ℓ1\ell_1-minimization method for link flow correction

A computational method, based on $\ell_1$-minimization, is proposed for the problem of link flow correction, when the available traffic flow data on many links in a road network are inconsistent with respect to the flow conservation law. Without extra information, the problem is generally ill-posed when a large portion of the link sensors are unhealthy. It is possible, however, to correct the corrupted link flows \textit{accurately} with the proposed method under a recoverability condition if there are only a few bad sensors which are located at certain links. We analytically identify the links that are robust to miscounts and relate them to the geometric structure of the traffic network by introducing the recoverability concept and an algorithm for computing it. The recoverability condition for corrupted links is simply the associated recoverability being greater than 1. In a more realistic setting, besides the unhealthy link sensors, small measurement noises may be present at the other sensors. Under the same recoverability condition, our method guarantees to give an estimated traffic flow fairly close to the ground-truth data and leads to a bound for the correction error. Both synthetic and real-world examples are provided to demonstrate the effectiveness of the proposed method

A tensor approximation method based on ideal minimal residual formulations for the solution of high-dimensional problems

In this paper, we propose a method for the approximation of the solution of high-dimensional weakly coercive problems formulated in tensor spaces using low-rank approximation formats. The method can be seen as a perturbation of a minimal residual method with residual norm corresponding to the error in a specified solution norm. We introduce and analyze an iterative algorithm that is able to provide a controlled approximation of the optimal approximation of the solution in a given low-rank subset, without any a priori information on this solution. We also introduce a weak greedy algorithm which uses this perturbed minimal residual method for the computation of successive greedy corrections in small tensor subsets. We prove its convergence under some conditions on the parameters of the algorithm. The residual norm can be designed such that the resulting low-rank approximations are quasi-optimal with respect to particular norms of interest, thus yielding to goal-oriented order reduction strategies for the approximation of high-dimensional problems. The proposed numerical method is applied to the solution of a stochastic partial differential equation which is discretized using standard Galerkin methods in tensor product spaces

Waveform Relaxation for the Computational Homogenization of Multiscale Magnetoquasistatic Problems

This paper proposes the application of the waveform relaxation method to the homogenization of multiscale magnetoquasistatic problems. In the monolithic heterogeneous multiscale method, the nonlinear macroscale problem is solved using the Newton--Raphson scheme. The resolution of many mesoscale problems per Gauss point allows to compute the homogenized constitutive law and its derivative by finite differences. In the proposed approach, the macroscale problem and the mesoscale problems are weakly coupled and solved separately using the finite element method on time intervals for several waveform relaxation iterations. The exchange of information between both problems is still carried out using the heterogeneous multiscale method. However, the partial derivatives can now be evaluated exactly by solving only one mesoscale problem per Gauss point.Comment: submitted to JC

Some Applications of Coding Theory in Computational Complexity

Error-correcting codes and related combinatorial constructs play an important role in several recent (and old) results in computational complexity theory. In this paper we survey results on locally-testable and locally-decodable error-correcting codes, and their applications to complexity theory and to cryptography. Locally decodable codes are error-correcting codes with sub-linear time error-correcting algorithms. They are related to private information retrieval (a type of cryptographic protocol), and they are used in average-case complexity and to construct ``hard-core predicates'' for one-way permutations. Locally testable codes are error-correcting codes with sub-linear time error-detection algorithms, and they are the combinatorial core of probabilistically checkable proofs

Quantum Error Correction beyond the Bounded Distance Decoding Limit

In this paper, we consider quantum error correction over depolarizing channels with non-binary low-density parity-check codes defined over Galois field of size $2^p$ . The proposed quantum error correcting codes are based on the binary quasi-cyclic CSS (Calderbank, Shor and Steane) codes. The resulting quantum codes outperform the best known quantum codes and surpass the performance limit of the bounded distance decoder. By increasing the size of the underlying Galois field, i.e., $2^p$, the error floors are considerably improved.Comment: To appear in IEEE Transactions on Information Theor

