Computational inference in systems biology

Parameter inference in mathematical models of biological pathways, expressed as coupled ordinary differential equations (ODEs), is a challenging problem. The computational costs associated with repeatedly solving the ODEs are often high. Aimed at reducing this cost, new concepts using gradient matching have been proposed. This paper combines current adaptive gradient matching approaches, using Gaussian processes, with a parallel tempering scheme, and conducts a comparative evaluation with current methods used for parameter inference in ODEs

Parallel Diagonally Implicit Runge-Kutta Methods For Solving Ordinary Differential Equations

This thesis focuses on the derivations of diagonally implicit Runge-Kutta (DIRK) methods with the capability to be implemented by parallel executions. A few new methods are proposed by having sparsity patterns which enable the parallelization of methods. In the first part of the thesis, a fifth order DIRK suitable for two processors parallel executions and DIRK methods of fourth and fifth orders suitable for three processors are proposed. The executions of these methods are done by using fixed stepsizes on a set of nonstiff problems. The regions of stability are presented and numerical results of the methods are compared to the existing methods. Parallel computations show significant time reduction when solving large systems of nonstiff ordinary differential equations (ODEs). The subsequent part of the thesis discusses on embedded DIRK methods suitable for two processors implementations. Two 4(3) and also two 5(4) embedded DIRK methods with adequate stability regions to solve stiff ODEs are proposed. Numerical experiments on stiff test problems are done based on variable stepsize strategy. An existing code for solving stiff ODEs suitable for embedded DIRK with equal diagonal elements is modified to accommodate the new methods with alternate diagonal elements. Comparisons on numerical results to existing methods show a competitive efficiency when solving small systems of stiff ODEs. A parallel code is developed with the same capability of the modified sequential code to handle stiff ODEs, linear and nonlinear problems. All algorithms are written in C language and the parallel code is implemented on Sun Fire V1280 distributed memory system. Three large scales of stiff ODEs are used to measure the parallel performances of the new embedded methods. Results show that speedups increased as the dimensions of the problems gets larger which is a significant contribution in reducing the cost of computations

Parallel Block Methods for Solving Ordinary Differential Equations

In this thesis, new and efficient codes are developed for solving Initial Value Problems (IVPs) of first and higher order Ordinary Differential Equations (ODEs) using variable step size. The new codes are based on the implicit multistep block methods formulae. Subsequently, a more structured and efficient algorithm comprising the block methods was constructed for solving systems of first order ODEs using variable step size and order. The new codes were then used for the parallel implementation in solving large systems of first and higher order ODEs. The sequential programs of these methods were executed on DYNIXlptx operating system. The parallel programs were run on a Sequent Symmetry SE30 parallel computer.The Cq stability in the multistep method was introduced and the focused was on the error propagation from a more practical angle. The numerical results showed that the sequential implementation of the new codes could reduce the total number of steps and execution times even when solving small systems of first and higher order ODEs compared with the 1-point method and the existing 2PBVSO code in Omar (1 999). The parallel implementation of the codes was found to be most appropriate in solving large systems of first and higher order ODEs. It was also discovered that the maximum speed up of the parallel methods improved as the dimension of the ODEs systems increased. In conclusion, the new codes developed in this thesis are suitable for solving systems of first and higher order ODE

Parallel Block Methods for Solving Higher Order Ordinary Differential Equations Directly

Numerous problems that are encountered in various branches of science and engineering involve ordinary differential equations (ODEs). Some of these problems require lengthy computation and immediate solutions. With the availability of parallel computers nowadays, the demands can be achieved. However, most of the existing methods for solving ODEs directly, particularly of higher order, are sequential in nature. These methods approximate numerical solution at one point at a time and therefore do not fully exploit the capability of parallel computers. Hence, the development of parallel algorithms to suit these machines becomes essential. In this thesis, new explicit and implicit parallel block methods for solving a single equation of ODE directly using constant step size and back values are developed. These methods, which calculate the numerical solution at more than one point simultaneously, are parallel in nature. The programs of the methods employed are run on a shared memory Sequent Symmetry S27 parallel computer. The numerical results show that the new methods reduce the total number of steps and execution time. The accuracy of the parallel block and 1-point methods is comparable particularly when finer step sizes are used. A new parallel algorithm for solving systems of ODEs using variable step size and order is also developed. The strategies used to design this method are based on both the Direct Integration (DI) and parallel block methods. The results demonstrate the superiority of the new method in terms of the total number of steps and execution times especially with finer tolerances. In conclusion, the new methods developed can be used as viable alternatives for solving higher order ODEs directly

Parallel implementation of explicit 2 and 3-point block methods for solving system of special second order ODEs directly

In this paper the explicit 2 and 3-point block method for solving large systems of special second order ODEs directly is discussed. Codes based on the methods are executed in sequential and parallel. The numerical results show that parallel to sequential counterpart for solving the large system of special second order ODEs

Approximate parameter inference in systems biology using gradient matching: a comparative evaluation

