bitstream/15000/22272/1/CD%2011764.pdf
3a661350ee62058a498a1872a14e671e
openAccess
CÁLCULO CIENTÍFICO
MODELIZACIÓN
oai:bibdigital.epn.edu.ec:15000/236672023-03-08T21:49:13Zcom_15000_17342com_15000_7554com_15000_7553com_15000_1col_15000_17343
Algoritmos de aproximación para un problema multi-periodo de calendarización de máquinas paralelas.
Jiménez Torres, Fernando Germán
Quito : EPN, 2023.
2023-02
masterThesis
spa
The aim of this thesis is to study a multi-period version of SRDM (Scheduling with Release times and Deadlines on a minimum number of Machines), in which some given jobs with release times and deadlines must be scheduled minimizing the number of required machine-periods. We investigate the application of an approximation algorithm used for the JISP (Job Interval Selection Problem) and two approximation algorithms for the SRDM to the solution of this new problem. We establish bounds on the approximability factor of the proposed algorithms and test their performance on computational experiments.
http://bibdigital.epn.edu.ec/handle/15000/23667
/var/dspace/bitstream/15000/23667/2/license.txt
8a4605be74aa9ea9d79846c1fba20a33
/var/dspace/bitstream/15000/23667/1/CD%2012968.pdf
bcb0da72ccb04d72282bdeb1c612585d
openAccess
MATEMÁTICAS
PROGRAMACIÓN LINEAL ENTERA
ALGORITMOS DE APROXIMACIÓN
CALENDARIZACIÓN DE INTERVALOS
JISP
SRDM
oai:bibdigital.epn.edu.ec:15000/236752023-03-14T13:57:45Zcom_15000_17342com_15000_7554com_15000_7553com_15000_1col_15000_17343
Algoritmo forward-backward multimalla con aplicación al problema de supresión de ruido en una imagen.
Guerra Urgilés, Edison Felipe
Throughout history, optimization problems have generated great interest in the scientific community, leading to
the development of theories and algorithms that determine the solution, or approximation of it, of this branch of
mathematics. Image processing is a field that involves optimization. This branch of research has been studied,
with greater intensity in recent decades, for the diversity of applications such as: satellite images, X -ray
images, computerized tomography, etc.
Within this diverse set of applications associated with optimization problems, those that due to their extensive
number of variables are called large -scale optimization problems arise. Multigrid methods were developed to
address large -scale problems. Then a natural question emerges, is it possible to design a multigrid algorithm
from the Forward-Backward Splitting Method method in such a way that they approach the solution of a large scale optimization problem?
With the aim of answering the question asked, we are preparing to develop a multigrid algorithm derived from
the Forward-backward Splitting Method method. To do this, we will first design the problem of lower
dimension and a certain condition that relates both the original problem and the problem of lower dimension.
Then, we will build a decrease address for the directional derivative of the original problem. Finally, we will
indicate the best way to perform a linear search that provides a quasi-Fejér-monotonicity in the line search
iteration, inherent property of the Forward-backward method.
Quito : EPN, 2023.
2023-01
bachelorThesis
spa
Valkonen, Tuomo, director.
http://bibdigital.epn.edu.ec/handle/15000/23675
/var/dspace/bitstream/15000/23675/2/license.txt
8a4605be74aa9ea9d79846c1fba20a33
/var/dspace/bitstream/15000/23675/1/CD%2012986.pdf
7c9fd856509dc0542def80879b68875a
openAccess
MÉTODO MULTIMALLA
IMÁGENES
OPTIMIZACIÓN
MATEMÁTICAS
oai:bibdigital.epn.edu.ec:15000/248662023-10-02T15:38:34Zcom_15000_17342com_15000_7554com_15000_7553com_15000_1col_15000_17343
Particionamiento Balanceado de Hipergrafos en un Número Fijo de Componentes.
Landázuri Mejía, Guillermo Andrés
In this paper, the partitioning hypergraph problem in a fixed number of components is studied. Hypergraphs are the generalization of graphs where, unlike edges, each hyperedge can connect two or more vertices. Given a hypergraph H = (V,E), where V is the vertices set and E the hyperedges set, we seek to partition its vertex set into k disjoint components such that each vertex is covered by hyperedges completely contained in some component, while minimizing the total cost of these hyperedges. Several Integer Programming formulations are proposed for the k-way equipartitioning, minimum size partitioning, balanced partitioning, and k-way equipartitioning in linear hyper-trees. Moreover, some types of valid inequalities are demonstrated and implemented for the different formulations. Finally, the computational experiments performed on the different formulations for different instances are discussed.
Quito : EPN, 2023.
2023-09-28
masterThesis
spa
http://bibdigital.epn.edu.ec/handle/15000/24866
/var/dspace/bitstream/15000/24866/2/license.txt
8a4605be74aa9ea9d79846c1fba20a33
/var/dspace/bitstream/15000/24866/1/CD%2013537.pdf
0b8b910b3edb025215686ef36ad92028
openAccess
PARTICIONAMIENTO DE HIPERGRAFOS
EQUIPARTICIONAMIENTO
DESIGUALDADES VÁLIDAS
PROGRAMACIÓN ENTERA
OPTIMIZACIÓN EN GRAFOS