An Interior-Point-Inspired algorithm for Linear Programs arising in Discrete Optimal Transport

Discrete Optimal Transport problems give rise to very large linear programs (LP) with a particular structure of the constraint matrix. In this paper we present a hybrid algorithm that mixes an interior point method (IPM) and column generation, specialized for the LP originating from the Kantorovich Optimal Transport problem. Knowing that optimal solutions of such problems display a high degree of sparsity, we propose a column-generation-like technique to force all intermediate iterates to be as sparse as possible. The algorithm is implemented nearly matrix-free. Indeed, most of the computations avoid forming the huge matrices involved and solve the Newton system using only a much smaller Schur complement of the normal equations. We prove theoretical results about the sparsity pattern of the optimal solution, exploiting the graph structure of the underlying problem. We use these results to mix iterative and direct linear solvers efficiently, in a way that avoids producing preconditioners or factorizations with excessive fill-in and at the same time guaranteeing a low number of conjugate gradient iterations. We compare the proposed method with two state-of-the-art solvers and show that it can compete with the best network optimization tools in terms of computational time and memory usage. We perform experiments with problems reaching more than four billion variables and demonstrate the robustness of the proposed method

A new stopping criterion for Krylov solvers applied in Interior Point Methods

A surprising result is presented in this paper with possible far reaching consequences for any optimization technique which relies on Krylov subspace methods employed to solve the underlying linear equation systems. In this paper the advantages of the new technique are illustrated in the context of Interior Point Methods (IPMs). When an iterative method is applied to solve the linear equation system in IPMs, the attention is usually placed on accelerating their convergence by designing appropriate preconditioners, but the linear solver is applied as a black box solver with a standard termination criterion which asks for a sufficient reduction of the residual in the linear system. Such an approach often leads to an unnecessary 'oversolving' of linear equations. In this paper a new specialized termination criterion for Krylov methods used in IPMs is designed. It is derived from a deep understanding of IPM needs and is demonstrated to preserve the polynomial worst-case complexity of these methods. The new criterion has been adapted to the Conjugate Gradient (CG) and to the Minimum Residual method (MINRES) applied in the IPM context. The new criterion has been tested on a set of linear and quadratic optimization problems including compressed sensing, image processing and instances with partial differential equation constraints. Evidence gathered from these computational experiments shows that the new technique delivers significant improvements in terms of inner (linear) iterations and those translate into significant savings of the IPM solution time

A regularized Interior Point Method for sparse Optimal Transport on Graphs

In this work, the authors address the Optimal Transport (OT) problem on graphs using a proximal stabilized Interior Point Method (IPM). In particular, strongly leveraging on the induced primal-dual regularization, the authors propose to solve large scale OT problems on sparse graphs using a bespoke IPM algorithm able to suitably exploit primal-dual regularization in order to enforce scalability. Indeed, the authors prove that the introduction of the regularization allows to use sparsified versions of the normal Newton equations to inexpensively generate IPM search directions. A detailed theoretical analysis is carried out showing the polynomial convergence of the inner algorithm in the proposed computational framework. Moreover, the presented numerical results showcase the efficiency and robustness of the proposed approach when compared to network simplex solvers

Efficient interior point algorithms for large scale convex optimization problems

Interior point methods (IPMs) are among the most widely used algorithms for convex optimization problems. They are applicable to a wide range of problems, including linear, quadratic, nonlinear, conic and semidefinite programming problems, requiring a polynomial number of iterations to find an accurate approximation of the primal-dual solution. The formidable convergence properties of IPMs come with a fundamental drawback: the numerical linear algebra involved becomes progressively more and more challenging as the IPM converges towards optimality. In particular, solving the linear systems to find the Newton directions requires most of the computational effort of an IPM. Proposed remedies to alleviate this phenomenon include regularization techniques, predictor-corrector schemes, purposely developed preconditioners, low-rank update strategies, to mention a few. For problems of very large scale, this unpleasant characteristic of IPMs becomes a more and more problematic feature, since any technique used must be efficient and scalable in order to maintain acceptable computational requirements. In this Thesis, we deal with convex linear and quadratic problems of large “dimension”: we use this term in a broader sense than just a synonym for “size” of the problem. The instances considered can be either problems with a large number of variables and/or constraints but with a sparse structure, or problems with a moderate number of variables and/or constraints but with a dense structure. Both these type of problems require very efficient strategies to be used during the algorithm, even though the corresponding difficulties arise for different reasons. The first application that we consider deals with a moderate size quadratic problem where the quadratic term is 100% dense; this problem arises from X-ray tomographic imaging reconstruction, in particular with the goal of separating the distribution of two materials present in the observed sample. A novel non-convex regularizer is introduced for this purpose; convexity of the overall problem is maintained by careful choice of the parameters. We derive a specialized interior point method for this problem and an appropriate preconditioner for the normal equations linear system, to be used without ever forming the fully dense matrices involved. The next major contribution is related to the issue of efficiently computing the Newton direction during IPMs. When an iterative method is applied to solve the linear equation system in IPMs, the attention is usually placed on accelerating their convergence by designing appropriate preconditioners, but the linear solver is applied as a black box with a standard termination criterion which asks for a sufficient reduction of the residual in the linear system. Such an approach often leads to an unnecessary “over-solving” of linear equations. We propose new indicators for the early termination of the inner iterations and test them on a set of large scale quadratic optimization problems. Evidence gathered from these computational experiments shows that the new technique delivers significant improvements in terms of inner (linear) iterations and those translate into significant savings of the IPM solution time. The last application considered is discrete optimal transport (OT) problems; these kind of problems give rise to very large linear programs with highly structured matrices. Solutions of such problems are expected to be sparse, that is only a small subset of entries in the optimal solution is expected to be nonzero. We derive an IPM for the standard OT formulation, which exploits a column-generation-like technique to force all intermediate iterates to be as sparse as possible. We prove theoretical results about the sparsity pattern of the optimal solution and we propose to mix iterative and direct linear solvers in an efficient way, to keep computational time and memory requirement as low as possible. We compare the proposed method with two state-of-the-art solvers and show that it can compete with the best network optimization tools in terms of computational time and memory usage. We perform experiments with problems reaching more than four billion variables and demonstrate the robustness of the proposed method. We consider also the optimal transport problem on sparse graphs and present a primal-dual regularized IPM to solve it. We prove that the introduction of the regularization allows us to use sparsified versions of the normal equations system to inexpensively generate inexact IPM directions. The proposed method is shown to have polynomial complexity and to outperform a very efficient network simplex implementation, for problems with up to 50 million variables

