Genetics and analysis of quantitative traits

Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

Numerical Initial Value Problems in Ordinary Differential Equations

Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus $\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i)$. Hopefully, the quadrature rule ${\{w_j, t_j\}}_{j=1}^{N}$ corresponding to the weight function $\omega$(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weight function are known or can be calculated.

Calculation of Gauss quadrature rules.

Interactions among loci or between genes and environmental factors make a substantial contribution to variation in complex traits such as disease susceptibility. Nonetheless, many studies that attempt to identify the genetic basis of complex traits ignore the possibility that loci interact. We argue that epistasis should be accounted for in complex trait studies; we critically assess current study designs for detecting epistasis and discuss how these might be adapted for use in additional populations, including humans.

/pdf/epistasis-too-often-neglected-in-complex-trait-studies-5dh6k52o1l.pdf

Epistasis: too often neglected in complex trait studies?

Motivation: A simultaneous search is necessary for maximizing the power to detect epistatic quantitative trait loci (QTL). The computational complexity demands that the traditional exhaustive search be replaced by a more efficient global optimization algorithm.

Results: We have the previously known algorithm adapted DIRECT, to the problem of simultaneous mapping of multiple QTL. We have compared DIRECT with standard exhaustive search and a genetic algorithm previously used for QTL mapping in two dimensions. In all two- and three-QTL test cases, DIRECT accurately finds the global optimum two to four orders of magnitude faster than when using an exhaustive search, and one order of magnitude faster than when using the genetic algorithm. Thus, randomization testing for determining empirical significance thresholds for at least three QTL is made feasible by the use of DIRECT.

Availability: The code of the prototype implementation is available at http://user.it.uu.se/~kl/qtl_software.html

/pdf/simultaneous-search-for-multiple-qtl-using-the-global-35h8hgh3iw.pdf

Simultaneous search for multiple QTL using the global optimization algorithm DIRECT

The non-uniform memory access times of modern cc-NUMA systems often impair performance for shared memory applications. This is especially true for applications exhibiting complex access patterns. To improve performance, a mechanism for co-locating threads and data during the execution is needed. In this paper, we study how an affinity-on-next-touch procedure can be used to attain this goal. Such a procedure can increase thread-data affinity by migrating data across nodes to better match the access pattern. The migration is triggered by a directive and it can often be implemented as a re-invocation of a standard first-touch page placement procedure. We study an industrial-class scientific application where the thread-data affinity is small due to serial initializations of data structures accessed indirectly. Adding a single affinity-on-next-touch directive, we observed a performance improvement of 69% for 22 threads. We also perform experiments to study the scalability of the affinity-on-next-touch procedure. Our results indicate that the overhead associated with the procedure is highly dependent on the efficiency of the mechanism used to keep TLBs consistent. Using larger but fewer memory pages in the virtual memory sub-system we measured a total performance improvement of 166% compared to the original code.

affinity-on-next-touch: increasing the performance of an industrial PDE solver on a cc-NUMA system

Several different numerical propagation techniques for explicitly time-dependent Hamiltonians are discussed and compared, with the focus on models of pump-probe experiments. The quality of the rotating wave approximation is analyzed analytically, and we point out under which circumstances the modeling becomes inaccurate. For calculations with the fully time-dependent Hamiltonian, we show that for multistate systems, with either time or space dependence in the interstate coupling, the fourth order truncated Magnus expansion can be reformulated so that no commutators appear. Our results show that the split-operator method should only be used when low accuracy is acceptable. For accurate and efficient time stepping, the Magnus-Lanczos approach appears to be the best choice.

Accurate time propagation for the Schrodinger equation with an explicitly time-dependent Hamiltonian.

Systems of equations arising from implicit time discretizations and finite difference space discretizations of systems of partial differential equations in two space dimensions are considered The 

Iterative solution methods and preconditioners for block-tridiagonal systems of equations

Efficient solution of partial differential equations require a match between the algorithm and the target architecture. Many recent chip multiprocessors, CMPs (a.k.a. multi-core), feature low intra-thread communication costs and smaller per-thread caches compared to previous shared memory multi-processor systems. From an algorithmic point of view this means that data locality issues become more important than communication overheads. A fact that may require a re-evaluation of many existing algorithms.We have investigated parallel implementations of multi-grid methods using a parallel temporally blocked, naturally ordered smoother. Compared to the standard multigrid solution based on a red-black ordering, we improve the data locality often as much as ten times, while our use of a fine-grained locking scheme keeps the parallel efficiency high.Our algorithm was initially inspired by CMPs and it was surprising to see that our OpenMP multigrid implementation ran up to 40 percent faster than the standard red-black algorithm on a contemporary 8-way SMP system. Thanks to the temporal blocking introduced, our smoother implementation often allowed us to apply the smoother two times at the same cost as a single application of a red-black smoother. By executing our smoother on a 32-thread UltraSPARC T1 (Niagara) SMT/CMP and a simulated 32-way CMP we demonstrate that such architectures can tolerate the increased communication costs implied by the tradeoffs made in our implementation.

https://www.it.uu.se/research/publications/reports/2006-018/2006-018-nc.pdf

Sverker Holmgren

Papers

Simultaneous search for multiple QTL using the global optimization algorithm DIRECT

affinity-on-next-touch: increasing the performance of an industrial PDE solver on a cc-NUMA system

Accurate time propagation for the Schrodinger equation with an explicitly time-dependent Hamiltonian.

Iterative solution methods and preconditioners for block-tridiagonal systems of equations

Multigrid and Gauss-Seidel smoothers revisited: parallelization on chip multiprocessors