8 results on '"Kolev, Tzanio"'
Search Results
2. MATRIX-FREE HIGH-PERFORMANCE SADDLE-POINT SOLVERS FOR HIGH-ORDER PROBLEMS IN H(div).
- Author
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PAZNER, WILL, KOLEV, TZANIO, and VASSILEVSKI, PANAYOT S.
- Subjects
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ALGEBRAIC multigrid methods , *RADIATION trapping , *SCHUR complement , *MULTIGRID methods (Numerical analysis) , *BENCHMARK problems (Computer science) , *LAPLACIAN matrices , *POROUS materials - Abstract
This work describes the development of matrix-free GPU-accelerated solvers for high-order finite element problems in if(div). The solvers are applicable to grad-div and Darcy problems in saddle-point formulation, and have applications in radiation diffusion and porous media flow problems, among others. Using the interpolation--histopolation basis (cf. [W. Pazner, T. Kolev, and C. R. Dohrmann, SIAM J. Sci. Comput., 45 (2023), pp. A675-A702]), efficient matrix-free preconditioners can be constructed for the (1,1)-block and Schur complement of the block system. With these approximations, block-preconditioned MINRES converges in a number of iterations that is independent of the mesh size and polynomial degree. The approximate Schur complement takes the form of an M-matrix graph Laplacian and therefore can be well-preconditioned by highly scalable algebraic multigrid methods. High-performance GPU-accelerated algorithms for all components of the solution algorithm are developed, discussed, and benchmarked. Numerical results are presented on a number of challenging test cases, including the "crooked pipe" grad-div problem, the SPE10 reservoir modeling benchmark problem, and a nonlinear radiation diffusion test case. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
3. End-to-end GPU acceleration of low-order-refined preconditioning for high-order finite element discretizations.
- Author
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Pazner, Will, Kolev, Tzanio, and Camier, Jean-Sylvain
- Subjects
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RADIATION trapping , *ELECTROMAGNETIC radiation , *COMPUTATIONAL complexity , *PERFORMANCE theory , *ALGORITHMS - Abstract
In this article, we present algorithms and implementations for the end-to-end GPU acceleration of matrix-free low-order-refined preconditioning of high-order finite element problems. The methods described here allow for the construction of effective preconditioners for high-order problems with optimal memory usage and computational complexity. The preconditioners are based on the construction of a spectrally equivalent low-order discretization on a refined mesh, which is then amenable to, for example, algebraic multigrid preconditioning. The constants of equivalence are independent of mesh size and polynomial degree. For vector finite element problems in H (curl) and H (div) (e.g., for electromagnetic or radiation diffusion problems), a specially constructed interpolation–histopolation basis is used to ensure fast convergence. Detailed performance studies are carried out to analyze the efficiency of the GPU algorithms. The kernel throughput of each of the main algorithmic components is measured, and the strong and weak parallel scalability of the methods is demonstrated. The different relative weighting and significance of the algorithmic components on GPUs and CPUs is discussed. Results on problems involving adaptively refined nonconforming meshes are shown, and the use of the preconditioners on a large-scale magnetic diffusion problem using all spaces of the finite element de Rham complex is illustrated. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
4. Matrix-free subcell residual distribution for Bernstein finite elements: Monolithic limiting
- Author
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Hajduk, Hennes, Kuzmin, Dmitri, Kolev, Tzanio, Tomov, Vladimir, Tomas, Ignacio, and Shadid, John. N.
- Subjects
high-order finite elements ,advection problems ,limiters ,residual distribution ,discrete maximum principles ,Bernstein polynomials ,matrix-free methods - Abstract
This paper is focused on the aspects of limiting in residual distribution (RD) schemes for high-order finite element approximations to advection problems. Both continuous and discontinuous Galerkin methods are considered in this work. Discrete maximum principles are enforced using algebraic manipula-tions of element contributions to the global nonlinear system. The required modifications can be carried out without calculating the element matrices and assembling their global counterparts. The components of element vec-tors associated with the standard Galerkin discretization are manipulated di-rectly using localized subcell weights to achieve optimal accuracy. Low-order nonlinear RD schemes of this kind were originally developed to calculate local extremum diminishing predictors for flux-corrected transport (FCT) algorithms. In the present paper, we incorporate limiters directly into the residual distribution procedure, which makes it applicable to stationary prob-lems and leads to well-posed nonlinear discrete problems. To circumvent the second-order accuracy barrier, the correction factors of monolithic limiting approaches and FCT schemes are adjusted using smoothness sensors based on second derivatives. The convergence behavior of presented methods is illustrated by numerical studies for two-dimensional test problems, Ergebnisberichte des Instituts für Angewandte Mathematik;621
