[ES] El acceso a una atención sanitaria de alta calidad es un marcador importante del desarrollo de las sociedades humanas. Los aportes tecnológicos a la medicina han mostrado un potencial relevante para descubrir procedimientos efectivos a nivel preventivo, diagnóstico y terapéutico. En particular, los métodos computacionales permiten el procesamiento eficaz de datos médicos y, por tanto, pueden modelar sistemas biológicos complejos. Esto ha influido en el desarrollo de la Medicina Personalizada (MP) durante las últimas décadas, donde la obtención de conocimiento específico de cada caso permite realizar intervenciones a medida, todo ello a un coste de recursos accesible. La simulación de remodelación ósea es un campo prometedor en el contexto de la MP. Predecir un proceso de adaptación ósea en un caso concreto puede dar lugar a numerosas aplicaciones en el campo de las enfermedades óseas, tanto a nivel clínico como experimental. Mediante la combinación del Método de Elementos Finitos (FEM) y los algoritmos de remodelación ósea, es posible obtener modelos numéricos de un hueso específico a partir de datos médicos (por ejemplo, una tomografía computarizada). Todo ello puede dar lugar a una revolución en la medicina personalizada. / [CA] L'accés a una atenció sanitària d'alta qualitat és un marcador important del desenvolupament de les societats humanes. Les aportacions tecnològiques a la medicina han mostrat un potencial rellevant per a descobrir procediments efectius a nivell preventiu, diagnòstic i terapèutic. En particular, els mètodes computacionals permeten el processament eficaç de dades mèdiques i, per tant, poden modelar sistemes biològics complexos. Això ha influït en el desenvolupament de la Medicina Personalitzada (MP) durant les últimes dècades, on l'obtenció de coneixement específic de cada cas permet realitzar intervencions a mesura, tot això a un cost de recursos accessible. La simulació de remodelació òssia és un camp prometedor en el context de la MP. Predir un procés d'adaptació òssia en un cas concret pot donar lloc a nombroses aplicacions en el camp de les malalties òssies, tant a nivell clínic com experimental. Mitjançant la combinació del Mètode d'Elements Finits (*FEM) i els algorismes de remodelació òssia, és possible obtindre models numèrics d'un os específic a partir de dades mèdiques (per exemple, una tomografia computada). Tot això pot donar lloc a una revolució en la medicina personalitzada. / [EN] Access to high-quality healthcare is an important marker of the development of human societies. Technological contributions to medicine have shown relevant potential to discover effective procedures at a preventive, diagnostic and therapeutic level. In particular, computational methods enable efficient processing of medical data and can therefore model complex biological systems. This has influenced the development of Personalized Medicine (PM) over recent decades, where obtaining specific knowledge of each case allows for tailored interventions, all at an affordable resource cost. Simulation of bone remodeling is a promising field in the context of PM. Predicting a bone adaptation process in a specific case can lead to numerous applications in the field of bone diseases, both clinically and experimentally. By combining the Finite Element Method (FEM) and bone remodeling algorithms, it is possible to obtain numerical models of a specific bone from medical data (for example, a CT scan). All of this can lead to a revolution in personalized medicine. / Thanks to the Valencian funding programme FDGENT/2018, for providing economic resources to develop this long-term work. / Gutiérrez Gil, J. (2023). Personalization of Bone Remodelling Simulation Models for Clinical Applications [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/202059

This thesis deals with regularization parameter selection methods in the context of Tikhonov-type regularization with Poisson distributed data, in particular the reconstruction of images, as well as with the identification of the volatility surface from observed option prices. In Part I we examine the choice of the regularization parameter when reconstructing an image, which is disturbed by Poisson noise, with Tikhonov-type regularization. This type of regularization is a generalization of the classical Tikhonov regularization in the Banach space setting and often called variational regularization. After a general consideration of Tikhonov-type regularization for data corrupted by Poisson noise, we examine the methods for choosing the regularization parameter numerically on the basis of two test images and real PET data. In Part II we consider the estimation of the volatility function from observed call option prices with the explicit formula which has been derived by Dupire using the Black-Scholes partial differential equation. The option prices are only available as discrete noisy observations so that the main difficulty is the ill-posedness of the numerical differentiation. Finite difference schemes, as regularization by discretization of the inverse and ill-posed problem, do not overcome these difficulties when they are used to evaluate the partial derivatives. Therefore we construct an alternative algorithm based on the weak formulation of the dual Black-Scholes partial differential equation and evaluate the performance of the finite difference schemes and the new algorithm for synthetic and real option prices.

This thesis deals with regularization parameter selection methods in the context of Tikhonov-type regularization with Poisson distributed data, in particular the reconstruction of images, as well as with the identification of the volatility surface from observed option prices. In Part I we examine the choice of the regularization parameter when reconstructing an image, which is disturbed by Poisson noise, with Tikhonov-type regularization. This type of regularization is a generalization of the classical Tikhonov regularization in the Banach space setting and often called variational regularization. After a general consideration of Tikhonov-type regularization for data corrupted by Poisson noise, we examine the methods for choosing the regularization parameter numerically on the basis of two test images and real PET data. In Part II we consider the estimation of the volatility function from observed call option prices with the explicit formula which has been derived by Dupire using the Black-Scholes partial differential equation. The option prices are only available as discrete noisy observations so that the main difficulty is the ill-posedness of the numerical differentiation. Finite difference schemes, as regularization by discretization of the inverse and ill-posed problem, do not overcome these difficulties when they are used to evaluate the partial derivatives. Therefore we construct an alternative algorithm based on the weak formulation of the dual Black-Scholes partial differential equation and evaluate the performance of the finite difference schemes and the new algorithm for synthetic and real option prices.