A lagrangian reconstruction of a class of local search methods.

by Choi Mo Fung Kenneth.Thesis (M.Phil.)--Chinese University of Hong Kong, 1998.Includes bibliographical references (leaves 105-112).Abstract also in Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Constraint Satisfaction Problems --- p.2Chapter 1.2 --- Constraint Satisfaction Techniques --- p.2Chapter 1.3 --- Motivation of the Research --- p.4Chapter 1.4 --- Overview of the Thesis --- p.5Chapter 2 --- Related Work --- p.7Chapter 2.1 --- Min-conflicts Heuristic --- p.7Chapter 2.2 --- GSAT --- p.8Chapter 2.3 --- Breakout Method --- p.8Chapter 2.4 --- GENET --- p.9Chapter 2.5 --- E-GENET --- p.9Chapter 2.6 --- DLM --- p.10Chapter 2.7 --- Simulated Annealing --- p.11Chapter 2.8 --- Genetic Algorithms --- p.12Chapter 2.9 --- Tabu Search --- p.12Chapter 2.10 --- Integer Programming --- p.13Chapter 3 --- Background --- p.15Chapter 3.1 --- GENET --- p.15Chapter 3.1.1 --- Network Architecture --- p.15Chapter 3.1.2 --- Convergence Procedure --- p.18Chapter 3.2 --- Classical Optimization --- p.22Chapter 3.2.1 --- Optimization Problems --- p.22Chapter 3.2.2 --- The Lagrange Multiplier Method --- p.23Chapter 3.2.3 --- Saddle Point of Lagrangian Function --- p.25Chapter 4 --- Binary CSP's as Zero-One Integer Constrained Minimization Prob- lems --- p.27Chapter 4.1 --- From CSP to SAT --- p.27Chapter 4.2 --- From SAT to Zero-One Integer Constrained Minimization --- p.29Chapter 5 --- A Continuous Lagrangian Approach for Solving Binary CSP's --- p.33Chapter 5.1 --- From Integer Problems to Real Problems --- p.33Chapter 5.2 --- The Lagrange Multiplier Method --- p.36Chapter 5.3 --- Experiment --- p.37Chapter 6 --- A Discrete Lagrangian Approach for Solving Binary CSP's --- p.39Chapter 6.1 --- The Discrete Lagrange Multiplier Method --- p.39Chapter 6.2 --- Parameters of CSVC --- p.43Chapter 6.2.1 --- Objective Function --- p.43Chapter 6.2.2 --- Discrete Gradient Operator --- p.44Chapter 6.2.3 --- Integer Variables Initialization --- p.45Chapter 6.2.4 --- Lagrange Multipliers Initialization --- p.46Chapter 6.2.5 --- Condition for Updating Lagrange Multipliers --- p.46Chapter 6.3 --- A Lagrangian Reconstruction of GENET --- p.46Chapter 6.4 --- Experiments --- p.52Chapter 6.4.1 --- Evaluation of LSDL(genet) --- p.53Chapter 6.4.2 --- Evaluation of Various Parameters --- p.55Chapter 6.4.3 --- Evaluation of LSDL(max) --- p.63Chapter 6.5 --- Extension of LSDL --- p.66Chapter 6.5.1 --- Arc Consistency --- p.66Chapter 6.5.2 --- Lazy Arc Consistency --- p.67Chapter 6.5.3 --- Experiments --- p.70Chapter 7 --- Extending LSDL for General CSP's: Initial Results --- p.77Chapter 7.1 --- General CSP's as Integer Constrained Minimization Problems --- p.77Chapter 7.1.1 --- Formulation --- p.78Chapter 7.1.2 --- Incompatibility Functions --- p.79Chapter 7.2 --- The Discrete Lagrange Multiplier Method --- p.84Chapter 7.3 --- A Comparison between the Binary and the General Formulation --- p.85Chapter 7.4 --- Experiments --- p.87Chapter 7.4.1 --- The N-queens Problems --- p.89Chapter 7.4.2 --- The Graph-coloring Problems --- p.91Chapter 7.4.3 --- The Car-Sequencing Problems --- p.92Chapter 7.5 --- Inadequacy of the Formulation --- p.94Chapter 7.5.1 --- Insufficiency of the Incompatibility Functions --- p.94Chapter 7.5.2 --- Dynamic Illegal Constraint --- p.96Chapter 7.5.3 --- Experiments --- p.97Chapter 8 --- Concluding Remarks --- p.100Chapter 8.1 --- Contributions --- p.100Chapter 8.2 --- Discussions --- p.102Chapter 8.3 --- Future Work --- p.103Bibliography --- p.10

Optimization Methods for Inverse Problems

Optimization plays an important role in solving many inverse problems. Indeed, the task of inversion often either involves or is fully cast as a solution of an optimization problem. In this light, the mere non-linear, non-convex, and large-scale nature of many of these inversions gives rise to some very challenging optimization problems. The inverse problem community has long been developing various techniques for solving such optimization tasks. However, other, seemingly disjoint communities, such as that of machine learning, have developed, almost in parallel, interesting alternative methods which might have stayed under the radar of the inverse problem community. In this survey, we aim to change that. In doing so, we first discuss current state-of-the-art optimization methods widely used in inverse problems. We then survey recent related advances in addressing similar challenges in problems faced by the machine learning community, and discuss their potential advantages for solving inverse problems. By highlighting the similarities among the optimization challenges faced by the inverse problem and the machine learning communities, we hope that this survey can serve as a bridge in bringing together these two communities and encourage cross fertilization of ideas.Comment: 13 page

