Appl Math Comput

The problem of recovering the ruin probability in the classical risk model based on the scaled Laplace transform inversion is studied. It is shown how to overcome the problem of evaluating the ruin probability at large values of an initial surplus process. Comparisons of proposed approximations with the ones based on the Laplace transform inversions using a fixed Talbot algorithm as well as on the ones using the Trefethen-Weideman-Schmelzer and maximum entropy methods are presented via a simulation study.CC999999/Intramural CDC HHS/United States2016-10-01T00:00:00Z26752796PMC470470

Laplace deconvolution on the basis of time domain data and its application to Dynamic Contrast Enhanced imaging

In the present paper we consider the problem of Laplace deconvolution with noisy discrete non-equally spaced observations on a finite time interval. We propose a new method for Laplace deconvolution which is based on expansions of the convolution kernel, the unknown function and the observed signal over Laguerre functions basis (which acts as a surrogate eigenfunction basis of the Laplace convolution operator) using regression setting. The expansion results in a small system of linear equations with the matrix of the system being triangular and Toeplitz. Due to this triangular structure, there is a common number $m$ of terms in the function expansions to control, which is realized via complexity penalty. The advantage of this methodology is that it leads to very fast computations, produces no boundary effects due to extension at zero and cut-off at $T$ and provides an estimator with the risk within a logarithmic factor of the oracle risk. We emphasize that, in the present paper, we consider the true observational model with possibly nonequispaced observations which are available on a finite interval of length $T$ which appears in many different contexts, and account for the bias associated with this model (which is not present when $T\rightarrow\infty$). The study is motivated by perfusion imaging using a short injection of contrast agent, a procedure which is applied for medical assessment of micro-circulation within tissues such as cancerous tumors. Presence of a tuning parameter $a$ allows to choose the most advantageous time units, so that both the kernel and the unknown right hand side of the equation are well represented for the deconvolution. The methodology is illustrated by an extensive simulation study and a real data example which confirms that the proposed technique is fast, efficient, accurate, usable from a practical point of view and very competitive.Comment: 36 pages, 9 figures. arXiv admin note: substantial text overlap with arXiv:1207.223

Model reconstruction for moment-based stochastic chemical kinetics

Based on the theory of stochastic chemical kinetics, the inherent randomness and stochasticity of biochemical reaction networks can be accurately described by discrete-state continuous-time Markov chains, where each chemical reaction corresponds to a state transition of the process. However, the analysis of such processes is computationally expensive and sophisticated numerical methods are required. The main complication comes due to the largeness problem of the state space, so that analysis techniques based on an exploration of the state space are often not feasible and the integration of the moments of the underlying probability distribution has become a very popular alternative. In this thesis we propose an analysis framework in which we integrate a number of moments of the process instead of the state probabilities. This results in a more timeefficient simulation of the time evolution of the process. In order to regain the state probabilities from the moment representation, we combine the moment-based simulation (MM) with a maximum entropy approach: the maximum entropy principle is applied to derive a distribution that fits best to a given sequence of moments. We further extend this approach by incorporating the conditional moments (MCM) which allows not only to reconstruct the distribution of the species present in high amount in the system, but also to approximate the probabilities of species with low molecular counts. For the given distribution reconstruction framework, we investigate the numerical accuracy and stability using case studies from systems biology, compare two different moment approximation methods (MM and MCM), examine if it can be used for the reaction rates estimation problem and describe the possible future applications. In this thesis we propose an analysis framework in which we integrate a number of moments of the process instead of the state probabilities. This results in a more time-efficient simulation of the time evolution of the process. In order to regain the state probabilities from the moment representation, we combine the moment-based simulation (MM) with a maximum entropy approach: the maximum entropy principle is applied to derive a distribution that fits best to a given sequence of moments. We further extend this approach by incorporating the conditional moments (MCM) which allows not only to reconstruct the distribution of the species present in high amount in the system, but also to approximate the probabilities of species with low molecular counts. For the given distribution reconstruction framework, we investigate the numerical accuracy and stability using case studies from systems biology, compare two different moment approximation methods (MM and MCM), examine if it can be used for the reaction rates estimation problem and describe the possible future applications.Basierend auf der Theorie der stochastischen chemischen Kinetiken können die inhärente Zufälligkeit und Stochastizität von biochemischen Reaktionsnetzwerken durch diskrete zeitkontinuierliche Markow-Ketten genau beschrieben werden, wobei jede chemische Reaktion einem Zustandsübergang des Prozesses entspricht. Die Analyse solcher Prozesse ist jedoch rechenaufwendig und komplexe numerische Verfahren sind erforderlich. Analysetechniken, die auf dem Abtasten des Zustandsraums basieren, sind durch dessen Größe oft nicht anwendbar. Als populäre Alternative wird heute häufig die Integration der Momente der zugrundeliegenden Wahrscheinlichkeitsverteilung genutzt. In dieser Arbeit schlagen wir einen Analyserahmen vor, in dem wir, anstatt der Zustandswahrscheinlichkeiten, zugrundeliegende Momente des Prozesses integrieren. Dies führt zu einer zeiteffizienteren Simulation der zeitlichen Entwicklung des Prozesses. Um die Zustandswahrscheinlichkeiten aus der Momentreprsentation wiederzugewinnen, kombinieren wir die momentbasierte Simulation (MM) mit Entropiemaximierung: Die Maximum- Entropie-Methode wird angewendet, um eine Verteilung abzuleiten, die am besten zu einer bestimmten Sequenz von Momenten passt. Wir erweitern diesen Ansatz durch das Einbeziehen bedingter Momente (MCM), die es nicht nur erlauben, die Verteilung der in großer Menge im System enthaltenen Spezies zu rekonstruieren, sondern es ebenso ermöglicht, sich den Wahrscheinlichkeiten von Spezies mit niedrigen Molekulargewichten anzunähern. Für das gegebene System zur Verteilungsrekonstruktion untersuchen wir die numerische Genauigkeit und Stabilität anhand von Fallstudien aus der Systembiologie, vergleichen zwei unterschiedliche Verfahren der Momentapproximation (MM und MCM), untersuchen, ob es für das Problem der Abschätzung von Reaktionsraten verwendet werden kann und beschreiben die mögliche zukünftige Anwendungen