Approximation Factor Research Articles

Target PCA: Transfer learning large dimensional panel data

This paper develops a novel method to estimate a latent factor model for a large target panel with missing observations by optimally using the information from auxiliary panel data sets. We refer to our estimator as target-PCA. Transfer learning from auxiliary panel data allows us to deal with a large fraction of missing observations and weak signals in the target panel. We show that our estimator is more efficient and can consistently estimate weak factors, which are not identifiable with conventional methods. We provide the asymptotic inferential theory for target-PCA under very general assumptions on the approximate factor model and missing patterns. In an empirical study of imputing data in a mixed-frequency macroeconomic panel, we demonstrate that target-PCA significantly outperforms all benchmark methods.

Boundary corrections on multi-dimensional PDEs

Two new boundary correction techniques are proposed in order to mitigate the order reduction phenomenon associated with the numerical solution of initial boundary value problems for parabolic partial differential equations in arbitrary spatial dimensions with time-dependent Dirichlet boundary conditions. The new techniques are based on the idea of discretizing the PDE problem at the boundary points as similar as possible to that of the interior points of the domain. These new techniques are considered for the time integration with W-methods based on approximate matrix factorization. By suitably modifying the internal stages of the methods on the boundary points, it is illustrated by numerical testing with time-dependent boundary conditions that the new boundary correction techniques are able to keep the same accuracy and order of convergence that the method reaches in the case of homogeneous boundary conditions.

A Robust Adaptive Mesh Generation Algorithm: A Solution for Simulating 2D Crack Growth Problems.

This paper introduces a robust algorithm that efficiently generates high-quality unstructured triangular meshes to model complex two-dimensional crack growth problems within the framework of linear elastic fracture mechanics (LEFM). The proposed Visual Fortran code aims to address key challenges in mesh generation including geometric complexity, required simulation accuracy, and computational resource constraints. The algorithm incorporates adaptive refinement and updates to the mesh structure near the crack tip, resulting in the formation of rosette elements that provide accurate approximations of stress intensity factors (SIFs). By utilizing the maximum circumferential stress theory, the algorithm predicts the new crack path based on these SIFs. Throughout the simulation of crack propagation, a node splitting approach was employed to represent the progression of the crack, while the crack growth path is determined by successive linear extensions for each crack growth increment. To compute stress intensity factors (SIFs) for each increment of crack extension, a displacement extrapolation method was used. The experimental and numerical results demonstrated the algorithm's effectiveness in accurately predicting crack growth and facilitating reliable stress analysis for complex crack growth problems in two dimensions. The obtained results for the SIF were found to be consistent with other analytical solutions for standard geometries.

Greedy versus optimal analysis of bounded size dictionary compression and on-the-fly distributed computing

Scalability and robustness are not an issue when compression is applied for massive data storage, in the context of distributed computing. Speeding up on-the-fly compression for data transmission is more controversial. In such case, a compression technique merging together an adaptive and a non-adaptive approach has to be considered. A practical implementation of LZW (Lempel, Ziv and Welch) compression, called LZC (C indicates the Unix command ’compress’), has this characteristic. The non-adaptive phases work with bounded size prefix dictionaries built by LZW factorizations during the adaptive ones. Both phases employ the greedy approach. We show a worst case analysis of the greedy approach with respect to the optimal solution decodable by the LZC decompressor and give worst case examples for which the solution cost produced by the distributed implementations reaches the Θ(d) theoretical upper bound to the optimal cost approximation factor, where d is the dictionary size. We prove that for the worst case examples of the totally adaptive version of LZW compression the approximation factor is Θ(d). Such analysis suggests that parallelization of on the fly compression is not suitable for highly disseminated data since the non-adaptive phases are too far from optimal. In order to improve the compression effectiveness, we suggest to apply LZMW (Lempel, Ziv, Miller and Wegman) factorization to LZC compression (LZCMW) during the adaptive phases. Such factorization was originally thought with a dictionary bounded by the least recently used strategy. We introduce LZCMW in order to have non-adaptive phases.