Efficient Quantum Transforms

Quantum mechanics requires the operation of quantum computers to be unitary, and thus makes it important to have general techniques for developing fast quantum algorithms for computing unitary transforms. A quantum routine for computing a generalized Kronecker product is given. Applications include re-development of the networks for computing the Walsh-Hadamard and the quantum Fourier transform. New networks for two wavelet transforms are given. Quantum computation of Fourier transforms for non-Abelian groups is defined. A slightly relaxed definition is shown to simplify the analysis and the networks that computes the transforms. Efficient networks for computing such transforms for a class of metacyclic groups are introduced. A novel network for computing a Fourier transform for a group used in quantum error-correction is also given.Comment: 30 pages, LaTeX2e, 7 figures include

Error Correction for Index Coding With Coded Side Information

Index coding is a source coding problem in which a broadcaster seeks to meet the different demands of several users, each of whom is assumed to have some prior information on the data held by the sender. If the sender knows its clients' requests and their side-information sets, then the number of packet transmissions required to satisfy all users' demands can be greatly reduced if the data is encoded before sending. The collection of side-information indices as well as the indices of the requested data is described as an instance of the index coding with side-information (ICSI) problem. The encoding function is called the index code of the instance, and the number of transmissions employed by the code is referred to as its length. The main ICSI problem is to determine the optimal length of an index code for and instance. As this number is hard to compute, bounds approximating it are sought, as are algorithms to compute efficient index codes. Two interesting generalizations of the problem that have appeared in the literature are the subject of this work. The first of these is the case of index coding with coded side information, in which linear combinations of the source data are both requested by and held as users' side-information. The second is the introduction of error-correction in the problem, in which the broadcast channel is subject to noise. In this paper we characterize the optimal length of a scalar or vector linear index code with coded side information (ICCSI) over a finite field in terms of a generalized min-rank and give bounds on this number based on constructions of random codes for an arbitrary instance. We furthermore consider the length of an optimal error correcting code for an instance of the ICCSI problem and obtain bounds on this number, both for the Hamming metric and for rank-metric errors. We describe decoding algorithms for both categories of errors

Exploring single-sample SNP and INDEL calling with whole-genome de novo assembly

Motivation: Eugene Myers in his string graph paper (Myers, 2005) suggested that in a string graph or equivalently a unitig graph, any path spells a valid assembly. As a string/unitig graph also encodes every valid assembly of reads, such a graph, provided that it can be constructed correctly, is in fact a lossless representation of reads. In principle, every analysis based on whole-genome shotgun sequencing (WGS) data, such as SNP and insertion/deletion (INDEL) calling, can also be achieved with unitigs. Results: To explore the feasibility of using de novo assembly in the context of resequencing, we developed a de novo assembler, fermi, that assembles Illumina short reads into unitigs while preserving most of information of the input reads. SNPs and INDELs can be called by mapping the unitigs against a reference genome. By applying the method on 35-fold human resequencing data, we showed that in comparison to the standard pipeline, our approach yields similar accuracy for SNP calling and better results for INDEL calling. It has higher sensitivity than other de novo assembly based methods for variant calling. Our work suggests that variant calling with de novo assembly be a beneficial complement to the standard variant calling pipeline for whole-genome resequencing. In the methodological aspects, we proposed FMD-index for forward-backward extension of DNA sequences, a fast algorithm for finding all super-maximal exact matches and one-pass construction of unitigs from an FMD-index. Availability: http://github.com/lh3/fermi Contact: [email protected]: Rev2: submitted version with minor improvements; 7 page

A multilevel algorithm for flow observables in gauge theories

We study the possibility of using multilevel algorithms for the computation of correlation functions of gradient flow observables. For each point in the correlation function an approximate flow is defined which depends only on links in a subset of the lattice. Together with a local action this allows for independent updates and consequently a convergence of the Monte Carlo process faster than the inverse square root of the number of measurements. We demonstrate the feasibility of this idea in the correlation functions of the topological charge and the energy density.Comment: Minor modifications to the text. Version accepted to be published in PRD. 18 pages, 5 figure