Regularization Tools

... solution is not unique or if it is not a continuous function of the data—i.e., if an arbitrarily small perturbation of the data can cause an arbitrarily large perturbation of the solution. Hadamard believed that ill-posed problems were “artificial” in that they would not describe physical systems. H ...

... solution is not unique or if it is not a continuous function of the data—i.e., if an arbitrarily small perturbation of the data can cause an arbitrarily large perturbation of the solution. Hadamard believed that ill-posed problems were “artificial” in that they would not describe physical systems. H ...

Limitations of Front-to-End Bidirectional Heuristic Search

... of its nodes deeper than the solution midpoint, we call its heuristic weak and show that a bidirectional heuristic search expands no fewer nodes than a bidirectional bruteforce search. If the majority are expanded at shallower depth than the solution midpoint, then we call it a strong heuristic and ...

... of its nodes deeper than the solution midpoint, we call its heuristic weak and show that a bidirectional heuristic search expands no fewer nodes than a bidirectional bruteforce search. If the majority are expanded at shallower depth than the solution midpoint, then we call it a strong heuristic and ...

E - Read

... – 2n+o(n) by Bent and John (STOC 1985) – (2+2-80)n by Dor and Zwick (FOCS 1996, SIAM Journal on Discrete Math 2001). ...

... – 2n+o(n) by Bent and John (STOC 1985) – (2+2-80)n by Dor and Zwick (FOCS 1996, SIAM Journal on Discrete Math 2001). ...

High order schemes based on operator splitting and - HAL

... In this work, the high order quadrature formulas over a time step ∆t, corresponding to an s– stage implicit Runge–Kutta scheme, are evaluated using the numerical approximations computed by a splitting solver at the s intermediate collocation nodes. Such a dedicated splitting solver for stiff PDEs ca ...

... In this work, the high order quadrature formulas over a time step ∆t, corresponding to an s– stage implicit Runge–Kutta scheme, are evaluated using the numerical approximations computed by a splitting solver at the s intermediate collocation nodes. Such a dedicated splitting solver for stiff PDEs ca ...

CS 372: Computational Geometry Lecture 14 Geometric

... The points are on a grid with side length 0 ; we will choose it such that 0 ' ∆0 . So in the worst case, there are about as many points in P 0 as in a (1/) × (1/) · · · × (1/) grid in Rd . There are O((1/)d ) such points. We can compute ∆0 by brute force in time O((1/)2d ). ...

... The points are on a grid with side length 0 ; we will choose it such that 0 ' ∆0 . So in the worst case, there are about as many points in P 0 as in a (1/) × (1/) · · · × (1/) grid in Rd . There are O((1/)d ) such points. We can compute ∆0 by brute force in time O((1/)2d ). ...

Fast Matrix Rank Algorithms and Applications - USC

... is a randomized algorithm to find a solution of size min{opt, k} in Õ(|A| + nkω−1 ) time. 3. Given a linear matroid union problem with an r × n matrix |A|, there is a randomized algorithm to find min{opt, k} disjoint bases in Õ(k|A|+min{kω+1 bω , k3 b3 }) time, where b denotes the size of a basis. ...

... is a randomized algorithm to find a solution of size min{opt, k} in Õ(|A| + nkω−1 ) time. 3. Given a linear matroid union problem with an r × n matrix |A|, there is a randomized algorithm to find min{opt, k} disjoint bases in Õ(k|A|+min{kω+1 bω , k3 b3 }) time, where b denotes the size of a basis. ...

# Travelling salesman problem

The travelling salesman problem (TSP) asks the following question: Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city? It is an NP-hard problem in combinatorial optimization, important in operations research and theoretical computer science.TSP is a special case of the travelling purchaser problem and the Vehicle routing problem.In the theory of computational complexity, the decision version of the TSP (where, given a length L, the task is to decide whether the graph has any tour shorter than L) belongs to the class of NP-complete problems. Thus, it is possible that the worst-case running time for any algorithm for the TSP increases superpolynomially (perhaps, specifically, exponentially) with the number of cities.The problem was first formulated in 1930 and is one of the most intensively studied problems in optimization. It is used as a benchmark for many optimization methods. Even though the problem is computationally difficult, a large number of heuristics and exact methods are known, so that some instances with tens of thousands of cities can be solved completely and even problems with millions of cities can be approximated within a small fraction of 1%.The TSP has several applications even in its purest formulation, such as planning, logistics, and the manufacture of microchips. Slightly modified, it appears as a sub-problem in many areas, such as DNA sequencing. In these applications, the concept city represents, for example, customers, soldering points, or DNA fragments, and the concept distance represents travelling times or cost, or a similarity measure between DNA fragments. The TSP also appears in astronomy, as astronomers observing many sources will want to minimise the time spent slewing the telescope between the sources. In many applications, additional constraints such as limited resources or time windows may be imposed.