Paszyński, Maciej
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Item type:Article, Access status: Open Access , One-dimensional fully automatic h-adaptive isogeometric finite element method package(Wydawnictwa AGH, 2016) Lipski, Paweł; Paszyński, MaciejThis paper deals with an adaptive finite element method originally developed by Prof. Leszek Demkowicz for hierarchical basis functions. In this paper, we investigate the extension of the adaptive algorithm for isogeometric analysis performed with $B$-spline basis functions. We restrict ourselves to $h$-adaptivity, since the polynomial order of approximation must be fixed in the isogeometric case. The classical variant of the adaptive FEM algorithm, as delivered by the group of Prof. Demkowicz, is based on a two-grid paradigm, with coarse and fine grids (the latter utilized as a reference solution). The problem is solved independently over a coarse mesh and a fine mesh. The fine-mesh solution is then utilized as a reference to estimate the relative error of the coarse-mesh solution and to decide which elements to refine. Prof. Demkowicz uses hierarchical basis functions, which (though locally providing $C^{p−1}$ continuity) ensure only $C^0$ on the interfaces between elements. The CUDA C library described in this paper switches the basis to $B$-spline functions and proposes a one-dimensional isogeometric version of the $h$-adaptive FEM algorithm to achieve global $C^{p−1}$ continuity of the solution.Item type:Article, Access status: Open Access , Grammar based multi-frontal solver for isogeometric analysis in 1D(Wydawnictwa AGH, 2013) Kuźnik, Krzysztof; Paszyński, Maciej; Calo, Victor ManuelIn this paper, we present a multi-frontal direct solver for one-dimensional isogeometric finite element method. The solver implementation is based on the graph grammar (GG) model. The GG model allows us to express the entire solver algorithm, including generation of frontal matrices, merging, and eliminations as a set of basic undividable tasks called graph grammar productions. Having the solver algorithm expressed as GG productions, we can find the partial order of execution and create a dependency graph, allowing for scheduling of tasks into shared memory parallel machine. We focus on the implementation of the solver with NVIDIA CUDA on the graphic processing unit (GPU). The solver has been tested for linear, quadratic, cubic, and higher-order B-splines, resulting in logarithmic scalability.Item type:Article, Access status: Open Access , Multi-frontal solver for simulations of linear elasticity coupled with acoustics(Wydawnictwa AGH, 2011) Paszyński, Maciej; Jurczyk, Tomasz; Pardo, DavidThis paper describes the concurrent multi-frontal direct solver algorithm for a multi-physics Finite Element Method (FEM). The multi-physics FEM utilizes different element sizes as well as polynomial orders of approximation over element edges, faces, and interiors (element nodes). The solver is based on the concept of a node, and management of unknowns is realized at the level of nodes. The solver is tested on a challenging multi-physis problem: acoustics coupled with linear elasticity over a 3D ball shape domain.Item type:Article, Access status: Open Access , Hypergrammar-based parallel multi-frontal solver for Grids with point singularities(Wydawnictwa AGH, 2015) Gurgul, Piotr; Paszyński, Maciej; Paszyńska, AnnaThis paper describes the application of hypergraph grammars to drive a linear computational cost solver for grids with point singularities. Such graph grammar productions are the first mathematical formalisms used to describe solver algorithms, and each indicates the smallest atomic task that can be executed in parallel, which is very useful in the case of parallel execution. In particular, the partial order of execution of graph grammar productions can be found, and the sets of independent graph grammar productions can be localized. They can be scheduled set by set into a shared memory parallel machine. The graphgrammar-based solver has been implemented with NVIDIA CUDA for GPU. Graph grammar productions are accompanied by numerical results for a 2D case. We show that our graph-grammar-based solver with a GPU accelerator is, by order of magnitude, faster than the state-of-the-art MUMPS solver.Item type:Book Chapter, Access status: Open Access , EXPBrain: Exponential Integrators for Glioblastoma Brain Tumor Simulations(Springer, 2025) Pabisz, Magdalena; Ciupek, Dominika; Vilkha, Askold; Paszyński, Maciej; Paszynski, M., Barnard, A.S., Zhang, Y.J. (eds)
Wydział InformatykiNOTE. This is a preprint of the paper with the same name in the Lecture Notes in Computer Science Journal. This preprint has not undergone peer review (when applicable) or any post-submission improvements or corrections. The Version of Record of this contribution is published in Paszynski, M., Barnard, A.S., Zhang, Y.J. (eds) Computational Science – ICCS 2025 Workshops. ICCS 2025. Lecture Notes in Computer Science, vol 15907, and is available online at: https://doi.org/10.1007/978-3-031-97554-7_10 In this paper we discuss a MATLAB implementation of the exponential integrators method employed for simulating of the brain tumor progression. As the input data we utilize publicly available T1-weighted magnetic resonance imaging dataset ds003826, representing healthy individuals. The data from these datasets are originally stored using NIfTI format. We select randomly one anonimized individual from the considered dataset. We normalize the brain scan data using min-max normalization to a range of 0 to 255. In the data from the dataset ds003826 the voxel resolution is not isotropic in all directions, so we interpolate the data from dimensions 176×248×256 into 194×248×256 in order to have proper proportions of the human brain. We set the data asa sequence of 256 PNG files with the resolution of 194 × 248. Having the MRI scan data, we run the exponential integrators method simulating the glioblastoma tumor growth using the Fisher-Kolmogorov diffusion-reaction model with logistic growth. We assume the initial tumor location and run the simulation predicting two years forward tumor growth. For the spatial discretization we employ the finite difference method, and for the temporal discretization we use the ultra-fast exponential integrators method. Our simulator generates the