Bisection method

Abstract: We establish a correspondence between the singular values of a transfer matrix evaluated along the imaginary axis and the imaginary eigenvalues of a related Hamiltonian matrix. We give a simple linear algebraic proof, and also a more intuitive explanation based on a certain indefinite quadratic optimal control problem. This result yields a simple bisection algorithm to compute the H∞ norm of a transfer matrix. The bisection method is far more efficient than algorithms which involve a search over frequencies, and the usual problems associated with such methods (such as determining how fine the search should be) do not arise. The method is readily extended to compute other quantities of system-theoretic interest, for instance, the minimum dissipation of a transfer matrix. A variation of the method can be used to solve the H∞ Armijo line-search problem with no more computation than is required to compute a single H∞ norm.

Abstract: This letter introduces a new iterative method for contact dynamics problems. The proposed method is based on an efficient bisection method which iterates over each contact. We compared our approach to two existing ones for the same model and found that it is about twice as fast as the existing ones. We also introduce four different robotic simulation experiments and compare the proposed method to the most common contact solver, the projected Gauss-Seidel (PGS) method. We show that, while both methods are very efficient in solving simple problems, the proposed method significantly outperforms the PGS method in more complicated contact scenarios. Simulating one time step of an 18-DOF quadruped robot with multiple contacts took less than 20 μs with a single core of a CPU. This is at least an order of magnitude faster than many other simulators which employ multiple relaxation methods to the major dynamic principles in order to boost their computational speed. The proposed simulation method is also stable at 50 Hz due to its strict adherence to the dynamical principles. Although the accuracy might be compromised at such a low update rate, this means that we can simulate an 18-DOF robot more than thousand times faster than the real time.

Abstract: The most commonly used p-way partitioning method is recursive bisection (RB). It first divides a graph or a mesh into two equal-sized pieces, by a "good" bisection algorithm, and then recursively divides the two pieces. Ideally, we would like to use an optimal bisection algorithm. Because the optimal bisection problem that partitions a graph into two equal-sized subgraphs to minimize the number of edges cut is NP-complete, practical RB algorithms use more efficient heuristics in place of an optimal bisection algorithm. Most such heuristics are designed to find the best possible bisection within allowed time. We show that the RB method, even when an optimal bisection algorithm is assumed, may produce a p-way partition that is very far way from the optimal one. Our negative result is complemented by two positive ones: first we show that for some important classes of graphs that occur in practical applications, such as well-shaped finite-element and finite-difference meshes, RB is within a constant factor of the optimal one "almost always." Second, we show that if the balance condition is relaxed so that each block in the p-way partition is bounded by 2n/p, where n is the number of vertices of the graph, then a modified RB finds an approximately balanced $p$-way partition whose cost is within an O(log p) factor of the cost of the optimal p-way partition.