Background: A challenging problem in current systems biology is that of parameter inference in biological pathways expressed as coupled ordinary differential equations (ODEs). Conventional methods that repeatedly numerically solve the ODEs have large associated computational costs. Aimed at reducing this cost, new concepts using gradient matching have been proposed, which bypass the need for numerical integration. This paper presents a recently established adaptive gradient matching approach, using Gaussian processes, combined with a parallel tempering scheme, and conducts a comparative evaluation with current state of the art methods used for parameter inference in ODEs. Among these contemporary methods is a technique based on reproducing kernel Hilbert spaces (RKHS). This has previously shown promising results for parameter estimation, but under lax experimental settings. We look at a range of scenarios to test the robustness of this method. We also change the approach of inferring the penalty parameter from AIC to cross validation to improve the stability of the method. Methodology: Methodology for the recently proposed adaptive gradient matching method using Gaussian processes, upon which we build our new method, is provided. Details of a competing method using reproducing kernel Hilbert spaces are also described here. Results: We conduct a comparative analysis for the methods described in this paper, using two benchmark ODE systems. The analyses are repeated under different experimental settings, to observe the sensitivity of the techniques. Conclusions: Our study reveals that for known noise variance, our proposed method based on Gaussian processes and parallel tempering achieves overall the best performance. When the noise variance is unknown, the RKHS method proves to be more robust

Parallel R-point implicit block method for solving higher order ordinary differential equations directly

Most of the existing methods for solving ordinary differential equations (ODEs) of higher order are sequential in nature. These methods approximate numerical solution at one point at a time and therefore do not fully exploit the capability of parallel computers. Hence, the development of parallel algorithms to suit these machines becomes essential. In this paper, a new method called parallel R-point implicit block method for solving higher order ODEs directly using constant step size is developed. This method calculates the numerical solution at more than one point simultaneously and is parallel in nature, thus suitable for parallel computation. Computational advantages are presented comparing the results obtained by the new method with that of conventional 1-point method. The numerical results show that the new method reduces the total number of steps and execution time. The accuracy of the parallel block and the conventional 1-point methods are comparable particularly when finer step sizes are used

Direct Block Methods for Solving Special Second Order Ordinary Differential Equations and Their Parallel Implementations

This thesis focuses mainly on deriving block methods of constant step size for solving special second order ODEs. The first part of the thesis is about the construction and derivation of block methods using linear difference operator. The regions of stability for both explicit and implicit block methods are presented. The numerical results of the methods are compared with existing methods. The results suggest a significant improvement in efficiency of the new methods. The second part of the thesis describes the derivation of the r-point block methods based on Newton-Gregory backward interpolation formula. The numerical results of explicit and implicit r-point block methods are presented to illustrate the effectiveness of the methods in terms of total number of steps taken, accuracy and execution time. Both the explicit and implicit methods are more efficient compare to the existing method. The r-point block methods that calculate the solution at r-point simultaneously are suitable for parallel implementation. The parallel codes of the block methods for the solution of large systems of ODEs are developed. Hence the last part of the thesis discusses the parallel execution of the codes. The parallel algorithms are written in C language and implemented on Sun Fire V1280 distributed memory system. The fine-grained strategy is used to divide a computation into smaller parts and assign them to different processors. The performances of the r-point block methods using sequential and parallel codes are compared in terms of the total steps, execution time, speedup and efficiency. The parallel implementation of the new codes produced better speedup as the number of equations increase. The parallel codes gain better speedup and efficiency compared to sequential codes

Implementation of four-point fully implicit block method for solving ordinary differential equations

This paper describes the development of a four-point fully implicit block method for solving first order ordinary differential equations (ODEs) using variable step size. This method will estimate the solutions of initial value problems (IVPs) at four points simultaneously. The method developed is suitable for the numerical integration of non-stiff and mildly stiff differential systems. The performances of the four-point block method are compared in terms of maximum error, total number of steps and execution times to the non-block method 1PVSO in [Z. Omar, Developing parallel block methods for solving higher order ODEs directly, Ph.D. Thesis, University Putra Malaysia, Malaysia, 1999]

Runge-Kutta-Nystrom Methods For Solving Oscillatory Problems

New Runge-Kutta-Nyström (RKN) methods are derived for solving system of second-order Ordinary Differential Equations (ODEs) in which the solutions are in the oscillatory form. The dispersion and dissipation relations are imposed to get methods with the highest possible order of dispersion and dissipation. The derivation of Embedded Explicit RKN (ERKN) methods for variable step size codes are also given. The strategies in choosing the free parameters are also discussed. We analyze the numerical behavior of the RKN and ERKN methods both theoretically and experimentally and comparisons are made over the existing methods. In the second part of this thesis, a Block Embedded Explicit RKN (BERKN) method are developed. The implementation of BERKN method is discussed. The numerical results are compared with non block method. We find that the new code on Block Embedded Explicit RKN (BERKN) method is more efficient for solving system of second-order ODEs directly. Next, we discussed the derivation of Diagonally Implicit RKN (DIRKN) methods for solving stiff second order ODEs in which the solutions are oscillating functions. The dispersion and dissipation relations are developed and again are imposed in the derivation of the methods. For solving oscillatory problems with high frequency, method with P-stability property is discussed. We also derive the Embedded Diagonally Implicit RKN (EDIRKN) methods for variable step size codes. To see the preciseness and effectiveness of the methods, the constant and variable step size codes are developed and numerical results are compared with current methods given in the literature. Finally, the Parallel Embedded Explicit RKN (PERKN) method is developed. The parallel implementation of PERKN on the parallel machine is discussed. The performance of the PERKN algorithm for solving large system of ODEs are presented. We observe that the PERKN gives the better performance when solving large system of ODEs. In conclusion, the new codes developed in this thesis are suitable for solving system of second-order ODEs in which the solutions are in the oscillatory form