A-stable time discretizations preserve maximal parabolic regularity

It is shown that for a parabolic problem with maximal $L^p$-regularity (for $1<p<\infty$), the time discretization by a linear multistep method or Runge--Kutta method has maximal $\ell^p$-regularity uniformly in the stepsize if the method is A-stable (and satisfies minor additional conditions). In particular, the implicit Euler method, the Crank-Nicolson method, the second-order backward difference formula (BDF), and the Radau IIA and Gauss Runge--Kutta methods of all orders preserve maximal regularity. The proof uses Weis' characterization of maximal $L^p$-regularity in terms of $R$-boundedness of the resolvent, a discrete operator-valued Fourier multiplier theorem by Blunck, and generating function techniques that have been familiar in the stability analysis of time discretization methods since the work of Dahlquist. The A($\alpha$)-stable higher-order BDF methods have maximal $\ell^p$-regularity under an $R$-boundedness condition in a larger sector. As an illustration of the use of maximal regularity in the error analysis of discretized nonlinear parabolic equations, it is shown how error bounds are obtained without using any growth condition on the nonlinearity or for nonlinearities having singularities

Initializing and stabilizing variational multistep algorithms for modeling dynamical systems

Backward error initialization and parasitic mode control are well-suited for use in algorithms that arise from a discrete variational principle on phase-space dynamics. Dynamical systems described by degenerate Lagrangians, such as those occurring in phase-space action principles, lead to variational multistep algorithms for the integration of first-order differential equations. As multistep algorithms, an initialization procedure must be chosen and the stability of parasitic modes assessed. The conventional selection of initial conditions using accurate one-step methods does not yield the best numerical performance for smoothness and stability. Instead, backward error initialization identifies a set of initial conditions that minimize the amplitude of undesirable parasitic modes. This issue is especially important in the context of structure-preserving multistep algorithms where numerical damping of the parasitic modes would violate the conservation properties. In the presence of growing parasitic modes, the algorithm may also be periodically re-initialized to prevent the undesired mode from reaching large amplitude. Numerical examples of variational multistep algorithms are presented in which the backward error initialized trajectories outperform those initialized using highly accurate approximations of the true solution

Galerkin v. least-squares Petrov--Galerkin projection in nonlinear model reduction

Least-squares Petrov--Galerkin (LSPG) model-reduction techniques such as the Gauss--Newton with Approximated Tensors (GNAT) method have shown promise, as they have generated stable, accurate solutions for large-scale turbulent, compressible flow problems where standard Galerkin techniques have failed. However, there has been limited comparative analysis of the two approaches. This is due in part to difficulties arising from the fact that Galerkin techniques perform optimal projection associated with residual minimization at the time-continuous level, while LSPG techniques do so at the time-discrete level. This work provides a detailed theoretical and computational comparison of the two techniques for two common classes of time integrators: linear multistep schemes and Runge--Kutta schemes. We present a number of new findings, including conditions under which the LSPG ROM has a time-continuous representation, conditions under which the two techniques are equivalent, and time-discrete error bounds for the two approaches. Perhaps most surprisingly, we demonstrate both theoretically and computationally that decreasing the time step does not necessarily decrease the error for the LSPG ROM; instead, the time step should be `matched' to the spectral content of the reduced basis. In numerical experiments carried out on a turbulent compressible-flow problem with over one million unknowns, we show that increasing the time step to an intermediate value decreases both the error and the simulation time of the LSPG reduced-order model by an order of magnitude.Comment: Submitted to Journal of Computational Physic

Analysis of the effect of Time Filters on the implicit method: increased accuracy and improved stability

This report considers linear multistep methods through time filtering. The approach has several advantages. It is modular and requires the addition of only one line of additional code. Error estimation and variable timesteps is straightforward and the individual effect of each step\ is conceptually clear. We present its development for the backward Euler method and a curvature reducing time filter leading to a 2-step, strongly A-stable, second order linear multistep method