Black esophagus

Black esophagus is an uncommon clinical entity and its pathogenesis remains unknown. Clinical presentation is usually characterized by the combination of hematemesis and circumferential darkness of the mucosa in the distal esophagus. This case illustrates an atypical presentation of the disease. Despite its rarity, black esophagus should be considered in the differential diagnosis of acute upper gastrointestinal bleeding, especially in patients with predisposing factors. \ua9 2015 Taiwan Society of Emergency Medicine

a multibody simulation of a human fall model creation and validation

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Material-separating regularizer for multi-energy x-ray tomography

Dual-energy x-ray tomography is considered in a context where the target under imaging consists of two distinct materials. The materials are assumed to be possibly intertwined in space, but at any given location there is only one material present. Further, two x-ray energies are chosen so that there is a clear difference in the spectral dependence of the attenuation coefficients of the two materials. A novel regularizer is presented for the inverse problem of reconstructing separate tomographic images for the two materials. A combination of two things, (a) non-negativity constraint, and (b) penalty term containing the inner product between the two material images, promotes the presence of at most one material in a given pixel. A preconditioned interior point method is derived for the minimization of the regularization functional. Numerical tests with digital phantoms suggest that the new algorithm outperforms the baseline method, joint total variation regularization, in terms of correctly material-characterized pixels. While the method is tested only in a two-dimensional setting with two materials and two energies, the approach readily generalizes to three dimensions and more materials. The number of materials just needs to match the number of energies used in imaging.Peer reviewe

AMFORM, a new mass-based model for the calculation of the unit formula of amphiboles from electron microprobe analyses

In this work, we have studied the relationships between mass concentration and unit formula of amphibole using 114 carefully selected high-quality experimental data, obtained by electron microprobe (EMP) + single-crystal X‑ray structure refinement (SREF) ± secondary-ion mass spectrometry (SIMS) analyses, of natural and synthetic Li-free monoclinic species belonging to the Ca and Na-Ca subgroups, and 75 Li-free and Mn-free C2/m end-members including oxo analogs of Ca amphiboles. Theoretical considerations and crystal-chemical driven regression analysis allowed us to obtain several equations that can be used to: (1) calculate from EMP analyses amphibole unit-formulas consistent with SREF±SIMS data, (2) discard unreliable EMP analyses, and (3) estimate WO2– and Fe3+ contents in Li-free C2/m amphiboles with relatively low Cl contents (≤1 wt%). The AMFORM approach mostly relies on the fact that while the cation mass in Cl-poor amphiboles increases with the content of heavy elements, its anion mass maintains a nearly constant value, i.e., 22O + 2(OH,F,O), resulting in a very well-defined polynomial correlation between the molecular mass and the cation mass per gram (R2 = 0.998). The precision of estimating the amphibole formula [e.g., TSi ± 0.02, CAl ± 0.02, A(Ca+Na+K) ± 0.04 apfu] is 2–4 times higher than when using methods published following the last IMA recommended scheme (2012). It is worth noting that most methods using IMA1997 recommendations (e.g., PROBE-AMPH) give errors that are about twice those of IMA2012-based methods. A linear relation between WO2– and the sum of C(Ti, Fe3+) and A(Na+K) contents, useful to estimate the iron oxidation state of highly oxidized amphiboles typical of post-magmatic processes, is also proposed. A step by step procedure (Appendix1 1) and a user-friendly spreadsheet (AMFORM.xlsx, provided as supplementary material1) allowing one to calculate amphibole unit-formulas from EMP analyses are presented. This work opens new perspectives on the unit-formula calculation of other minerals containing OH and structural vacancies (e.g., micas)

Inner product regularized multi-energy X-ray tomography for material decomposition

Multi-energy X-ray tomography is studied for decomposing three materials using three X-ray energies and a classical energy-integrating detector. A novel regularization term comprises inner products between the material distribution functions, penalizing any overlap of different materials. The method is tested on real data measured of a phantom embedded with Na$_2$SeO$_3$, Na$_2$SeO$_4$, and elemental selenium. It is found that the two-dimensional distributions of selenium in different oxidation states can be mapped and distinguished from each other with the new algorithm. The results have applications in material science, chemistry, biology and medicine

Nanoscale electro-structural characterisation of ohmic contacts formed on p-type implanted 4H-SiC

This work reports a nanoscale electro-structural characterisation of Ti/Al ohmic contacts formed on p-type Al-implanted silicon carbide (4H-SiC). The morphological and the electrical properties of the Al-implanted layer, annealed at 1700°C with or without a protective capping layer, and of the ohmic contacts were studied using atomic force microscopy [AFM], transmission line model measurements and local current measurements performed with conductive AFM