Search

In this article on variational regularization for ill-posed nonlinear problems, we are once again discussing the consequences of an oversmoothing penalty term. This means in our model that the searched-for solution of the considered nonlinear operator equation does not belong to the domain of definition of the penalty functional. In the past years, such variational regularization has been investigated comprehensively in Hilbert scales, but rarely in a Banach space setting. Our present results try to establish a theoretical justification of oversmoothing regularization in Banach scales. This new study includes convergence rates results for a priori and a posteriori choices of the regularization parameter, both for H\"older-type smoothness and low order-type smoothness. An illustrative example is intended to indicate the specificity of occurring non-reflexive Banach spaces., Comment: arXiv admin note: substantial text overlap with arXiv:2206.02010

Numerical differentiation of a function, contaminated with noise, over the unit interval $[0,1] \subset \mathbb{R}$ by inverting the simple integration operator $J:L^2([0,1]) \to L^2([0,1])$ defined as $[Jx](s):=\int_0^s x(t) dt$ is discussed extensively in the literature. The complete singular system of the compact operator $J$ is explicitly given with singular values $\sigma_n(J)$ asymptotically proportional to $1/n$, which indicates a degree {\sl one} of ill-posedness for this inverse problem. We recall the concept of the degree of ill-posedness for linear operator equations with compact forward operators in Hilbert spaces. In contrast to the one-dimensional case with operator $J$, there is little material available about the analysis of the d-dimensional case, where the compact integral operator $J_d:L^2([0,1]^d) \to L^2([0,1]^d)$ defined as $[J_d\,x](s_1,\ldots,s_d):=\int_0^{s_1}\ldots\int_0^{s_d} x(t_1,\ldots,t_d)\, dt_d\ldots dt_1$ over unit $d$-cube is to be inverted. This inverse problem of mixed differentiation $x(s_1,\ldots,s_d)=\frac{\partial^d}{\partial s_1 \ldots \partial s_d} y(s_1,\ldots ,s_d)$ is of practical interest, for example when in statistics copula densities have to be verified from empirical copulas over $[0,1]^d \subset \mathbb{R}^d$. In this note, we prove that the non-increasingly ordered singular values $\sigma_n(J_d)$ of the operator $J_d$ have an asymptotics of the form $\frac{(\log n)^{d-1}}{n}$, which shows that the degree of ill-posedness stays at one, even though an additional logarithmic factor occurs. Some more discussion refers to the special case $d=2$ for characterizing the range $\mathcal{R}(J_2)$ of the operator $J_2$., Comment: One author has left team of authors

We study the application of Tikhonov regularization to ill-posed nonlinear operator equations. The objective of this work is to prove low order convergence rates for the discrepancy principle under low order source conditions of logarithmic type. We work within the framework of Hilbert scales and extend existing studies on this subject to the oversmoothing case. The latter means that the exact solution of the treated operator equation does not belong to the domain of definition of the penalty term. As a consequence, the Tikhonov functional fails to have a finite value.

In the present work, we discuss variational regularization for ill-posed nonlinear problems with focus on an oversmoothing penalty term. This means in our model that the searched-for solution of the considered nonlinear operator equation does not belong to the domain of definition of the penalty functional. In the past years, such variational regularization has been investigated comprehensively in Hilbert scales. Our present study tries to continue and to extend those investigations to Banach scales. This new study includes convergence rates results for a priori choices of the regularization parameter, both for H\"older-type smoothness and low order-type smoothness. The necessary tools for low order smoothness in the Banach space setting are provided.

The present thesis addresses both theoretical as well as numerical aspects of the treatment of nonlinear inverse problems. The first part considers Tikhonov regularization for nonlinear ill-posed operator equations in Hilbert scales with oversmoothing penalties. Sufficient as well as necessary conditions to establish convergence are introduced and convergence rate results are given for various parameter choice rules under a two sided nonlinearity constraint. Ultimately, both a posteriori as well as certain a priori parameter choice rules lead to identical converce rates. The theoretical results are supported and augmented by extensive numerical case studies. In particular it is shown, that the localization of the above mentioned nonlinearity constraint is not trivial. Incorrect localization will prevent convergence of the regularized to the exact solution. The second part of the thesis considers two open problems in inverse option pricing and electrical impedance tomography. While regularization through discretization is sufficient to overcome ill-posedness of the latter, the first requires a more sophisticated approach. It is shown, that the recovery of time dependent volatility and interest rate functions from observed option prices is everywhere locally ill-posed. This motivates Tikhonov-type or variational regularization with two parameters and penalty terms to simultaneously recover these functions. Two parameter choice rules using the L-hypersurface as well as a combination of L-curve and quasi-optimality are introduced. The results are again supported by extensive numerical case studies.

