Constrained Formulations Research Articles

Iterative regularization for constrained minimization formulations of nonlinear inverse problems

In this paper we study the formulation of inverse problems as constrained minimization problems and their iterative solution by gradient or Newton type methods. We carry out a convergence analysis in the sense of regularization methods and discuss applicability to the problem of identifying the spatially varying diffusivity in an elliptic PDE from different sets of observations. Among these is a novel hybrid imaging technology known as impedance acoustic tomography, for which we provide numerical experiments.

Taming the Duplication-Loss-Coalescence Model with Integer Linear Programming.

The duplication-loss-coalescence (DLC) parsimony model is invaluable for analyzing the complex scenarios of concurrent duplication loss and deep coalescence events in the evolution of gene families. However, inferring such scenarios for already moderately sized families is prohibitive owing to the computational complexity involved. To overcome this stringent limitation, we make the first step by describing a flexible integer linear programming (ILP) formulation for inferring DLC evolutionary scenarios. Then, to make the DLC model more scalable, we introduce four sensibly constrained versions of the model and describe modified versions of our ILP formulation reflecting these constraints. Our simulation studies showcase that our constrained ILP formulations compute evolutionary scenarios that are substantially larger than scenarios computable under our original ILP formulation and the original dynamic programming algorithm by Wu et al. Furthermore, scenarios computed under our constrained DLC models are remarkably accurate compared with corresponding scenarios under the original DLC model, which we also confirm in an empirical study with thousands of gene families.

Beating Level-Set Methods for 5-D Seismic Data Interpolation: A Primal-Dual Alternating Approach

Acquisition cost is a crucial bottleneck for seismic workflows, and low-rank formulations for data interpolation allow practitioners to “fill in” data volumes from critically subsampled data acquired in the field. Tremendous size of seismic data volumes required for seismic processing remains a major challenge for these techniques. Residual-constrained formulations require less parameter tuning when the target noise floor is known. We propose a new approach to solve residual constrained formulations for interpolation. We represent the data volume in a compressed manner using low-rank matrix factors, and build a block-coordinate algorithm with constrained convex subproblems that are solved with a primal-dual splitting scheme. The develop optimization framework works on the whole seismic temporal frequency slices and does not require windowing or nontrivial sorting of seismic data. The new approach is competitive with state of the art level-set algorithms that interchange the role of objectives with constraints. We use the new algorithm to successfully interpolate a large scale 5-D seismic data volume (upto $10^{10}$ data points), generated from the geologically complex synthetic 3-D Compass velocity model, where 80% of the data have been removed. We also develop a robust extension of the primal-dual approach to deal with the outliers (or noise) in the data.

A note on higher-derivative actions for free higher-spin fields

Higher-derivative theories of free higher-spin fields are investigated focusing on their symmetries. Generalizing familiar two-derivative constrained formulations, we first construct less-constrained Einstein-like and Maxwell-like higher-derivative actions. Then, we construct Weyl-like actions — the actions admitting constrained Weyl symmetries —with different numbers of derivatives. They are presented in a factorized form making use of Einstein-like and Maxwell-like tensors. The last (highest-derivative) member of the hierarchy of the Weyl-like actions coincides with the Fradkin-Tseytlin conformal higher-spin action in four dimensions.

Energy-consistent numerical integration of mechanical systems with mixed holonomic and nonholonomic constraints

The present work deals with the energy-consistent numerical integration of mechanical systems with mixed holonomic and nonholonomic constraints. The underlying differential-algebraic equations (DAEs) with index three are directly discretized. This approach makes possible the development of a new energy-consistent time-stepping scheme for general nonholonomic systems. In particular, both nonholonomic problems from rigid body dynamics as well as flexible multibody dynamics can be treated in a unified manner. In this connection specific constrained formulations of rigid bodies and geometrically exact beams are presented. Moreover, the newly developed discrete null space method is applied to achieve a size-reduction and an improved conditioning of the discrete system. The numerical examples deal with a nonholonomic rigid body system and a flexible multibody system.