simulational results suitable for visualization using the ParaView tool.Item type:Article, Access status: Open Access , On the computational cost and complexity of stochastic inverse solvers(Wydawnictwa AGH, 2016) Faliszewski, Piotr; Smołka, Maciej; Schaefer, Robert; Paszyński, MaciejThe goal of this paper is to provide a starting point for investigations into a mainly underdeveloped area of research regarding the computational cost analysis of complex stochastic strategies for solving parametric inverse problems. This area has two main components: solving global optimization problems and solving forward problems (to evaluate the misfit function that we try to minimize). For the first component, we pay particular attention to genetic algorithms with heuristics and to multi-deme algorithms that can be modeled as ergodic Markov chains. We recall a simple method for evaluating the first hitting time for the single-deme algorithm and we extend it to the case of HGS, a multi-deme hierarchic strategy. We focus on the case in which at least the demes in the leaves are well tuned. Finally, we also express the problems of finding local and global optima in terms of a classic complexity theory. We formulate the natural result that finding a local optimum of a function is an NP-complete task, and we argue that finding a global optimum is a much harder, DP-complete, task. Furthermore, we argue that finding all global optima is, possibly, even harder (#P-hard) task. Regarding the second component of solving parametric inverse problems (i.e., regarding the forward problem solvers), we discuss the computational cost of hp-adaptive Finite Element solvers and their rates of convergence with respect to the increasing number of degrees of freedom. The presented results provide a useful taxonomy of problems and methods of studying the computational cost and complexity of various strategies for solving inverse parametric problems. Yet, we stress that our goal was not to deliver detailed evaluations for particular algorithms applied to particular inverse problems, but rather to try to identify possible ways of obtaining such results.Item type:Article, Access status: Open Access , Application of projection-based interpolation algorithm for non-stationary problem(Wydawnictwa AGH, 2016) Woźniak, Maciej; Paszyński, MaciejIn this paper, we present a solver for non-stationary problems using $L^2$ projection and h-adaptations. The solver utilizes the Euler time integration scheme for time evolution mixed with projection-based interpolation techniques for solving the $L^2$ projection problem at every time step. The solver is tested on the model problem of a heat transfer in an L-shape domain. We show that our solver delivers linear computational cost at every time step.Item type:Article, Access status: Open Access , Linear computational cost implicit solver for parabolic problems(Wydawnictwa AGH, 2020) Gurgul, Grzegorz; Łoś, Marcin Mateusz; Paszyński, Maciej; Calo, VictorIn this paper, we use the alternating direction method for isogeometric finite elements to simulate transient problems. Namely, we focus on a parabolic problem and use B-spline basis functions in space and an implicit time-marching method to fully discretize the problem. We introduce intermediate time-steps and separate our differential operator into a summation of the blocks that act along a particular coordinate axis in the intermediate time-steps. We show that the resulting stiffness matrix can be represented as a multiplication of two (in 2D) or three (in 3D) multi-diagonal matrices, each one with B-spline basis functions along the particular axis of the spatial system of coordinates. As a result of these algebraic transformations, we get a system of linear equations that can be factorized in a linear O(N) computational cost at every time-step of the implicit method. We use our method to simulate the heat transfer problem. We demonstrate theoretically and verify numerically that our implicit method is unconditionally stable for heat transfer problems (i.e., parabolic). We conclude our presentation with a discussion on the limitations of the method.Item type:Article, Access status: Open Access , Convergence of iterative solvers for non-linear step-and-flash imprint lithography simulations(Wydawnictwa AGH, 2011) Paszyński, MaciejThe paper presents the analysis of the iterative solvers utilized to solve the non-linear problem of Step-and-Flash Imprint Lithography (SFIL) a modern patterning process. The simulations consists in solving molecular statics problem for the polymer network, with quadratic potentials. The model distinguishes the strong interparticle interactions between particles forming a polymer network, and weak interactions between remaining particles. It also allows for large deformations, which all together implies the non-linear model. To illustrate the convergence of the iterative solvers, we present snapshots of the deformation of the sample being subject to the iterative solution. We claim that the animation is an interesting way of illustrating the convergence of the iterative solvers.Item type:Article, Access status: Open Access , Overview of adaptive and low-rank approximation algorithms for modeling influence of electromagnetic waves generated by cellphone antenna on human head(Wydawnictwa AGH, 2021) Głut, Barbara; Paszyński, MaciejThis paper presents an overview of formulations and algorithms that are dedicated to modeling the influence of electromagnetic waves on the human head. We start from h adaptive approximation of a three-dimensional MRI scan of the human head. Next, we solve the time-harmonic Maxwell equations with a 1.8 GHz cellphone antenna. We compute the specific absorption rate used as the heat source for the Pennes bioheat equation modeling the heat generated by EM waves inside the head. We propose an adaptive algorithm mixed with time-stepping iterations where we simultaneously refine the computational mesh, solve the Maxwell and Pennes equations, and iterate the time steps. We employ the sparse Gaussian elimination algorithm with the low-rank compres-sion of the off-diagonal matrix blocks for the factorization of the matrices. We conclude with the statement that 15 minutes of talking with a 1.8 GHz antenna with one watt of power results in increased brain tissue temperatures (up to 38.4$^{\circ}$C).