Reconstruction of the primordial Universe by a Monge--Ampere--Kantorovich optimisation scheme

A method for the reconstruction of the primordial density fluctuation field is presented. Various previous approaches to this problem rendered {\it non-unique} solutions. Here, it is demonstrated that the initial positions of dark matter fluid elements, under the hypothesis that their displacement is the gradient of a convex potential, can be reconstructed uniquely. In our approach, the cosmological reconstruction problem is reformulated as an assignment problem in optimisation theory. When tested against numerical simulations, our scheme yields excellent reconstruction on scales larger than a few megaparsecs.Comment: 14 pages, 10 figure

Dependent Nonparametric Bayesian Group Dictionary Learning for online reconstruction of Dynamic MR images

In this paper, we introduce a dictionary learning based approach applied to the problem of real-time reconstruction of MR image sequences that are highly undersampled in k-space. Unlike traditional dictionary learning, our method integrates both global and patch-wise (local) sparsity information and incorporates some priori information into the reconstruction process. Moreover, we use a Dependent Hierarchical Beta-process as the prior for the group-based dictionary learning, which adaptively infers the dictionary size and the sparsity of each patch; and also ensures that similar patches are manifested in terms of similar dictionary atoms. An efficient numerical algorithm based on the alternating direction method of multipliers (ADMM) is also presented. Through extensive experimental results we show that our proposed method achieves superior reconstruction quality, compared to the other state-of-the- art DL-based methods

Arbitrary-Lagrangian-Eulerian One-Step WENO Finite Volume Schemes on Unstructured Triangular Meshes

In this article we present a new class of high order accurate Arbitrary-Eulerian-Lagrangian (ALE) one-step WENO finite volume schemes for solving nonlinear hyperbolic systems of conservation laws on moving two dimensional unstructured triangular meshes. A WENO reconstruction algorithm is used to achieve high order accuracy in space and a high order one-step time discretization is achieved by using the local space-time Galerkin predictor. For that purpose, a new element--local weak formulation of the governing PDE is adopted on moving space--time elements. The space-time basis and test functions are obtained considering Lagrange interpolation polynomials passing through a predefined set of nodes. Moreover, a polynomial mapping defined by the same local space-time basis functions as the weak solution of the PDE is used to map the moving physical space-time element onto a space-time reference element. To maintain algorithmic simplicity, the final ALE one-step finite volume scheme uses moving triangular meshes with straight edges. This is possible in the ALE framework, which allows a local mesh velocity that is different from the local fluid velocity. We present numerical convergence rates for the schemes presented in this paper up to sixth order of accuracy in space and time and show some classical numerical test problems for the two-dimensional Euler equations of compressible gas dynamics.Comment: Accepted by "Communications in Computational Physics

OPTIMASS: A Package for the Minimization of Kinematic Mass Functions with Constraints

Reconstructed mass variables, such as $M_2$, $M_{2C}$, $M_T^\star$, and $M_{T2}^W$, play an essential role in searches for new physics at hadron colliders. The calculation of these variables generally involves constrained minimization in a large parameter space, which is numerically challenging. We provide a C++ code, OPTIMASS, which interfaces with the MINUIT library to perform this constrained minimization using the Augmented Lagrangian Method. The code can be applied to arbitrarily general event topologies and thus allows the user to significantly extend the existing set of kinematic variables. We describe this code and its physics motivation, and demonstrate its use in the analysis of the fully leptonic decay of pair-produced top quarks using the $M_2$ variables.Comment: 39 pages, 12 figures, (1) minor revision in section 3, (2) figure added in section 4.3, (3) reference added and (4) matched with published versio

Lagrangian ADER-WENO Finite Volume Schemes on Unstructured Triangular Meshes Based On Genuinely Multidimensional HLL Riemann Solvers

In this paper we use the genuinely multidimensional HLL Riemann solvers recently developed by Balsara et al. to construct a new class of computationally efficient high order Lagrangian ADER-WENO one-step ALE finite volume schemes on unstructured triangular meshes. A nonlinear WENO reconstruction operator allows the algorithm to achieve high order of accuracy in space, while high order of accuracy in time is obtained by the use of an ADER time-stepping technique based on a local space-time Galerkin predictor. The multidimensional HLL and HLLC Riemann solvers operate at each vertex of the grid, considering the entire Voronoi neighborhood of each node and allows for larger time steps than conventional one-dimensional Riemann solvers. The results produced by the multidimensional Riemann solver are then used twice in our one-step ALE algorithm: first, as a node solver that assigns a unique velocity vector to each vertex, in order to preserve the continuity of the computational mesh; second, as a building block for genuinely multidimensional numerical flux evaluation that allows the scheme to run with larger time steps compared to conventional finite volume schemes that use classical one-dimensional Riemann solvers in normal direction. A rezoning step may be necessary in order to overcome element overlapping or crossing-over. We apply the method presented in this article to two systems of hyperbolic conservation laws, namely the Euler equations of compressible gas dynamics and the equations of ideal classical magneto-hydrodynamics (MHD). Convergence studies up to fourth order of accuracy in space and time have been carried out. Several numerical test problems have been solved to validate the new approach