Peter Lory

Comparing routines for the numerical solution of initial value problems of ordinary differential equations in multiple shooting

SummaryThe numerical solution of two-point boundary value problems and problems of optimal control by shooting techniques requires integration routines. By solving 15 real-life problems four well-known intergrators are compared relative to reliability, fastness and precision. Hints are given, which routines could be used for a problem.

On oxygen diffusion in a spherical cell with michaelis-menten oxygen uptake kinetics

This paper demonstrates that there is one and only one solution to a non-linear singular two-point boundary-value problem which describes oxygen diffusion in a spherical cell. Previous authors have calculated numerical results that differ substantially. Numerical computations using the multiple shooting method support the results of McElwain.

Renal countercurrent system: Role of collecting duct convergence and pelvic urea predicted from a mathematical model

A differential equation model of the renal countercurrent system has been developed and physiological data from nephron segments were incorporated together with recently suggested urea recycling from renal pelvis to inner medulla and, particularly, an exponential reduction in the number of collecting tubules towards the renal papilla. The role of these features for the countercurrent concentrating mechanism has been studied by simulation runs. The computations, using the multiple shooting method, provide predictions about concentration profiles for salt and urea in tubes (nephron segments) and in the central core along the entire medullary countercurrent system. The results indicate that this model, without active salt or urea transport in the inner medulla, yields concentration gradients along the medullary axis compatible with those measured in the tissue.

Numerical solution of a kidney model by multiple shooting

Abstract The mathematical modeling of the renal counterflow system involves a two-point boundary-value problem for a system of nonlinear differential equations. In this paper the multiple shooting technique is used for the numerical solution of this problem. This method ensures highly precise and reliable computations. Special consideration is given to the treatment of the border between outer and inner medulla for various widths of the outer zone.

Secure Distributed Multiplication of Two Polynomially Shared Values: Enhancing the Efficiency of the Protocol

In view of practical applications, it is a high priority to optimize the efficiency of methods for secure multiparty computations. These techniques enable, for instance, truly practical double auctions and distributed signatures. The multiplication protocol for the secure multiparty multiplication of two polynomially shared values over Z_q with a public prime number q is an important module in these computations. The protocol of Gennaro, Rabin and Rabin (1998) is a well known and efficient protocol for this purpose. It requires one round of communication and O(n^2 k \log n + n k^2) bit-operations per player, where k is the bit size of the prime q and n is the number of players. In a previous paper (2007), the author has presented a modification of this protocol, that reduces its complexity to O(n^2k + nk^2). The present paper reduces this complexity further to O(n^2 k). This reduction is profitable in situations where n is smaller than k. The new protocol requires the same amount of communication as the original one and is unconditionally secure, as well.

Reducing the Complexity in the Distributed Multiplication Protocol of Two Polynomially Shared Values

The multiparty multiplication of two polynomially shared values over Zq with a public prime number q is an important module in distributed computations. The multiplication protocol of Gennaro, Rabin and Rabin (1998) is considered as the best protocol for this purpose. It requires a complexity of O(n2k log n + nk2) bit-operations per player, where k is the bit size of the prime q andn is the number of players. The present paper reduces this complexity to O(n2k + nk2) by using Newtons classical interpolation formula. The impact of the new method on distributed signatures is outlined.

Enlarging the domain of convergence for multiple shooting by the homotopy method

SummaryThe homotopy method is a frequently used technique in overcoming the local convergence nature of multiple shooting. In this paper sufficient conditions are given that guarantee the homotopy process to be feasible. The results are applicable to a class of two-point boundary value problems. Finally, the numerical solution of two practical problems arising in physiology is described.

The kidney model as an inverse problem

The mammalian kidney is modeled by a multipoint boundary-value problem for a system of nonlinear ordinary differential equations. A corresponding inverse problem is presented, which allows the rigorous judgement of the potential of the given modeling technique. For its numerical solution a discretization is proposed, which is tailor-made for kidney models. It leads to a nonlinear-programming problem with nonlinear equality and inequality constraints. The suggested methods are applied to current research problems in renal physiology.

Enhancing the efficiency in privacy preserving learning of decision trees in partitioned databases

This paper considers a scenario where two parties having private databases wish to cooperate by computing a data mining algorithm on the union of their databases without revealing any unnecessary information. In particular, they want to apply the decision tree learning algorithm ID3 in a privacy preserving manner. Lindell and Pinkas (2002) have presented a protocol for this purpose, which enjoys a formal proof of privacy and is considerably more efficient than generic solutions. The crucial point of their protocol is the approximation of the logarithm function by a truncated Taylor series. The present paper improves this approximation by using a suitable Chebyshev expansion. This approach results in a considerably more efficient new version of the protocol.

A coloured Petri net trust model

Public-key infrastructures are a prerequisite for security in distributed systems and for reliable electronic commerce. It is their goal to provide the authenticity of public keys. Formal models for public-key infrastructures (trust models) contribute decisively to a deeper understanding of the desirable design principles of these infrastructures. The trust model of the presented paper is based on the modelling technique of coloured Petri nets. These are a special class of high-level Petri nets with an intuitively appealing graphical representation and a few powerful primitives. Elaborate and well tested software is available.