Eero Vainikko | University of Tartu (original) (raw)

Papers by Eero Vainikko

arXiv (Cornell University), Jul 8, 2015

In this paper we give new results on domain decomposition preconditioners for GM-RES when computi... more In this paper we give new results on domain decomposition preconditioners for GM-RES when computing piecewise-linear finite-element approximations of the Helmholtz equation −∆u − (k 2 + iε)u = f , with absorption parameter ε ∈ R. Multigrid approximations of this equation with ε = 0 are commonly used as preconditioners for the pure Helmholtz case (ε = 0). However a rigorous theory for such (so-called "shifted Laplace") preconditioners, either for the pure Helmholtz equation, or even the absorptive equation (ε = 0), is still missing. We present a new theory for the absorptive equation that provides rates of convergence for (left-or right-) preconditioned GMRES, via estimates of the norm and field of values of the preconditioned matrix. This theory uses a k-and ε-explicit coercivity result for the underlying sesquilinear form and shows, for example, that if |ε| ∼ k 2 , then classical overlapping additive Schwarz will perform optimally for the absorptive problem, provided the subdomain and coarse mesh diameters are carefully chosen. Extensive numerical experiments are given that support the theoretical results. While the theory applies to a certain weighted variant of GMRES, the experiments for both weighted and classical GMRES give comparable results. The theory for the absorptive case gives insight into how its domain decomposition approximations perform as preconditioners for the pure Helmholtz case ε = 0. At the end of the paper we propose a (scalable) multilevel preconditioner for the pure Helmholtz problem that has an empirical computation time complexity of about O(n 4/3) for solving finite element systems of size n = O(k 3), where we have chosen the mesh diameter h ∼ k −3/2 to avoid the pollution effect. Experiments on problems with h ∼ k −1 , i.e. a fixed number of grid points per wavelength, are also given.

Automatic Control 1990, 1991

The ordinary systems and the systems including elements of new kind (to describe these e l ements... more The ordinary systems and the systems including elements of new kind (to describe these e l ements the new mathemati cal methods have been developed) was studied. In this paper we state some ne\' results about periodic oscillations: existence of oscillations, number of distinct conditions, sta bility conditions, new numerical procedures for control probl e ms, new kinds of nonlinearities.

SN computer science, Aug 17, 2022

Elsevier eBooks, 1991

In this paper, we propose a secret sharing based secure multiparty computation (SMC) protocol for... more In this paper, we propose a secret sharing based secure multiparty computation (SMC) protocol for computing the minimum spanning trees in dense graphs. The challenges in the design of the protocol arise from the necessity to access memory according to private addresses, as well as from the need to reduce the round complexity. In our implementation, we use the single-instruction-multiple-data (SIMD) operations to reduce the round complexity of the SMC protocol; the SIMD instructions reduce the latency of the network among the three servers of the SMC platform. We present a state-of-the-art parallel privacy-preserving minimum spanning tree algorithm which is based on Prim's algorithm for finding a minimum spanning tree (MST) in dense graphs. Performing permutation of the graph with sharemind to be able to perform the calculation of the MST on the shuffled graph outside the environment. We compare our protocol to the state of the art and find that its performance exceeds the existing protocols when being applied to dense graphs.

Mathematics of Computation, Feb 8, 2017

European Conference on Parallel Processing, Aug 27, 2002

We describe the construction of parallel iterative solvers for finite element approximations of t... more We describe the construction of parallel iterative solvers for finite element approximations of the Navier-Stokes equations on unstructured grids using domain decomposition methods. The iterative method used is FGMRES, preconditioned by a parallel adaptation of a recent block preconditioner proposed by Kay, Loghin and Wathen. The parallelisation is achieved by adapting the technology of our domain decomposition solver DOUG (previously used for scalar problems) to block-systems. An application of the resultant linear solver to the stability assessment of flows is briefly indicated.

The aim of Elliptic Curve Cryptosystems (ECC) is to achieve the same security level as RSA but wi... more The aim of Elliptic Curve Cryptosystems (ECC) is to achieve the same security level as RSA but with shorter key size. The basic operation in the ECC is scalar multiplication which is an expensive operation. In this paper, we focus on optimizing ECC scalar multiplication based on Non-Adjacent Form (NAF). A new algorithm is introduced that combines an Add-Subtract Scalar Multiplication Algorithm with NAF representation to accelerate the performance of the ECC calculation. Parallelizing the new algorithm shows an efficient method to calculate ECC. The proposed method has sped up the calculation up to 60% compared with the standard method.

