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Quarteroni Alfio - One of the best experts on this subject based on the ideXlab platform.
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MATHICSE Technical Report : Multi space reduced basis preconditioners for parametrized Stokes equations
Écublens MATHICSE, 2019Co-Authors: Dal Santo Niccolò, Deparis Simone, Manzoni Andrea, Quarteroni AlfioAbstract:In this work we introduce a two-level preconditioner for the efficient solution of large scale saddlepoint linear systems arising from the finite element (FE) discretization of parametrized Stokes equations.The proposed preconditioner extends the Multi Space Reduced Basis (MSRB) preconditioning methodproposed in [12], and relies on the combination of an approximated block (fine grid) preconditioner witha reduced basis solver, which plays the role of coarse component. A sequence of RB spaces, constructedeither with an enriched velocity formulation or a Petrov-Galerkin projection, is built. As a matter offact, each RB coarse component is tailored to perform a single Iteration of the iterative method at hand.The exible GMRES (FGMRES) algorithm is employed to solve the resulting preconditioned systemand targets small tolerances with a very small Iteration Count and in a very short time. Numerical testcases dealing with Stokes flows in three dimensional parameter-ependent geometries are consideredto assess the numerical performance of the proposed technique in different large scale computationalsettings. A detailed comparison with both the current state of the art of i) standard RB methodsand ii) preconditioning techniques for Stokes equations highlights the better efficiency of the proposedmethodology
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MATHICSE Technical Report : Multi space reduced basis preconditioners for large-scale parametrized PDEs
Écublens MATHICSE, 2019Co-Authors: Dal Santo Niccolò, Deparis Simone, Manzoni Andrea, Quarteroni AlfioAbstract:In this work we introduce a new two-level preconditioner for the efficient solution of large scale linear systems arising from the discretization of parametrized PDEs. The proposed preconditioner combines in a multiplicative way a reduced basis solver, which plays the role of coarse component, and a "traditional" fine grid preconditioner, such as one-level Additive Schwarz, block Gauss-Seidel or block Jacobi preconditioners. The coarse component is built up on a new Multi Space Reduced Basis (MSRB) method that we introduce for the first time in this paper, where are reduced basis space is built through the proper orthogonal decomposition (POD) algorithm at each step of the iterative method at hand, like the flexible GMRES method. MSRB strategy consists in building reduced basis (RB) spaces that are well-suited to perform a single Iteration, by addressing the error components which have not been treated yet. The Krylov Iterations employed to solve the resulting preconditioned system targets small tolerances with a very small Iteration Count and in a very short time, showing good optimality and scalability properties. Simulations are carried out to evaluate the performance of the proposed preconditioner indifferent large scale computational settings related to parametrized advection diffusion equations and compared with the current state of the art algebraic multigrid preconditioners
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Multi space reduced basis preconditioners for large-scale parametrized PDEs
'Society for Industrial & Applied Mathematics (SIAM)', 2018Co-Authors: Dal Santo Niccolò, Deparis Simone, Manzoni Andrea, Quarteroni AlfioAbstract:In this work we introduce a new two-level preconditioner for the efficient solution of large-scale linear systems arising from the discretization of parametrized PDEs. The proposed preconditioner combines in a multiplicative way a reduced basis solver, which plays the role of coarse component, and a “traditional” fine-grid preconditioner, such as one-level additive Schwarz, block Gauss--Seidel, or block Jacobi preconditioners. The coarse component is built upon a new multi space reduced basis (MSRB) method that we introduce for the first time in this paper, where a reduced basis space is built through the proper orthogonal decomposition algorithm at each step of the iterative method at hand, like the flexible GMRES method. MSRB strategy consists in building reduced basis spaces that are well suited to perform a single Iteration, by addressing the error components which have not been treated yet. The Krylov Iterations employed to solve the resulting preconditioned system target small tolerances with a very small Iteration Count and in a very short time, showing good optimality and scalability properties. Simulations are carried out to evaluate the performance of the proposed preconditioner in different large-scale computational settings related to parametrized advection diffusion equations and compared with the current state-of-the-art algebraic multigrid preconditioners