Silent error detection in numerical time-stepping schemes

Errors due to hardware or low level software problems, if detected, can be fixed by various schemes, such as recomputation from a checkpoint. Silent errors are errors in application state that have escaped low-level error detection. At extreme scale, where machines can perform astronomically many operations per second, silent errors threaten the validity of computed results. We propose a new paradigm for detecting silent errors at the application level. Our central idea is to frequently compare computed values to those provided by a cheap checking computation, and to build error detectors based on the difference between the two output sequences. Numerical analysis provides us with usable checking computations for the solution of initial-value problems in ODEs and PDEs, arguably the most common problems in computational science. Here, we provide, optimize, and test methods based on Runge-Kutta and linear multistep methods for ODEs, and on implicit and explicit finite difference schemes for PDEs. We take the heat equation and Navier-Stokes equations as examples. In tests with artificially injected errors, this approach effectively detects almost all meaningful errors, without significant slowdown

A new embedded variable stepsize, variable order family of low computational complexity

Variable Stepsize Variable Order (VSVO) methods are the methods of choice to efficiently solve a wide range of ODEs with minimal work and assured accuracy. However, VSVO methods have limited impact in timestepping methods in complex applications due to their computational complexity and the difficulty to implement them in legacy code. We introduce a family of implicit, embedded, VSVO methods that require only one BDF solve at each time step followed by adding linear combinations of the solution at previous time levels. In particular, we construct implicit and linearly implicit VSVO methods of orders two, three and four with the same computational complexity as variable stepsize BDF3. The choice of changing the order of the method is simple and does not require additional solves of linear or nonlinear systems.Comment: 32 pages,8 figure

An explicit multistep method for the Wigner problem

An explicit multistep scheme is proposed for solving the initial-value Wigner problem. In this scheme, the integrated form of the Wigner equation is approximated by extrapolation or interpolation polynomials on backwards characteristics, and the pseudo-differential operator is tackled by the spectral collocation method. Since it exploits the exact Lagrangian advection, the time stepping of the multistep scheme is not restricted by the CFL-type condition. It is also demonstrated that the calculations of the Wigner potential can be carried out by two successive FFTs, thereby reducing the computational complexity dramatically. Numerical examples illustrating its accuracy are presented

Emergent properties of the G1/S network

Tato práce se zabývá buněčným cyklem kvasinky Saccgaromyces cerevisiae. Oblastí našeho zájmu je přechod mezi G1 a S fází, kde je naším cílem identifikovat velikosti buňky v době počátku DNA replikace. Nejprve se věnujeme nedávno publikovanému matematickému modelu, který popisuje mechanismy vedoucí k S fázi. Práce poskytuje detailní popis tohoto modelu, stejně jako časový průběh některých důležitých proteinů či jejich sloučenin. Dále se zabýváme pravděpodobnostním modelem aktivace replikačních počátků DNA. Nově uvažujeme vliv šíření DNA replikace mezi sousedícími počátky a analyzujeme jeho důsledky. Poskytujeme také senzitivní analýzu kritické velikosti buňky vzhledem ke konstantám popisujícím dynamiku reakcí v modelu G1/S přechodu.In this thesis we deal with the cell cycle of the yeast Saccharomyces cerevisiae. We are interested in its G1 to S transition, and our main goal is to determine the cell size at the onset of its DNA replication. At first, we study a recent mathematical model describing the mechanisms leading to the S phase, we provide its detailed description and present the dynamics of some significant protein and protein complexes. Further, we take a closer look at the probabilistic model for firing of DNA replication origins. We newly consider the influence of DNA replication spreading among neighboring origins, and we analyze its consequences. We also provide a sensitivity analysis of the critical cell size with respect to rate constants of G1 to S transition model.

High order finite difference methods

Graduation date: 199

Discrete-Time Accelerated Block Successive Overrelaxation Methods for Time-Dependent Stokes Equations

To further study the application of waveform relaxation methods in fluid dynamics in actual computation, this paper provides a general theoretical analysis of discrete-time waveform relaxation methods for solving linear DAEs. A class of discrete-time waveform relaxation methods, named discrete-time accelerated block successive overrelaxation (DABSOR) methods, is proposed for solving linear DAEs derived from discretizing time-dependent Stokes equations in space by using "Method of Lines". The analysis of convergence property and optimality of the DABSOR method are presented in detail. The theoretical results and the efficiency of the DABSOR method are verified by numerical experiments.Comment: 31 pages, 12 figures, 10 table