Lecture Notes in Computer Science, 2002

ABSTRACT We describe the construction of parallel iterative solvers for finite element approximat... more ABSTRACT We describe the construction of parallel iterative solvers for finite element approximations of the Navier-Stokes equations on unstructured grids using domain decomposition methods. The iterative method used is FGMRES, preconditioned by a parallel adaptation of a recent block preconditioner proposed by Kay, Loghin and Wathen. The parallelisation is achieved by adapting the technology of our domain decomposition solver DOUG (previously used for scalar problems) to block-systems. An application of the resultant linear solver to the stability assessment of flows is briefly indicated.

Proceedings of the 8th International Conference on Information Systems Security and Privacy

Reducing the round complexity in secure multiparty computation (SMC) protocols is a worthy goal d... more Reducing the round complexity in secure multiparty computation (SMC) protocols is a worthy goal due to the latency of the network. The SIMD approach is considered an efficient strategy to reduce the round complexity of an SMC protocol. This paper studies the secure multiparty computation (SMC) protocol for the shortest path problem in sparse and dense graphs, building upon the breadth-first search algorithm. The sensitivity of operations in processing the algorithms led us to produce two different structural algorithms for computing the shortest path. We present state-of-the-art parallel privacy-preserving shortest path algorithms for weighted and unweighted graphs based on the breadth-first search. We have implemented the proposed algorithms on top of the Sharemind SMC protocol set and tested it for different graphs, dense and sparse, represented as the adjacency matrix.

In this talk we discuss the use of domain decomposition parallel iterative solvers for highly het... more In this talk we discuss the use of domain decomposition parallel iterative solvers for highly heterogeneous problems of flow in porous media, in both the deterministic and (Monte-Carlo simulated) stochastic cases. We are particularly interested in the case of highly unstructured coefficient variation where standard periodic or stochastic homogenisation theory is not applicable, and where there is no a priori scale separation. We will restrict attention to the important model elliptic problem

Proceedings of the 6th International Conference on Information Systems Security and Privacy, 2020

In this paper, we study the parallel implementations of elliptic curve scalar multiplication over... more In this paper, we study the parallel implementations of elliptic curve scalar multiplication over prime fields using signed binary representations. Our implementation speeds up the calculation of scalar multiplication in comparison with the standard case. We introduce parallel algorithms for computing elliptic curve scalar multiplication based on representing the scalar by the Complementary Recoding Technique (CRT) and the Direct Recording Method (DRM). Both implementations of the proposed algorithms show speed-ups reaching up to 60% in comparison with execution time for sequential cases of the algorithms. We find that ECC-DRM is faster than ECC-CRT in both parallel and sequential counterparts.

Modern Solvers for Helmholtz Problems, 2017

In this paper we present an overview of recent progress on the development and analysis of domain... more In this paper we present an overview of recent progress on the development and analysis of domain decomposition preconditioners for discretised Helmholtz problems, where the preconditioner is constructed from the corresponding problem with added absorption. Our preconditioners incorporate local subproblems that can have various boundary conditions, and include the possibility of a global coarse mesh. While the rigorous analysis describes preconditioners for the Helmholtz problem with added absorption, this theory also informs the development of efficient multilevel solvers for the "pure" Helmholtz problem without absorption. For this case, 2D experiments for problems containing up to about 50 wavelengths are presented. The experiments show iteration counts of order about O(n 0.2) and times (on a serial machine) of order about O(n α), with α ∈ [1.3, 1.4] for solving systems of dimension n. This holds both in the pollution-free case corresponding to meshes with grid size O(k −3/2) (as the wavenumber k increases), and also for discretisations with a fixed number of grid points per wavelength, commonly used in applications. Parallelisation of the algorithms is also briefly discussed.

