44,188 research outputs found
Enclosure of all index-1 saddle points of general nonlinear functions
Transition states (index-1 saddle points) play a crucial role in determining the rates of chemical transformations but their reliable identification remains challenging in many applications. Deterministic global optimization methods have previously been employed for the location of transition states (TSs) by initially finding all stationary points and then identifying the TSs among the set of solutions. We propose several regional tests, applicable to general nonlinear, twice continuously differentiable functions, to accelerate the convergence of such approaches by identifying areas that do not contain any TS or that may contain a unique TS. The tests are based on the application of the interval extension of theorems from linear algebra to an interval Hessian matrix. They can be used within the framework of global optimization methods with the potential of reducing the computational time for TS location. We present the theory behind the tests, discuss their algorithmic complexity and show via a few examples that significant gains in computational time can be achieved by using these tests
An Overview of Polynomially Computable Characteristics of Special Interval Matrices
It is well known that many problems in interval computation are intractable,
which restricts our attempts to solve large problems in reasonable time. This
does not mean, however, that all problems are computationally hard. Identifying
polynomially solvable classes thus belongs to important current trends. The
purpose of this paper is to review some of such classes. In particular, we
focus on several special interval matrices and investigate their convenient
properties. We consider tridiagonal matrices, {M,H,P,B}-matrices, inverse
M-matrices, inverse nonnegative matrices, nonnegative matrices, totally
positive matrices and some others. We focus in particular on computing the
range of the determinant, eigenvalues, singular values, and selected norms.
Whenever possible, we state also formulae for determining the inverse matrix
and the hull of the solution set of an interval system of linear equations. We
survey not only the known facts, but we present some new views as well
Efficient Genomic Interval Queries Using Augmented Range Trees
Efficient large-scale annotation of genomic intervals is essential for
personal genome interpretation in the realm of precision medicine. There are 13
possible relations between two intervals according to Allen's interval algebra.
Conventional interval trees are routinely used to identify the genomic
intervals satisfying a coarse relation with a query interval, but cannot
support efficient query for more refined relations such as all Allen's
relations. We design and implement a novel approach to address this unmet need.
Through rewriting Allen's interval relations, we transform an interval query to
a range query, then adapt and utilize the range trees for querying. We
implement two types of range trees: a basic 2-dimensional range tree (2D-RT)
and an augmented range tree with fractional cascading (RTFC) and compare them
with the conventional interval tree (IT). Theoretical analysis shows that RTFC
can achieve the best time complexity for interval queries regarding all Allen's
relations among the three trees. We also perform comparative experiments on the
efficiency of RTFC, 2D-RT and IT in querying noncoding element annotations in a
large collection of personal genomes. Our experimental results show that 2D-RT
is more efficient than IT for interval queries regarding most of Allen's
relations, RTFC is even more efficient than 2D-RT. The results demonstrate that
RTFC is an efficient data structure for querying large-scale datasets regarding
Allen's relations between genomic intervals, such as those required by
interpreting genome-wide variation in large populations.Comment: 4 figures, 4 table
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