Combining spectral and shock-capturing methods: A new numerical approach for 3D relativistic core collapse simulations

We present a new three-dimensional general relativistic hydrodynamics code which is intended for simulations of stellar core collapse to a neutron star, as well as pulsations and instabilities of rotating relativistic stars. Contrary to the common approach followed in most existing three-dimensional numerical relativity codes which are based in Cartesian coordinates, in this code both the metric and the hydrodynamics equations are formulated and solved numerically using spherical polar coordinates. A distinctive feature of this new code is the combination of two types of accurate numerical schemes specifically designed to solve each system of equations. More precisely, the code uses spectral methods for solving the gravitational field equations, which are formulated under the assumption of the conformal flatness condition (CFC) for the three-metric. Correspondingly, the hydrodynamics equations are solved by a class of finite difference methods called high-resolution shock-capturing schemes, based upon state-of-the-art Riemann solvers and third-order cell-reconstruction procedures. We demonstrate that the combination of a finite difference grid and a spectral grid, on which the hydrodynamics and metric equations are, respectively, solved, can be successfully accomplished. This approach, which we call Mariage des Maillages (French for grid wedding), results in high accuracy of the metric solver and, in practice, allows for fully three-dimensional applications using computationally affordable resources, along with ensuring long-term numerical stability of the evolution. We compare our new approach to two other, finite difference based, methods to solve the metric equations which we already employed in earlier axisymmetric simulations of core collapse. A variety of tests in two and three dimensions is presented, involving highly perturbed neutron star spacetimes and (axisymmetric) stellar core collapse, which demonstrate the ability of the code to handle spacetimes with and without symmetries in strong gravity. These tests are also employed to assess the gravitational waveform extraction capabilities of the code, which is based on the Newtonian quadrupole formula. The code presented here is not limited to approximations of the Einstein equations such as CFC, but it is also well suited, in principle, to recent constrained formulations of the metric equations where elliptic equations have a preeminence over hyperbolic equations.

Penalty Formulations and Trap-Avoidance Strategies for Solving Hard Satisfiability Problems

In this paper we study the solution of SAT problems formulated as discrete decision and discrete constrained optimization problems. Constrained formulations are better than traditional unconstrained formulations because violated constraints may provide additional forces to lead a search towards a satisfiable assignment. We summarize the theory of extended saddle points in penalty formulations for solving discrete constrained optimization problems and the associated discrete penalty method (DPM). We then examine various formulations of the objective function, choices of neighborhood in DPM, strategies for updating penalties, and heuristics for avoiding traps. Experimental evaluations on hard benchmark instances pinpoint that traps contribute significantly to the inefficiency of DPM and force a trajectory to repeatedly visit the same set of or nearby points in the original variable space. To address this issue, we propose and study two trap-avoidance strategies. The first strategy adds extra penalties on unsatisfied clauses inside a trap, leading to very large penalties for unsatisfied clauses that are trapped more often and making these clauses more likely to be satisfied in the future. The second strategy stores information on points visited before, whether inside traps or not, and avoids visiting points that are close to points visited before. It can be implemented by modifying the penalty function in such a way that, if a trajectory gets close to points visited before, an extra penalty will take effect and force the trajectory to a new region. It specializes to the first strategy because traps are special cases of points visited before. Finally, we show experimental results on evaluating benchmarks in the DIMACS and SATLIB archives and compare our results with existing results on GSAT, WalkSAT, LSDL, and Grasp. The results demonstrate that DPM with trap avoidance is robust as well as effective for solving hard SAT problems.

A Unified Approach to the Energy‐Consistent Numer‐ical Integration of Nonholonomic Mechanical Systems and Flexible Multibody Dynamics

AbstractThe present work deals with the energy‐consistent numerical integration of mechanical systems with mixed holonomic and nonholonomic constraints. The underlying differentialalgebraic equations (DAEs) with index three are directly discretized. This approach makes possible the development of a new energy‐consistent time‐stepping scheme for general nonholonomic systems. In particular, both nonholonomic problems from rigid body dynamics as well as flexible multibody dynamics can be treated in a unified manner. In this connection specific constrained formulations of rigid bodies and geometrically exact beams are presented. Moreover, the newly developed discrete null space method is applied to achieve a size‐reduction and an improved conditioning of the discrete system. The numerical examples deal with a nonholonomic rigid body system and a flexible multibody system. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Geometry ofN=1 supergravity

A new geometrical formalism is suggested for the non-minimal and alternative minimal supergravities. This formalism connects the constrained superspace formulations with the unconstrained ones and is based on the notion of induced geometry. The relevant mathematical technique is that ofG-structures. A clear-cut geometrical content of the torsion and curvature constraints is revealed on the basis of a general theorem about the necessary and sufficient properties of induced geometry.