Bernhard A. Schmitt

ROWMAP—a ROW-code with Krylov techniques for large stiff ODEs

Abstract We present a Krylov-W-code ROWMAP for the integration of stiff initial value problems. It is based on the ROW-methods of the code ROS4 of Hairer and Wanner and uses Krylov techniques for the solution of linear systems. A special multiple Arnoldi process ensures order p = 4 already for fairly low dimensions of the Krylov subspaces independently of the dimension of the differential equations. Numerical tests and comparisons with the multistep code VODPK illustrate the efficiency of ROWMAP for large stiff systems. Furthermore, the application to nonautonomous systems is discussed in more detail.

Explicit two-step peer methods

We present a new class of explicit two-step peer methods for the solution of nonstiff differential systems. A construction principle for methods of order p=s, s the number of stages, with optimal zero-stability is given. Two methods of order p=6, found by numerical search, are tested in Matlab on several representative nonstiff problems. The comparison with ODE45 confirms the high potential of the new class of methods.

Matrix-free W-methods using a multiple Arnoldi iteration

The standard Arnoldi method approximates the solution of one single linear system through projection onto a Krylov space constructed from the right-hand side vector. The different stage equations of a W-method use the same matrix but different right-hand sides, which are, in general, not part of the original Krylov space. For the case of two right-hand sides we discuss two well-known deterministic strategies that extend the Arnoldi process from the first stage and present a new one that performs Krylov steps adaptively to minimize the residual of the next approximation. An implementation of these methods is tested on three different parabolic problems and compared with the code VODPK.

Design, analysis and testing of some parallel two-step W-methods for stiff systems

Parallel two-step W-methods are linearly-implicit integration methods where the s stage values can be computed in parallel We construct methods of stage order q = s and order p = s with favourable stability properties. Generalizations for the concepts of A- and L-stability are proposed and conditions for stiff accuracy are given. Numerical comparisons on a shared memory computer show the efficiency of the methods, especially in combination with Krylov-techniques for large stiff systems.

Norm bounds for rational matrix functions

SummaryIf the field of values of a matrixA is contained in the left complex halfplaneH and a functionf mapsH into the unit disc then ∥f(A)∥2≦1 by a theorem of J.v. Neumann. We prove a theorem of this type, only the field of values ofA is used for functions which are absolutely bounded by one in only part ofH. An extension can be used to show norm-stability of single step methods for stiff differential equations. The results are applicable among others to several subdiagonal Padé approximations which are notA-stable.

An algebraic approximation for the matrix exponential in singularly perturbed boundary value problems

A symmetric difference scheme for linear, stiff, or singularly perturbed boundary value problems of first order with constant coefficients is constructed, being based on a stability function containing a matrix square root. Its essential feature is the unconditional stability in the absence of purely imaginary eigenvalues of the coefficient matrix. Local damping of errors, uniform stability, and uniform second-order convergence are proved. The computation of the specific matrix square root by a well-known, stable variant of Newton’s method is discussed. A numerical example confirming the results is given.

Rapid turnover of DnaA at replication origin regions contributes to initiation control of DNA replication

DnaA is a conserved key regulator of replication initiation in bacteria, and is homologous to ORC proteins in archaea and in eukaryotic cells. The ATPase binds to several high affinity binding sites at the origin region and upon an unknown molecular trigger, spreads to several adjacent sites, inducing the formation of a helical super structure leading to initiation of replication. Using FRAP analysis of a functional YFP-DnaA allele in Bacillus subtilis, we show that DnaA is bound to oriC with a half-time of 2.5 seconds. DnaA shows similarly high turnover at the replication machinery, where DnaA is bound to DNA polymerase via YabA. The absence of YabA increases the half time binding of DnaA at oriC, showing that YabA plays a dual role in the regulation of DnaA, as a tether at the replication forks, and as a chaser at origin regions. Likewise, a deletion of soj (encoding a ParA protein) leads to an increase in residence time and to overinitiation, while a mutation in DnaA that leads to lowered initiation frequency, due to a reduced ATPase activity, shows a decreased residence time on binding sites. Finally, our single molecule tracking experiments show that DnaA rapidly moves between chromosomal binding sites, and does not arrest for more than few hundreds of milliseconds. In Escherichia coli, DnaA also shows low residence times in the range of 200 ms and oscillates between spatially opposite chromosome regions in a time frame of one to two seconds, independently of ongoing transcription. Thus, DnaA shows extremely rapid binding turnover on the chromosome including oriC regions in two bacterial species, which is influenced by Soj and YabA proteins in B. subtilis, and is crucial for balanced initiation control, likely preventing fatal premature multimerization and strand opening of DnaA at oriC.

Perturbation bounds for matrix square roots and pythagorean sums

Abstract Perturbation estimates for the square root R:= A 1 2 and Pythagorean sum P:=(I +A 2 ) 1 2 of complex matrices are proved. We present bounds in the spectral norm for the cases that R and P are accretive, i.e. have positive definite Hermitian parts, with special attention to the case of a large condition of A. The results are based on bounds for the solution of the matrix Sylvester equation and for the separation of two matrices.

Order results for Krylov-W-methods

We consider ROW-methods for stiff initial value problems, where the stage equations are solved by Krylov techniques. By using a certain ‘multiple Arnoldi process’ over all stages the order of the fully-implicit one-step scheme can be preserved with low Krylov dimensions. Explicit estimates for minimal order preserving dimensions are derived. They depend on the parameters of the method only, not on the dimension of the ODE. Stability restrictions usually require larger dimensions, of course, but this can be done adaptively. These results justify to adopt the step size control of the underlying ROW-method. The widely used ROW-methods of order 4 are discussed in detail and numerical illustrations are given. For the special class of semilinear systems with stiffness in a constant linear part we establish the order 2 of B-consistency for these Krylov-W-methods.

Numerical experiments with Krylov integrators

We discuss the use of preconditioning in Krylov-W-methods. The preconditioning is based on different operator splitting schemes for reaction-diffusion problems. Comparison of various Krylov codes show that the preconditioned version of ROWMAP works efficient and reliable, especially for large dimensions. Furthermore, parallelism can easily be exploited in the preconditioning.