The present thesis addresses both theoretical as well as numerical aspects of the treatment of nonlinear inverse problems. The first part considers Tikhonov regularization for nonlinear ill-posed operator equations in Hilbert scales with oversmoothing penalties. Sufficient as well as necessary conditions to establish convergence are introduced and convergence rate results are given for various parameter choice rules under a two sided nonlinearity constraint. Ultimately, both a posteriori as well as certain a priori parameter choice rules lead to identical converce rates. The theoretical results are supported and augmented by extensive numerical case studies. In particular it is shown, that the localization of the above mentioned nonlinearity constraint is not trivial. Incorrect localization will prevent convergence of the regularized to the exact solution. The second part of the thesis considers two open problems in inverse option pricing and electrical impedance tomography. While regularization through discretization is sufficient to overcome ill-posedness of the latter, the first requires a more sophisticated approach. It is shown, that the recovery of time dependent volatility and interest rate functions from observed option prices is everywhere locally ill-posed. This motivates Tikhonov-type or variational regularization with two parameters and penalty terms to simultaneously recover these functions. Two parameter choice rules using the L-hypersurface as well as a combination of L-curve and quasi-optimality are introduced. The results are again supported by extensive numerical case studies.

The present thesis addresses both theoretical as well as numerical aspects of the treatment of nonlinear inverse problems. The first part considers Tikhonov regularization for nonlinear ill-posed operator equations in Hilbert scales with oversmoothing penalties. Sufficient as well as necessary conditions to establish convergence are introduced and convergence rate results are given for various parameter choice rules under a two sided nonlinearity constraint. Ultimately, both a posteriori as well as certain a priori parameter choice rules lead to identical converce rates. The theoretical results are supported and augmented by extensive numerical case studies. In particular it is shown, that the localization of the above mentioned nonlinearity constraint is not trivial. Incorrect localization will prevent convergence of the regularized to the exact solution. The second part of the thesis considers two open problems in inverse option pricing and electrical impedance tomography. While regularization through discretization is sufficient to overcome ill-posedness of the latter, the first requires a more sophisticated approach. It is shown, that the recovery of time dependent volatility and interest rate functions from observed option prices is everywhere locally ill-posed. This motivates Tikhonov-type or variational regularization with two parameters and penalty terms to simultaneously recover these functions. Two parameter choice rules using the L-hypersurface as well as a combination of L-curve and quasi-optimality are introduced. The results are again supported by extensive numerical case studies.

The present thesis addresses both theoretical as well as numerical aspects of the treatment of nonlinear inverse problems. The first part considers Tikhonov regularization for nonlinear ill-posed operator equations in Hilbert scales with oversmoothing penalties. Sufficient as well as necessary conditions to establish convergence are introduced and convergence rate results are given for various parameter choice rules under a two sided nonlinearity constraint. Ultimately, both a posteriori as well as certain a priori parameter choice rules lead to identical converce rates. The theoretical results are supported and augmented by extensive numerical case studies. In particular it is shown, that the localization of the above mentioned nonlinearity constraint is not trivial. Incorrect localization will prevent convergence of the regularized to the exact solution. The second part of the thesis considers two open problems in inverse option pricing and electrical impedance tomography. While regularization through discretization is sufficient to overcome ill-posedness of the latter, the first requires a more sophisticated approach. It is shown, that the recovery of time dependent volatility and interest rate functions from observed option prices is everywhere locally ill-posed. This motivates Tikhonov-type or variational regularization with two parameters and penalty terms to simultaneously recover these functions. Two parameter choice rules using the L-hypersurface as well as a combination of L-curve and quasi-optimality are introduced. The results are again supported by extensive numerical case studies.

The analysis of Tikhonov regularization for nonlinear ill-posed equations with smoothness promoting penalties is an important topic in inverse problem theory. With focus on Hilbert scale models, the case of oversmoothing penalties, i.e., when the penalty takes an infinite value at the true solution gained increasing interest. The considered nonlinearity structure is as in the study B. Hofmann and P. Math\'{e}. Tikhonov regularization with oversmoothing penalty for non-linear ill-posed problems in Hilbert scales. Inverse Problems, 2018. Such analysis can address two fundamental questions. When is it possible to achieve order optimal reconstruction? How to select the regularization parameter? The present study complements previous ones by two main facets. First, an error decomposition into a smoothness dependent and a (smoothness independent) noise propagation term is derived, covering a large range of smoothness conditions. Secondly, parameter selection by balancing principles is presented. A detailed discussion, covering some history and variations of the parameter choice by balancing shows under which conditions such balancing principles yield order optimal reconstruction. A numerical case study, based on some exponential growth model, provides additional insights., Comment: 3 Figures

For the Tikhonov regularization of ill-posed nonlinear operator equations, convergence is studied in a Hilbert scale setting. We include the case of oversmoothing penalty terms, which means that the exact solution does not belong to the domain of definition of the considered penalty functional. In this case, we try to close a gap in the present theory, where H\"older-type convergence rates results have been proven under corresponding source conditions, but assertions on norm convergence of regularized solutions without source conditions are completely missing. A result of the present work is to provide sufficient conditions for convergence under a priori and a posteriori regularization parameter choice strategies, without any additional smoothness assumption on the solution. The obtained error estimates moreover allow us to prove low order convergence rates under associated (for example logarithmic) source conditions. Some numerical illustrations are also given.

