On Reduced Models For The Chemical Master Equation

We develop a general convergence analysis for a class of inexact Newton-type regularizations for stably solving nonlinear ill-posed problems. Each of the methods under consideration consists of two components: the outer Newton iteration and an inner regularization scheme which, applied to the linearized system, provides the update. In this paper we give a novel and unified convergence analysis which is not confined to a specific inner regularization scheme but applies to a multitude of schemes including Landweber and steepest decent iterations, iterated Tikhonov method, and method of conjugate gradients

Order Reduction of the Chemical Master Equation via Balanced Realisation

We consider a Markov process in continuous time with a finite number of discrete states. The time-dependent probabilities of being in any state of the Markov chain are governed by a set of ordinary differential equations, whose dimension might be large even for trivial systems. Here, we derive a reduced ODE set that accurately approximates the probabilities of subspaces of interest with a known error bound. Our methodology is based on model reduction by balanced truncation and can be considerably more computationally efficient than the Finite State Projection Algorithm (FSP) when used for obtaining transient responses. We show the applicability of our method by analysing stochastic chemical reactions. First, we obtain a reduced order model for the infinitesimal generator of a Markov chain that models a reversible, monomolecular reaction. In such an example, we obtain an approximation of the output of a model with 301 states by a reduced model with 10 states. Later, we obtain a reduced order model for a catalytic conversion of substrate to a product; and compare its dynamics with a stochastic Michaelis-Menten representation. For this example, we highlight the savings on the computational load obtained by means of the reduced-order model. Finally, we revisit the substrate catalytic conversion by obtaining a lower-order model that approximates the probability of having predefined ranges of product molecules.Comment: 12 pages, 6 figure

Jump-Diffusion Approximation of Stochastic Reaction Dynamics: Error bounds and Algorithms

Biochemical reactions can happen on different time scales and also the abundance of species in these reactions can be very different from each other. Classical approaches, such as deterministic or stochastic approach, fail to account for or to exploit this multi-scale nature, respectively. In this paper, we propose a jump-diffusion approximation for multi-scale Markov jump processes that couples the two modeling approaches. An error bound of the proposed approximation is derived and used to partition the reactions into fast and slow sets, where the fast set is simulated by a stochastic differential equation and the slow set is modeled by a discrete chain. The error bound leads to a very efficient dynamic partitioning algorithm which has been implemented for several multi-scale reaction systems. The gain in computational efficiency is illustrated by a realistically sized model of a signal transduction cascade coupled to a gene expression dynamics.Comment: 32 pages, 7 figure

Numerical Optimization of a Waveguide Transition Using Finite Element Beam Propagation

We develop a general convergence analysis for a class of inexact Newton-type regularizations for stably solving nonlinear ill-posed problems. Each of the methods under consideration consists of two components: the outer Newton iteration and an inner regularization scheme which, applied to the linearized system, provides the update. In this paper we give a novel and unified convergence analysis which is not confined to a specific inner regularization scheme but applies to a multitude of schemes including Landweber and steepest decent iterations, iterated Tikhonov method, and method of conjugate gradients

Data-driven modelling of biological multi-scale processes

Biological processes involve a variety of spatial and temporal scales. A holistic understanding of many biological processes therefore requires multi-scale models which capture the relevant properties on all these scales. In this manuscript we review mathematical modelling approaches used to describe the individual spatial scales and how they are integrated into holistic models. We discuss the relation between spatial and temporal scales and the implication of that on multi-scale modelling. Based upon this overview over state-of-the-art modelling approaches, we formulate key challenges in mathematical and computational modelling of biological multi-scale and multi-physics processes. In particular, we considered the availability of analysis tools for multi-scale models and model-based multi-scale data integration. We provide a compact review of methods for model-based data integration and model-based hypothesis testing. Furthermore, novel approaches and recent trends are discussed, including computation time reduction using reduced order and surrogate models, which contribute to the solution of inference problems. We conclude the manuscript by providing a few ideas for the development of tailored multi-scale inference methods.Comment: This manuscript will appear in the Journal of Coupled Systems and Multiscale Dynamics (American Scientific Publishers

On the Linearization of Operators Related to the Full Waveform Inversion in Seismology

We develop a general convergence analysis for a class of inexact Newton-type regularizations for stably solving nonlinear ill-posed problems. Each of the methods under consideration consists of two components: the outer Newton iteration and an inner regularization scheme which, applied to the linearized system, provides the update. In this paper we give a novel and unified convergence analysis which is not confined to a specific inner regularization scheme but applies to a multitude of schemes including Landweber and steepest decent iterations, iterated Tikhonov method, and method of conjugate gradients

An hp-Efficient Residual-Based A Posteriori Error Estimator for Maxwell\u27s Equations

We develop a general convergence analysis for a class of inexact Newton-type regularizations for stably solving nonlinear ill-posed problems. Each of the methods under consideration consists of two components: the outer Newton iteration and an inner regularization scheme which, applied to the linearized system, provides the update. In this paper we give a novel and unified convergence analysis which is not confined to a specific inner regularization scheme but applies to a multitude of schemes including Landweber and steepest decent iterations, iterated Tikhonov method, and method of conjugate gradients