Better-than-43\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{4}{3}$$\\end{document}-approximations for leaf-to-leaf tree and connectivity augmentation

The Connectivity Augmentation Problem (CAP) together with a well-known special case thereof known as the Tree Augmentation Problem (TAP) are among the most basic Network Design problems. There has been a surge of interest recently to find approximation algorithms with guarantees below 2 for both TAP and CAP, culminating in the currently best approximation factor for both problems of 1.393 through quite sophisticated techniques. We present a new and arguably simple matching-based method for the well-known special case of leaf-to-leaf instances. Combining our work with prior techniques, we readily obtain a (4/3+ε)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(4/3+\\varepsilon )$$\\end{document}-approximation for Leaf-to-Leaf CAP by returning the better of our solution and one of an existing method. Prior to our work, a 4/3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4/3$$\\end{document}-guarantee was only known for Leaf-to-Leaf TAP instances on trees of height 2. Moreover, when combining our technique with a recently introduced stack analysis approach, which is part of the above-mentioned 1.393-approximation, we can further improve the approximation factor to 1.29, obtaining for the first time a factor below 43\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{4}{3}$$\\end{document} for a nontrivial class of TAP/CAP instances.

Approximating Nash Social Welfare under Submodular Valuations through (Un)Matchings

We study the problem of approximating maximum Nash social welfare (NSW) when allocating m indivisible items among n asymmetric agents with submodular valuations. The NSW is a well-established notion of fairness and efficiency, defined as the weighted geometric mean of agents’ valuations. For special cases of the problem with symmetric agents and additive(-like) valuation functions, approximation algorithms have been designed using approaches customized for these specific settings, and they fail to extend to more general settings. Hence, no approximation algorithm with a factor independent of m was known either for asymmetric agents with additive valuations or for symmetric agents beyond additive(-like) valuations before this work. In this article, we extend our understanding of the NSW problem to far more general settings. Our main contribution is two approximation algorithms for asymmetric agents with additive and submodular valuations. Both algorithms are simple to understand and involve non-trivial modifications of a greedy repeated matchings approach. Allocations of high-valued items are done separately by un-matching certain items and re-matching them by different processes in both algorithms. We show that these approaches achieve approximation factors of O ( n ) and O ( n log n ) for additive and submodular cases, independent of the number of items. For additive valuations, our algorithm outputs an allocation that also achieves the fairness property of envy-free up to one item ( EF1 ). Furthermore, we show that the NSW problem under submodular valuations is strictly harder than all currently known settings with an \(\frac{\mathrm{e}}{\mathrm{e}-1}\) factor of the hardness of approximation, even for constantly many agents. For this case, we provide a different approximation algorithm that achieves a factor of \(\frac{\mathrm{e}}{\mathrm{e}-1}\) , hence resolving it completely.

Realization of Unity Power Factor Wireless Power Transfer System under Subnormal Operation Conditions

The power factor of wireless power transfer system, determined by its compensation network part, is easily affected by parameter detuning, coil misalignment, and load variation. In this paper, a mathematical model for the compensation network part is established. Theoretical analysis shows that the inverter part can be considered as a negative resistor by deducing the inherent static-state frequency solution of the compensation network part. Therefore, the unity power factor wireless power transfer system can be maintained under any possible operation conditions by tracking the inherent static-state frequency solution. More importantly, no digital controller or parameter identification or information interactions between the primary and secondary coils are needed during the tracking process. Compared with previous unity power factor realization methods, the proposed tracking strategy has the advantages of fewer sampling variables, a faster response time, and a simpler regulation process. Finally, an experimental platform is built to test the practical performance of the proposed tracking strategy under many subnormal operation conditions. Our experimental results show that approximate unity power factor can be realized at 10–15 cm coil misalignment distance and 30–90 Ω load variation range.

Sparse Approximate Multifrontal Factorization with Composite Compression Methods

This article presents a fast and approximate multifrontal solver for large sparse linear systems. In a recent work by Liu et al., we showed the efficiency of a multifrontal solver leveraging the butterfly algorithm and its hierarchical matrix extension, HODBF (hierarchical off-diagonal butterfly) compression to compress large frontal matrices. The resulting multifrontal solver can attain quasi-linear computation and memory complexity when applied to sparse linear systems arising from spatial discretization of high-frequency wave equations. To further reduce the overall number of operations and especially the factorization memory usage to scale to larger problem sizes, in this article we develop a composite multifrontal solver that employs the HODBF format for large-sized fronts, a reduced-memory version of the nonhierarchical block low-rank format for medium-sized fronts, and a lossy compression format for small-sized fronts. This allows us to solve sparse linear systems of dimension up to 2.7 × larger than before and leads to a memory consumption that is reduced by 70% while ensuring the same execution time. The code is made publicly available in GitHub.