There are various ways to regularize ill-posed operator equations in Hilbert space. If the underlying operator is accretive then Lavrentiev regularization (singular perturbation) is an immediate choice. The corresponding convergence rates for the regularization error depend on the given smoothness assumptions, and for general accretive operators these may be both with respect to the operator or its adjoint. Previous analysis revealed different convergence rates, and their optimality was unclear, specifically for adjoint source conditions. Based on the fundamental study by T. Kato, Fractional powers of dissipative operators. J. Math. Soc. Japan, 13(3):247--274, 1961, we establish power type convergence rates for this case. By measuring the optimality of such rates in terms on limit orders we exhibit optimality properties of the convergence rates, for general accretive operators under direct and adjoint source conditions, but also for the subclass of nonnegative selfadjoint operators.

There are various ways to regularize ill-posed operator equations in Hilbert space. If the underlying operator is accretive then Lavrentiev regularization (singular perturbation) is an immediate choice. The corresponding convergence rates for the regularization error depend on the given smoothness assumptions, and for general accretive operators these may be both with respect to the operator or its adjoint. Previous analysis revealed different convergence rates, and their optimality was unclear, specifically for adjoint source conditions. Based on the fundamental study by T. Kato, Fractional powers of dissipative operators. J. Math. Soc. Japan, 13(3):247--274, 1961, we establish power type convergence rates for this case. By measuring the optimality of such rates in terms on limit orders we exhibit optimality properties of the convergence rates, for general accretive operators under direct and adjoint source conditions, but also for the subclass of nonnegative selfadjoint operators.

There are various ways to regularize ill-posed operator equations in Hilbert space. If the underlying operator is accretive then Lavrentiev regularization (singular perturbation) is an immediate choice. The corresponding convergence rates for the regularization error depend on the given smoothness assumptions, and for general accretive operators these may be both with respect to the operator or its adjoint. Previous analysis revealed different convergence rates, and their optimality was unclear, specifically for adjoint source conditions. Based on the fundamental study by T. Kato, Fractional powers of dissipative operators. J. Math. Soc. Japan, 13(3):247--274, 1961, we establish power type convergence rates for this case. By measuring the optimality of such rates in terms on limit orders we exhibit optimality properties of the convergence rates, for general accretive operators under direct and adjoint source conditions, but also for the subclass of nonnegative selfadjoint operators.

There are various ways to regularize ill-posed operator equations in Hilbert space. If the underlying operator is accretive then Lavrentiev regularization (singular perturbation) is an immediate choice. The corresponding convergence rates for the regularization error depend on the given smoothness assumptions, and for general accretive operators these may be both with respect to the operator or its adjoint. Previous analysis revealed different convergence rates, and their optimality was unclear, specifically for adjoint source conditions. Based on the fundamental study by T. Kato, Fractional powers of dissipative operators. J. Math. Soc. Japan, 13(3):247--274, 1961, we establish power type convergence rates for this case. By measuring the optimality of such rates in terms on limit orders we exhibit optimality properties of the convergence rates, for general accretive operators under direct and adjoint source conditions, but also for the subclass of nonnegative selfadjoint operators.

The dissetation deals with the inverse problem of identification of local volatilities from given option price data. The used separation between purely time- and purely price-dependent volatilities enables a detailed mathematical analysis of the corresponding inverse problems. Those are formulated in proper Banach spaces (Hilbert spaces) as operator equations. The unique solvability of these equations are examined. Because the solutions doesn't depend continuously form the given data, possibilities of regularization are discussed. In particular the nonlinear Tikhonov regularization and its applicability to the corresponding problems plays the leading part in these investigations. Detailed numerical studies illustrate these considerations and top this disseration off., Die Dissertation beschäftigt sich mit dem inversen Problem der Identifikation lokaler Volatilitäten aus gegebenen Optionspreisen. Die dabei benutzte Trennung zwischen rein zeit- und rein preisabhängigen Volatilitäten erlaubt eine tiefgehende mathematische Analyse der entsprechend formulierten inversen Probleme. Diese werden in geeigneten Banachräumen (Hilberträumen) als Operatorgleichung angegeben und auf die eindeutige Lösbarkeit hin untersucht. Da sich die Lösungen als instabil gegenüber Störungen in den Daten erweisen, werden Möglichkeiten der Regularisierung diskutiert. Insbesondere steht dabei die Untersuchung der Anwendbarkeit der Theorie der nichtlinearen Tikhonov-Regularisierung auf die entsprechenden Probleme im Vordergrund. Ausführliche numerische Studien illustrieren die diese Überlegungen und runden die Arbeit ab.