A. Visconti - One of the best experts on this subject based on the ideXlab platform.
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Examining PBKDF2 security margin—Case study of LUKS
'Elsevier BV', 2019Co-Authors: A. Visconti, Milan Brož, O. Mosnáček, Vashek MatyášAbstract:Passwords are widely used to protect our sensitive information or to gain access to specific resources. They should be changed frequently and be strong enough to prevent well-known attacks. Unfortunately, user-chosen passwords are usually short and lack sufficient entropy. A possible solution to these problems is to adopt a Key Derivation Function (KDF) that allows legitimate users to spend a moderate amount of time on key derivation, while imposing CPU/memory-intensive operations on the attacker side. In this paper, we focus on long-term passwords secured by the Password-Based Key Derivation Function 2 (PBKDF2) and present the case study of Linux Unified Key Setup (LUKS), a disk-encryption specification commonly implemented in Linux based operating systems. In particular, we describe how LUKS protects long-term keys by means of Iteration Counts defined at runtime, and analyze how external factors may affect the Iteration Counts computation. In doing so, we provide means of evaluating the Iteration Count values defined at run-time and experimentally show to what level PBKDF2 is still capable of providing sufficient security margin for a LUKS implementation
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Examining PBKDF2 security margin—Case study of LUKS
'Elsevier BV', 2019Co-Authors: A. Visconti, O. Mosná, M. Brož, V. MatyáAbstract:Passwords are widely used to protect our sensitive information or to gain access to specific resources. They should be changed frequently and be strong enough to prevent well-known attacks. Unfortunately, user-chosen passwords are usually short and lack sufficient entropy. A possible solution to these problems is to adopt a Key Derivation Function (KDF) that allows legitimate users to spend a moderate amount of time on key derivation, while imposing CPU/memory-intensive operations on the attacker side. In this paper, we focus on long-term passwords secured by the Password-Based Key Derivation Function 2 (PBKDF2) and present the case study of Linux Unified Key Setup (LUKS), a disk-encryption specification commonly implemented in Linux based operating systems. In particular, we describe how LUKS protects long-term keys by means of Iteration Counts defined at runtime, and analyze how external factors may affect the Iteration Counts computation. In doing so, we provide means of evaluating the Iteration Count values defined at run-time and experimentally show to what level PBKDF2 is still capable of providing sufficient security margin for a LUKS implementation
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Understanding Optimizations and Measuring Performances of PBKDF2
'Springer Science and Business Media LLC', 2019Co-Authors: A.f. Iuorio, A. ViscontiAbstract:Password-based key derivation functions (KDFs) are used to generate secure keys of arbitrary length implemented in many security-related systems. The strength of these KDFs is the ability to provide Countermeasures against brute-force/dictionary attacks. One of the most implemented KDFs is PBKDF2. In order to slow attackers down, PBKDF2 uses a salt and introduces computational intensive operations based on an iterated pseudorandom function. Since passwords are widely used to protect personal data and to authenticate users to access specific resources, if an application uses a small Iteration Count value, the strength of PBKDF2 against attacks performed on low-cost commodity hardware may be reduced. In this paper we introduce the cryptographic algorithms involved in the key derivation process, describing the optimization techniques used to speed up PBKDF2-HMAC-SHA1 in a GPU/CPU context. Finally, a testing activity has been executed on consumer-grade hardware, and experimental results are reported
Dal Santo Niccolò - One of the best experts on this subject based on the ideXlab platform.