Flexible scheduling of transactional memory on trees

We study the efficiency of executing transactions in a distributed transactional memory system. The system is modeled as a static network with the topology of a tree. Contrary to previous approaches, we allow the flexibility for both transactions and their requested objects to move simultaneously among the nodes in the tree. Given a batch of transactions and shared objects, the goal is to produce a schedule of executing the transactions that minimizes the cost of moving the transactions and the objects in the tree. We consider both techniques for accessing a remote object with respect to a transaction movement. In the first technique, instead of moving, transactions send control messages to remote nodes where the requested objects are gathered. In the second technique, the transactions migrate to the remote nodes where the objects are gathered to access them. When all the transactions use a single object, we give an offline algorithm that produces optimal schedules for both techniques. For the general case of multiple objects per transaction, in the first technique, we obtain a schedule with a constant-factor approximation of optimal. In the second technique, with transactions migrating, we give a k factor approximation where k is the maximum number of objects per transaction.

Adaptive robust large volatility matrix estimation based on high-frequency financial data

Several novel statistical methods have been developed to estimate large integrated volatility matrices based on high-frequency financial data. To investigate their asymptotic behaviors, they require a sub-Gaussian or finite high-order moment assumption for observed log-returns, which cannot account for the heavy-tail phenomenon of stock-returns. Recently, a robust estimator was developed to handle heavy-tailed distributions with some bounded fourth-moment assumption. However, we often observe that log-returns have heavier tail distribution than the finite fourth-moment and that the degrees of heaviness of tails are heterogeneous across asset and over time. In this paper, to deal with the heterogeneous heavy-tailed distributions, we develop an adaptive robust integrated volatility estimator that employs pre-averaging and truncation schemes based on jump-diffusion processes. We call this an adaptive robust pre-averaging realized volatility (ARP) estimator. We show that the ARP estimator has a sub-Weibull tail concentration with only finite 2α-th moments for any α>1. In addition, we establish matching upper and lower bounds to show that the ARP estimation procedure is optimal. To estimate large integrated volatility matrices using the approximate factor model, the ARP estimator is further regularized using the principal orthogonal complement thresholding (POET) method. The numerical study is conducted to check the finite sample performance of the ARP estimator.

On a fully-implicit VMS-stabilized FE formulation for low Mach number compressible resistive MHD with application to MCF

This study presents the development and evaluation of a fully-implicit variational multiscale (VMS) stabilized unstructured finite element (FE) formulation for compressible magnetohydrodynamics (MHD) model, at low Mach number regime. The model describes the dynamics of a compressible conducting fluid in the low Mach number limit in the presence of electromagnetic fields and can be used to study aspects of astrophysical phenomena, important science and technology applications, and basic plasma physics phenomena. The specific applications that motivate this study are macroscopic simulations of the longer time-scale stability and disruptions of magnetic confinement fusion (MCF) devices, specifically the ITER tokamak. The discussion considers the development of the VMS FE representation, the structure of the stabilizing terms that deal with significant convective flows, the stabilization of the nearly incompressible response of the fluid flow, and the stabilization of the constraint that enforces the solenoidal involution on the magnetic field. The nonlinear discretized system is solved with scalable preconditioned Newton–Krylov iterative methods, which employs a multiphysics block preconditioning method based on approximate block factorizations and Schur complements. The study presents an evaluation of the VMS method on a 2D cartesian tearing mode instability, and illustrates the scalability of the solvers on MCF relevant problems. A set of results are also presented for longer time-scale stability and disruptions for the ITER tokamak. These include a vertical displacement event (VDE), and a (1,1) internal kink mode. The formulation is demonstrated to be scalable and also reasonably robust with respect to the Lundquist number scaling.

In Congestion Games, Taxes Achieve Optimal Approximation

The Power of Traffic Tolls The paper “In Congestion Games, Taxes Achieve Optimal Approximation” by Paccagnan and Gairing investigates the power and limitations of taxes as interventions in congestion games. Perhaps surprisingly, they find that efficiently computed taxes can achieve the same performance as the best polynomial time algorithm, even when the algorithm has full control over the agents’ actions. The authors establish three main results to support this claim. First, they prove that minimizing congestion is NP-hard to approximate within a given factor based on the latency functions. Second, they demonstrate how convex optimization tools can be used to design optimal taxes. Third, they provide a polynomial time algorithm with an approximation factor matching the hardness barrier. The upshot of their contribution can be summarized as follows: Judiciously designed taxes achieve optimal approximation, and no other tractable intervention can improve upon this result.