The dissetation deals with the inverse problem of identification of local volatilities from given option price data. The used separation between purely time- and purely price-dependent volatilities enables a detailed mathematical analysis of the corresponding inverse problems. Those are formulated in proper Banach spaces (Hilbert spaces) as operator equations. The unique solvability of these equations are examined. Because the solutions doesn't depend continuously form the given data, possibilities of regularization are discussed. In particular the nonlinear Tikhonov regularization and its applicability to the corresponding problems plays the leading part in these investigations. Detailed numerical studies illustrate these considerations and top this disseration off., Die Dissertation beschäftigt sich mit dem inversen Problem der Identifikation lokaler Volatilitäten aus gegebenen Optionspreisen. Die dabei benutzte Trennung zwischen rein zeit- und rein preisabhängigen Volatilitäten erlaubt eine tiefgehende mathematische Analyse der entsprechend formulierten inversen Probleme. Diese werden in geeigneten Banachräumen (Hilberträumen) als Operatorgleichung angegeben und auf die eindeutige Lösbarkeit hin untersucht. Da sich die Lösungen als instabil gegenüber Störungen in den Daten erweisen, werden Möglichkeiten der Regularisierung diskutiert. Insbesondere steht dabei die Untersuchung der Anwendbarkeit der Theorie der nichtlinearen Tikhonov-Regularisierung auf die entsprechenden Probleme im Vordergrund. Ausführliche numerische Studien illustrieren die diese Überlegungen und runden die Arbeit ab.

The dissetation deals with the inverse problem of identification of local volatilities from given option price data. The used separation between purely time- and purely price-dependent volatilities enables a detailed mathematical analysis of the corresponding inverse problems. Those are formulated in proper Banach spaces (Hilbert spaces) as operator equations. The unique solvability of these equations are examined. Because the solutions doesn't depend continuously form the given data, possibilities of regularization are discussed. In particular the nonlinear Tikhonov regularization and its applicability to the corresponding problems plays the leading part in these investigations. Detailed numerical studies illustrate these considerations and top this disseration off., Die Dissertation beschäftigt sich mit dem inversen Problem der Identifikation lokaler Volatilitäten aus gegebenen Optionspreisen. Die dabei benutzte Trennung zwischen rein zeit- und rein preisabhängigen Volatilitäten erlaubt eine tiefgehende mathematische Analyse der entsprechend formulierten inversen Probleme. Diese werden in geeigneten Banachräumen (Hilberträumen) als Operatorgleichung angegeben und auf die eindeutige Lösbarkeit hin untersucht. Da sich die Lösungen als instabil gegenüber Störungen in den Daten erweisen, werden Möglichkeiten der Regularisierung diskutiert. Insbesondere steht dabei die Untersuchung der Anwendbarkeit der Theorie der nichtlinearen Tikhonov-Regularisierung auf die entsprechenden Probleme im Vordergrund. Ausführliche numerische Studien illustrieren die diese Überlegungen und runden die Arbeit ab.

The dissetation deals with the inverse problem of identification of local volatilities from given option price data. The used separation between purely time- and purely price-dependent volatilities enables a detailed mathematical analysis of the corresponding inverse problems. Those are formulated in proper Banach spaces (Hilbert spaces) as operator equations. The unique solvability of these equations are examined. Because the solutions doesn't depend continuously form the given data, possibilities of regularization are discussed. In particular the nonlinear Tikhonov regularization and its applicability to the corresponding problems plays the leading part in these investigations. Detailed numerical studies illustrate these considerations and top this disseration off., Die Dissertation beschäftigt sich mit dem inversen Problem der Identifikation lokaler Volatilitäten aus gegebenen Optionspreisen. Die dabei benutzte Trennung zwischen rein zeit- und rein preisabhängigen Volatilitäten erlaubt eine tiefgehende mathematische Analyse der entsprechend formulierten inversen Probleme. Diese werden in geeigneten Banachräumen (Hilberträumen) als Operatorgleichung angegeben und auf die eindeutige Lösbarkeit hin untersucht. Da sich die Lösungen als instabil gegenüber Störungen in den Daten erweisen, werden Möglichkeiten der Regularisierung diskutiert. Insbesondere steht dabei die Untersuchung der Anwendbarkeit der Theorie der nichtlinearen Tikhonov-Regularisierung auf die entsprechenden Probleme im Vordergrund. Ausführliche numerische Studien illustrieren die diese Überlegungen und runden die Arbeit ab.