2016 12th IEEE International Conference on Control and Automation (ICCA), 2016

The growing number of users in microblogging sites such as Twitter has created the problem of sea... more The growing number of users in microblogging sites such as Twitter has created the problem of searching useful followees among millions of users in a reasonable time. One way to address this problem is using a recommender system, which is aimed at providing a list of useful followees in a reasonable time. Although Twitter provides a functionality what it calls `Who to Follow', neither is it configurable by the user, nor its accuracy is of the highest level. Several approaches have been proposed in literature to recommend followees in Twitter. However, their accuracy and efficiency have been limited, given several Twitter-specific and natural language processing challenges. In this paper, we propose a semantic followee recommender in Twitter based on Topicmodel and Kalman filter, leveraging publicly available knowledge-bases. In particular, we aim to address the (1) word sense disambiguation problem in tweets using Wikipedia and WordNet, (2) classify users in multiple-labels using Topicmodel and a modified Normalized Google Distance, and (3) remove noise and predict future multi-label classes using the results obtained in step (2) above using Kalman filter. As an application, we conduct a case study to evaluate the efficacy of our model to recommend followees in six predefined classes: politics, sports, business, entertainment, science, and travel. Preliminary analysis show that the model can effectively recommend useful followees in Twitter.

2021 29th Euromicro International Conference on Parallel, Distributed and Network-Based Processing (PDP), 2021

The radius-stepping algorithm is an efficient, parallelizable algorithm for finding the shortest ... more The radius-stepping algorithm is an efficient, parallelizable algorithm for finding the shortest paths in graphs. It solved the problem in triangle\triangletriangle-Stepping algorithm, which has no known theoretical bounds for general graphs. In this paper, we describe a parallel privacy-preserving method for finding Single-Source Shortest Paths (SSSP). Our optimized method is based on the Radius-Stepping algorithm. The method is implemented on top of the Secure Multiparty Computation (SMC) Sharemind platform. We have re-shaped the radius-stepping algorithm to work on vectors representing the graph in a SIMD manner, in order to enable a fast execution using the secret-sharing based SMC protocol set of Sharemind. The results of the real implementation show an efficient method that reduced the execution time hundreds of times in comparison with a standard case of the privacy-preserving radius-stepping and triangle\triangletriangle-Stepping algorithms.

One of the problems that are encountered in recommender systems applications is the high sparsity... more One of the problems that are encountered in recommender systems applications is the high sparsity of the available data. In this paper we investigate the effect of the sparsity of datasets to the performance of a parallel implementation of the Collaborative Filtering Slope One algorithm. To represent the sparse data the Compressed Sparse Row (CSR) format is used and the implementation’s performance is evaluated on a Graphics Processing Unit using the MovieLens and artificially created datasets.

2018 10th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), 2018

The usage of mobile phones has become an important activity in our lives. The passive mobile posi... more The usage of mobile phones has become an important activity in our lives. The passive mobile positioning of mobiles provides large-scale data about human mobility. Hence, in this paper, we are presenting a technique based on continuous time switching Kalman filter to efficiently detect stop and move episodes. The technique has practical and theoretical advantages as the model is more closely related to measure human mobility characteristics and less sensitive to variations in radio network operations. The technique was tested on real radio network data and the results indicated significant improvement with respect to the model performance and the literature.

Proceedings of the Fourth ACM SIGSPATIAL International Workshop on Mobile Geographic Information Systems, 2015

The detection of stay-jump-and-moving movement episodes using only cellular data is a big challen... more The detection of stay-jump-and-moving movement episodes using only cellular data is a big challenge due to the nature of the data. In this article, we propose a method to automatically detect the movement episodes (stay-jump-and-moving) from sparsely sampled spatio-temporal data, in our case Call Detail Records (CDRs), using switching Kalman filter with a new integrated movement model and cellular coverage optimization approach. The algorithm is capable of estimating the movement episodes and classifying the trajectory sequences associated to a stay, a jump or a moving action. The result of this approach can be beneficial for applications using cellular data related to traffic management, mobility profiling, and semantic enrichment.

Cryptography, 2021

In this paper, we propose and present secure multiparty computation (SMC) protocols for single-so... more In this paper, we propose and present secure multiparty computation (SMC) protocols for single-source shortest distance (SSSD) and all-pairs shortest distance (APSD) in sparse and dense graphs. Our protocols follow the structure of classical algorithms—Bellman–Ford and Dijkstra for SSSD; Johnson, Floyd–Warshall, and transitive closure for APSD. As the computational platforms offered by SMC protocol sets have performance profiles that differ from typical processors, we had to perform extensive changes to the structure (including their control flow and memory accesses) and the details of these algorithms in order to obtain good performance. We implemented our protocols on top of the secret sharing based protocol set offered by the Sharemind SMC platform, using single-instruction-multiple-data (SIMD) operations as much as possible to reduce the round complexity. We benchmarked our protocols under several different parameters for network performance and compared our performance figures ...