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MATHICSE Technical Report : Multi space reduced basis preconditioners for parametrized Stokes equations
Écublens MATHICSE, 2019Co-Authors: Dal Santo Niccolò, Deparis Simone, Manzoni Andrea, Quarteroni AlfioAbstract:In this work we introduce a two-level preconditioner for the efficient solution of large scale saddlepoint linear systems arising from the finite element (FE) discretization of parametrized Stokes equations.The proposed preconditioner extends the Multi Space Reduced Basis (MSRB) preconditioning methodproposed in [12], and relies on the combination of an approximated block (fine grid) preconditioner witha reduced basis solver, which plays the role of coarse component. A sequence of RB spaces, constructedeither with an enriched velocity formulation or a Petrov-Galerkin projection, is built. As a matter offact, each RB coarse component is tailored to perform a single Iteration of the iterative method at hand.The exible GMRES (FGMRES) algorithm is employed to solve the resulting preconditioned systemand targets small tolerances with a very small Iteration Count and in a very short time. Numerical testcases dealing with Stokes flows in three dimensional parameter-ependent geometries are consideredto assess the numerical performance of the proposed technique in different large scale computationalsettings. A detailed comparison with both the current state of the art of i) standard RB methodsand ii) preconditioning techniques for Stokes equations highlights the better efficiency of the proposedmethodology
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MATHICSE Technical Report : Multi space reduced basis preconditioners for large-scale parametrized PDEs
Écublens MATHICSE, 2019Co-Authors: Dal Santo Niccolò, Deparis Simone, Manzoni Andrea, Quarteroni AlfioAbstract:In this work we introduce a new two-level preconditioner for the efficient solution of large scale linear systems arising from the discretization of parametrized PDEs. The proposed preconditioner combines in a multiplicative way a reduced basis solver, which plays the role of coarse component, and a "traditional" fine grid preconditioner, such as one-level Additive Schwarz, block Gauss-Seidel or block Jacobi preconditioners. The coarse component is built up on a new Multi Space Reduced Basis (MSRB) method that we introduce for the first time in this paper, where are reduced basis space is built through the proper orthogonal decomposition (POD) algorithm at each step of the iterative method at hand, like the flexible GMRES method. MSRB strategy consists in building reduced basis (RB) spaces that are well-suited to perform a single Iteration, by addressing the error components which have not been treated yet. The Krylov Iterations employed to solve the resulting preconditioned system targets small tolerances with a very small Iteration Count and in a very short time, showing good optimality and scalability properties. Simulations are carried out to evaluate the performance of the proposed preconditioner indifferent large scale computational settings related to parametrized advection diffusion equations and compared with the current state of the art algebraic multigrid preconditioners
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Multi space reduced basis preconditioners for large-scale parametrized PDEs
'Society for Industrial & Applied Mathematics (SIAM)', 2018Co-Authors: Dal Santo Niccolò, Deparis Simone, Manzoni Andrea, Quarteroni AlfioAbstract:In this work we introduce a new two-level preconditioner for the efficient solution of large-scale linear systems arising from the discretization of parametrized PDEs. The proposed preconditioner combines in a multiplicative way a reduced basis solver, which plays the role of coarse component, and a “traditional” fine-grid preconditioner, such as one-level additive Schwarz, block Gauss--Seidel, or block Jacobi preconditioners. The coarse component is built upon a new multi space reduced basis (MSRB) method that we introduce for the first time in this paper, where a reduced basis space is built through the proper orthogonal decomposition algorithm at each step of the iterative method at hand, like the flexible GMRES method. MSRB strategy consists in building reduced basis spaces that are well suited to perform a single Iteration, by addressing the error components which have not been treated yet. The Krylov Iterations employed to solve the resulting preconditioned system target small tolerances with a very small Iteration Count and in a very short time, showing good optimality and scalability properties. Simulations are carried out to evaluate the performance of the proposed preconditioner in different large-scale computational settings related to parametrized advection diffusion equations and compared with the current state-of-the-art algebraic multigrid preconditioners
Lexing Ying - One of the best experts on this subject based on the ideXlab platform.
