Surface networks capture the topological relations between passes of a continuous surface, the paths of steepest descent and ascent starting at the passes, and the pits and peaks where the steepest paths end. This paper extends the topology of the network in three ways. Objects at the edge of the surface model are introduced. Horizontal areas may represent passes, pits, or peaks, and therefore ...

The possibilities inherent in steepest descent methods have been considerably amplified by the introduction of the Barzilai-Borwein choice of step-size, and other related ideas. These methods have proved to be competitive with conjugate gradient methods for the minimization of large dimension unconstrained minimization problems. This paper suggests a method which is able to take advantage of th...

We consider the special case of the restarted Arnoldi method for approximating the product of a function of a Hermitian matrix with a vector which results when the restart length is set to one. When applied to the solution of a linear system of equations, this approach coincides with the method of steepest descent. We show that the method is equivalent with an interpolation process in which the...

Recently there has been much attention to MDL and its effectiveness in automatic shape modelling. One problem of this technique has been the slow convergence of the optimization step. In this paper the Jacobian of the objective function is derived. Being able to calculate the Jacobian, a variety of optimization techniques can be considered. In this paper we apply steepest descent and show that ...

We consider (a subclass of) convex quadratic optimization problems and analyze decomposition algorithms that perform, at least approximately, steepest-descent exact line search. We show that these algorithms, when implemented properly, are within ǫ of optimality after O(log 1/ǫ) iterations for strictly convex cost functions, and after O(1/ǫ) iterations in the general case. Our analysis is gener...

Let E be a real Banach space and let A : E → E be a Lipschitzian generalized strongly accretive operator. Let z ∈ E and x0 be an arbitrary initial value in E for which the steepest descent approximation scheme is defined by xn+1 = xn − αn(Ayn − z), yn = xn − βn(Axn − z), n = 0, 1, 2 . . . , where the sequences {αn} and {βn} satisfy the following conditions: (i) 0 ≤ αn, βn ≤ 1, (ii) ∞

We enumerate possible topologies of pseudoknots in single-stranded RNA molecules. We use a steepest-descent approximation in the large N matrix field theory, and a Feynman diagram formalism to describe the resulting pseudoknot structure. An RNA molecule is a heteropolymer strand made up of four types of nucleotides, uracil (U), adenine (A), guanine (G), and cytosine (C). The sequence of these n...