The One-Way Communication Complexity of Submodular Maximization with Applications to Streaming and Robustness

We consider the classical problem of maximizing a monotone submodular function subject to a cardinality constraint, which, due to its numerous applications, has recently been studied in various computational models. We consider a clean multiplayer model that lies between the offline and streaming model, and study it under the aspect of one-way communication complexity. Our model captures the streaming setting (by considering a large number of players), and, in addition, two-player approximation results for it translate into the robust setting. We present tight one-way communication complexity results for our model, which, due to the connections mentioned previously, have multiple implications in the data stream and robust setting. Even for just two players, a prior information-theoretic hardness result implies that no approximation factor above 1/2 can be achieved in our model, if only queries to feasible sets (i.e., sets respecting the cardinality constraint) are allowed. We show that the possibility of querying infeasible sets can actually be exploited to beat this bound, by presenting a tight 2/3-approximation taking exponential time, and an efficient 0.514-approximation. To the best of our knowledge, this is the first example where querying a submodular function on infeasible sets leads to provably better results. Through the link to the (non-streaming) robust setting mentioned previously, both of these algorithms improve on the current state of the art for robust submodular maximization, showing that approximation factors beyond 1/2 are possible. Moreover, exploiting the link of our model to streaming, we settle the approximability for streaming algorithms by presenting a tight 1/2+ɛ hardness result, based on the construction of a new family of coverage functions. This improves on a prior 0.586 hardness and matches, up to an arbitrarily small margin, the best-known approximation algorithm.

Anisotropic Nature of Lightweight Wooden Metamaterials with Mechanical/Thermomechanical Multistability

AbstractNanoengineered wood provides a renewable structural material with 3D micro and nanoarchitectures, exhibiting many beneficial characteristics such as being lightweight in nature, mechanically strong, eco‐friendly, thermally insulation, and low carbon footprint. Most nanocellulose aerogels lack sufficient mechanical strength, while nanowood involves a trade‐off between mechanical strength and insulation performance. Here, a nanowood‐derived product with mechanical/thermomechanical multistability called a wooden metamaterial, which is ultrastiff yet lightweight, is designed and synthesized. The self‐healing behaviors of cellulose nanofibrils originally present in the cell walls and their combination with microscale mechanical constraints are utilized to form directional porous frameworks (porosity ≥98%) and encapsulated empty fiber lumen in predesigned macroscopic architectures. The wooden metamaterials are lightweight, showing ultrahigh specific strength (207.7 MPa cm3 g−1), and ultrahigh anisotropy with an approximate factor of 4. Wooden metamaterials have overcome the mechanical/thermomechanical deficiencies of existing building materials and advanced aerospace thermal insulators, and have great potential for revolutionizing architecture and manufacturing industries, particularly as a lightweight, eco‐friendly, scalable, energy‐efficient, and cost‐effective.

Laguerre-Based Low-Rank Balanced Truncation of Discrete-Time Systems

In this brief, a new model order reduction method based on low-rank Gramian approximation is proposed for discrete-time systems. The main task is to use the Laguerre functions expansions to calculate the approximate low-rank decomposition factors of the controllability and observability Gramians, which are calculated by the recurrence formulas. Then, the reduced-order systems are obtained by the low-rank square root method which approximate the original system well. In addition, due to the disadvantages of the above method that it may produce an unstable system even if the original system is stable, a modified reduction procedure is proposed, which can preserve stability in some cases. It combines the dominant subspace projection method to use the sum of both subspaces for a state space projection. The effectiveness of the proposed methods are illustrated by two numerical examples.

Faster TKD: Towards Lightweight Decomposition for Large-Scale Tensors With Randomized Block Sampling

The Tucker Decomposition (TKD) is able to provide the low-dimensional and informative representations of real-world large-scale tensorial data, which are necessary to extract potential features and enhance the original data. However, computing such decomposition directly for a dense tensor is usually computationally elusive, due to the repetitive operations of computing large-scale tensor-matrix product. Instead of direct decomposition, this paper proposes an efficient algorithm for seeking the Faster TKD of the large-scale tensor, which is a lightweight decomposition approach based on the technique of randomized sampling. The proposed algorithm first converts the original large-scale tensor into a small-scale subtensor via full-mode sampling operation, and then the core tensor of TKD can be computed directly based on the subtensor with low complexity. Finally, an approximate TKD of the original large-scale tensor can be obtained after sequentially computing approximate full-mode factor matrices. A theoretical error analysis is provided to show that the approximation error approximates zero with high probability, and the proposed algorithm is verified based on real tensorial data of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{23821.24}~GB$</tex-math></inline-formula> .