- Published
- 2019
5. CONSERVATIVE AND ACCURATE SOLUTION TRANSFER BETWEEN HIGH-ORDER AND LOW-ORDER REFINED FINITE ELEMENT SPACES.
- Author
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KOLEV, TZANIO and PAZNER, WILL
- Subjects
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AFFINE geometry , *LINEAR operators , *CONSERVATIVES - Abstract
In this paper we introduce general transfer operators between high-order and loworder refined finite element spaces that can be used to couple high-order and low-order simulations. Under natural restrictions on the low-order refined space we prove that both the high-to-low-order and low-to-high-order linear mappings are conservative, constant preserving, and high-order accurate. While the proof holds for affine geometries, numerical experiments indicate that the results hold for more general curved meshes. We present several numerical results confirming our analysis and demonstrate the utility of the new mappings in the context of adaptive mesh refinement and conservative multidiscretization coupling. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
6. THE TARGET-MATRIX OPTIMIZATION PARADIGM FOR HIGH-ORDER MESHES.
- Author
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DOBREV, VESELIN, KOLEV, TZANIO, TOMOV, VLADIMIR, KNUPP, PATRICK, and MITTAL, KETAN
- Subjects
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MATRICES (Mathematics) , *FINITE element method , *MATHEMATICAL optimization - Abstract
We describe a framework for controlling and improving the quality of high-order finite element meshes based on extensions of the Target-Matrix Optimization Paradigm (TMOP) of [P. Knupp, Eng. Comput., 28 (2012), pp. 419--429]. This approach allows high-order applications to have a very precise control over local mesh quality, while still improving the mesh globally. We address the adaption of various TMOP components to the settings of general isoparametric element mappings, including the mesh quality metric in 2D and 3D, the selection of sample points and the solution of the resulting mesh optimization problem. We also investigate additional practical concerns, such as tangential relaxation and restricting the deviation from the original mesh. The benefits of the new high-order TMOP algorithms are illustrated on a number of test problems and examples from a high-order arbitrary Lagrangian--Eulerian (ALE) application [BLAST: High-order curvilinear finite elements for shock hydrodynamics. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
7. Simulation-driven optimization of high-order meshes in ALE hydrodynamics.
- Author
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Dobrev, Veselin, Knupp, Patrick, Kolev, Tzanio, Mittal, Ketan, Rieben, Robert, and Tomov, Vladimir
- Subjects
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HYDRODYNAMICS , *LAGRANGIAN functions , *ALE - Abstract
• Mesh adaptivity is used to control numerical dissipation in ALE hydrodynamics. • Algebraic problem formulation applicable to curved meshes in any dimension. • Adaptivity to discrete simulation features is done through node movement. • General target construction methods can adapt all geometric properties. • Adaptive ALE triggers can improve both accuracy and simulation time. In this paper we propose tools for high-order mesh optimization and demonstrate their benefits in the context of multi-material Arbitrary Lagrangian-Eulerian (ALE) compressible shock hydrodynamic applications. The mesh optimization process is driven by information provided by the simulation which uses the optimized mesh, such as shock positions, material regions, known error estimates, etc. These simulation features are usually represented discretely, for instance, as finite element functions on the Lagrangian mesh. The discrete nature of the input is critical for the practical applicability of the algorithms we propose and distinguishes this work from approaches that strictly require analytical information. Our methods are based on node movement through a high-order extension of the Target-Matrix Optimization Paradigm (TMOP) of [1]. The proposed formulation is fully algebraic and relies only on local Jacobian matrices, so it is applicable to all types of mesh elements, in 2D and 3D, and any order of the mesh. We discuss the notions of constructing adaptive target matrices and obtaining their derivatives, reconstructing discrete data in intermediate meshes, node limiting that enables improvement of global mesh quality while preserving space-dependent local mesh features, and appropriate normalization of the objective function. The adaptivity methods are combined with automatic ALE triggers that can provide robustness of the mesh evolution and avoid excessive remap procedures. The benefits of the new high-order TMOP technology are illustrated on several simulations performed in the high-order ALE application BLAST [2]. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
8. Matrix-free subcell residual distribution for Bernstein finite elements: Monolithic limiting.
- Author
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Hajduk, Hennes, Kuzmin, Dmitri, Kolev, Tzanio, Tomov, Vladimir, Tomas, Ignacio, and Shadid, John N.
- Subjects
- *
CORRECTION factors , *NONLINEAR equations , *GALERKIN methods , *NONLINEAR systems , *ADVECTION , *BERNSTEIN polynomials - Abstract
• Matrix-free methods for transport and remap problems based on residual distribution. • High-order Bernstein finite element discretizations. • Theoretical foundation that guarantees reasonable, CFL-like time step restrictions. • Comparison of new monolithic limiters to Flux-Corrected-Transport. • Optimal convergence rates for smooth solutions achieved by nodal smoothness indicators. This paper is focused on the aspects of limiting in residual distribution (RD) schemes for high-order finite element approximations to advection problems. Both continuous and discontinuous Galerkin methods are considered in this work. Discrete maximum principles are enforced using algebraic manipulations of element contributions to the global nonlinear system. The required modifications can be carried out without calculating the element matrices and assembling their global counterparts. The components of element vectors associated with the standard Galerkin discretization are manipulated directly using localized subcell weights to achieve optimal accuracy. Low-order nonlinear RD schemes of this kind were originally developed to calculate local extremum diminishing predictors for flux-corrected transport (FCT) algorithms. In the present paper, we incorporate limiters directly into the residual distribution procedure, which makes it applicable to stationary problems and leads to well-posed nonlinear discrete problems. To circumvent the second-order accuracy barrier, the correction factors of monolithic limiting approaches and FCT schemes are adjusted using smoothness sensors based on second derivatives. The convergence behavior of presented methods is illustrated by numerical studies for two-dimensional test problems. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
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