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recursive sweeping preconditioner for the three dimensional helmholtz equation
SIAM Journal on Scientific Computing, 2016Co-Authors: Fei Liu, Lexing YingAbstract:This paper introduces the recursive sweeping preconditioner for the numerical solution of the Helmholtz equation in three dimensions. This is based on the earlier work of the sweeping preconditioner with the moving perfectly matched layers. The key idea is to apply the sweeping preconditioner recursively to the quasi-two-dimensional auxiliary problems introduced in the three-dimensional (3D) sweeping preconditioner. Compared to the nonrecursive 3D sweeping preconditioner, the setup cost of this new approach drops from $O(N^{4/3})$ to $O(N)$, the application cost per Iteration drops from $O(N\log N)$ to $O(N)$, and the Iteration Count increases only mildly when combined with the standard GMRES solver. Several numerical examples are tested and the results are compared with the nonrecursive sweeping preconditioner to demonstrate the efficiency of the new approach.
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recursive sweeping preconditioner for the 3d helmholtz equation
arXiv: Numerical Analysis, 2015Co-Authors: Fei Liu, Lexing YingAbstract:This paper introduces the recursive sweeping preconditioner for the numerical solution of the Helmholtz equation in 3D. This is based on the earlier work of the sweeping preconditioner with the moving perfectly matched layers (PMLs). The key idea is to apply the sweeping preconditioner recursively to the quasi-2D auxiliary problems introduced in the 3D sweeping preconditioner. Compared to the non-recursive 3D sweeping preconditioner, the setup cost of this new approach drops from $O(N^{4/3})$ to $O(N)$, the application cost per Iteration drops from $O(N\log N)$ to $O(N)$, and the Iteration Count only increases mildly when combined with the standard GMRES solver. Several numerical examples are tested and the results are compared with the non-recursive sweeping preconditioner to demonstrate the efficiency of the new approach.
R N Slaybaugh - One of the best experts on this subject based on the ideXlab platform.
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Eigenvalue solvers for modeling nuclear reactors on leadership class machines
eScholarship University of California, 2018Co-Authors: R N Slaybaugh, Ramirez-zweiger M, Pandya T, Hamilton S, Tm EvansAbstract:Three complementary methods have been implemented in the code Denovo that accelerate neutral particle transport calculations with methods that use leadership-class computers fully and effectively: a multigroup block (MG) Krylov solver, a Rayleigh quotient Iteration (RQI) eigenvalue solver, and a multigrid in energy (MGE) preconditioner. The MG Krylov solver converges more quickly than Gauss Seidel and enables energy decomposition such that Denovo can scale to hundreds of thousands of cores. RQI should converge in fewer Iterations than power Iteration (PI) for large and challenging problems. RQI creates shifted systems that would not be tractable without the MG Krylov solver. It also creates ill-conditioned matrices. The MGE preconditioner reduces Iteration Count significantly when used with RQI and takes advantage of the new energy decomposition such that it can scale efficiently. Each individual method has been described before, but this is the first time they have been demonstrated to work together effectively. The combination of solvers enables the RQI eigenvalue solver to work better than the other available solvers for large reactors problems on leadership-class machines. Using these methods together, RQI converged in fewer Iterations and in less time than PI for a full pressurized water reactor core. These solvers also performed better than an Arnoldi eigenvalue solver for a reactor benchmark problem when energy decomposition is needed. The MG Krylov, MGE preconditioner, and RQI solver combination also scales well in energy. This solver set is a strong choice for very large and challenging problems
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Eigenvalue solvers for modeling nuclear reactors on leadership class machines