Distributed domination on sparse graph classes

We show that the dominating set problem admits a constant factor approximation in a constant number of rounds in the LOCAL model of distributed computing on graph classes with bounded expansion. This generalizes a result of Czygrinow et al. for graphs with excluded topological minors to very general classes of uniformly sparse graphs. We demonstrate how our general algorithm can be modified and fine-tuned to compute an (11+ɛ)-approximation (for any ɛ>0) of a minimum dominating set on planar graphs. This improves on the previously best known approximation factor of 52 on planar graphs, which was achieved by an elegant and simple algorithm of Lenzen et al.

Bayesian Network Design for Crude Palm Oil (CPO) Price Prediction Driven by Fluctuation Patterns and Trends

Crude Palm Oil (CPO) becomes the alternative to petroleum because it can be refined into different products, such as biodiesel. The price of crude palm oil can be considered the future direction of the world economy. CPO prices are based on market demand and the oil supply from the CPO producer, where they fluctuate and have a huge influence on economy. The price pattern of CPO requires a variety of factors to predict accurately. It needs a lot of data analytics to predict and respond in quick time to the highly variable CPO market. In this work, a dynamic probability model is proposed to predict CPO prices using the Bayesian rule. The rule is formed to be effective in responding to changeable market demands and supplies. It formulates the approximation factors for indicating the expected price of the CPO for 12 months. The dynamic training data has been modelled to fit the Bayesian rule. The process of Prediction and hypothesizing price fluctuations is identified according to several level of Bayesian rule. It is then used to generate a measure of deviation between the actual and probable prices. Our Bayesian-based prediction pricing model is able to predict the price pattern that are comparable to existing benchmark data and shows a lower standard error of regression. The prediction approach can help the traders have a better analysis of price fluctuations in CPO demand and supply. Using Bayesian probability not only improves prediction rules, but it can also forecast CPO trades in high fluctuation situations.

The Traveling Salesman Problem with Drones: The Benefits of Retraversing the Arcs

In the traveling salesman problem with drones (TSP-mD), a truck and multiple drones cooperate to serve customers in the minimum amount of time. The drones are launched and retrieved by the truck at customer locations, and each of their flights must not consume more energy than allowed by their batteries. Most problem settings in the literature restrict the feasible truck routes to cycles (i.e., closed paths), which never visit a node more than once. Revisiting a node, however, may lower the time required to serve all the customers. Additionally, we observe that optimal solutions for the TSP-mD may retraverse arcs (i.e., optimal truck routes may contain the same arcs multiple times). We refer to such solutions as arc retraversing and include them in our solution space by modeling the truck route as a closed walk. We describe Euclidean instances where all the optimal solutions are arc retraversing. The necessity of arc retraversals does not seem to have been investigated in previous studies, and those that allow node revisits seem to assume that there always exists an optimal solution without arc retraversals. We prove that under certain conditions, which are commonly met in the literature, this assumption is correct. When these conditions are not met, however, excluding arc-retraversing solutions might result in an increase of the optimal value; we identify cases where a priori and a posteriori upper bounds hold on such increase. Finally, we prove that there is no polynomial-time heuristic that can approximate the metric TSP-mD within a constant factor, unless P = NP. We identify a (nonconstant) approximation factor explicitly when the truck can visit all the nodes.

Approximating the Bundled Crossing Number

Bundling crossings is a strategy which can enhance the readability of graph drawings. In this paper we consider good drawings, i.e., we require that any two edges have at most one common point which can be a common vertex or a crossing. Our main result is that there is a polynomial-time algorithm to compute an 8-approximation of the bundled crossing number of a good drawing with no toothed hole. In general the number of toothed holes has to be added to the 8-approximation. In the special case of circular drawings the approximation factor is 8, this improves upon the 10-approximation of Fink et al.[Fink et al., LATIN 2016]. Our approach also works with the same approximation factor for families of pseudosegments, i.e., curves intersecting at most once. We also show how to compute a $\frac{9}{2}$-approximation when the intersection graph of the pseudosegments is bipartite and has no toothed hole.