eScholarship University of California, 2018Co-Authors: R N Slaybaugh, Ramirez-zweiger M, Pandya T, Hamilton S, Tm EvansAbstract:© 2018 American Nuclear Society. Three complementary methods have been implemented in the code Denovo that accelerate neutral particle transport calculations with methods that use leadership-class computers fully and effectively: a multigroup block (MG) Krylov solver, a Rayleigh quotient Iteration (RQI) eigenvalue solver, and a multigrid in energy (MGE) preconditioner. The MG Krylov solver converges more quickly than Gauss Seidel and enables energy decomposition such that Denovo can scale to hundreds of thousands of cores. RQI should converge in fewer Iterations than power Iteration (PI) for large and challenging problems. RQI creates shifted systems that would not be tractable without the MG Krylov solver. It also creates ill-conditioned matrices. The MGE preconditioner reduces Iteration Count significantly when used with RQI and takes advantage of the new energy decomposition such that it can scale efficiently. Each individual method has been described before, but this is the first time they have been demonstrated to work together effectively. The combination of solvers enables the RQI eigenvalue solver to work better than the other available solvers for large reactors problems on leadership-class machines. Using these methods together, RQI converged in fewer Iterations and in less time than PI for a full pressurized water reactor core. These solvers also performed better than an Arnoldi eigenvalue solver for a reactor benchmark problem when energy decomposition is needed. The MG Krylov, MGE preconditioner, and RQI solver combination also scales well in energy. This solver set is a strong choice for very large and challenging problems
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rayleigh quotient Iteration with a multigrid in energy preconditioner for massively parallel neutron transport
arXiv: Computational Engineering Finance and Science, 2017Co-Authors: R N Slaybaugh, Thomas M Evans, Gregory G Davidson, Paul P H WilsonAbstract:Three complementary methods have been implemented in the code Denovo that accelerate neutral particle transport calculations with methods that use leadership-class computers fully and effectively: a multigroup block (MG) Krylov solver, a Rayleigh quotient Iteration (RQI) eigenvalue solver, and a multigrid in energy preconditioner. The multigroup Krylov solver converges more quickly than Gauss Seidel and enables energy decomposition such that Denovo can scale to hundreds of thousands of cores. The new multigrid in energy preconditioner reduces Iteration Count for many problem types and takes advantage of the new energy decomposition such that it can scale efficiently. These two tools are useful on their own, but together they enable the RQI eigenvalue solver to work. Each individual method has been described before, but this is the first time they have been demonstrated to work together effectively. RQI should converge in fewer Iterations than power Iteration (PI) for large and challenging problems. RQI creates shifted systems that would not be tractable without the MG Krylov solver. It also creates ill-conditioned matrices that cannot converge without the multigrid in energy preconditioner. Using these methods together, RQI converged in fewer Iterations and in less time than all PI calculations for a full pressurized water reactor core. It also scaled reasonably well out to 275,968 cores.
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rayleigh quotient Iteration with a multigrid in energy preconditioner for massively parallel neutron transport
Mathematics and Computations Supercomputing in Nuclear Applications and Monte Carlo International Conference M and C+SNA+MC 2015, 2015Co-Authors: R N Slaybaugh, Thomas M Evans, Gregory G Davidson, Paul P H WilsonAbstract:Author(s): Slaybaugh, RN; Evans, TM; Davidson, GG; Wilson, PPH | Abstract: Copyright © (2015) by the American Nuclear Society All rights reserved. Three complementary methods have been implemented in the code Denovo that accelerate neutral particle transport calculations with methods that use leadership-class computers fully and effectively: a multigroup block (MG) Krylov solver, a Rayleigh quotient Iteration (RQI) eigenvalue solver, and a multigrid in energy preconditioner. The multigroup Krylov solver converges more quickly than Gauss Seidel and enables energy decomposition such that Denovo can scale to hundreds of thousands of cores. The new multigrid in energy preconditioner reduces Iteration Count for many problem types and takes advantage of the new energy decomposition such that it can scale efficiently. These two tools are useful on their own, but together they enable the RQI eigenvalue solver to work. Each individual method has been described before, but this is the first time they have been demonstrated to work together effectively. RQI should converge in fewer Iterations than power Iteration (PI) for large and challenging problems. RQI creates shifted systems that would not be tractable without the MG Krylov solver. It also creates ill-conditioned matrices that cannot converge without the multigrid in energy preconditioner. Using these methods together, RQI converged in fewer Iterations and in less time than all PI calculations for a full pressurized water reactor core. It also scaled reasonably well out to 275,968 cores.
