Xing Cai
University of Oslo, Department of Informatics, Faculty Member
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We want to be able to perform accurate simulations of a large number of cardiac cells based on mathematical models where each individual cell is represented in the model. This implies that the computational mesh has to have a typical... more
We want to be able to perform accurate simulations of a large number of cardiac cells based on mathematical models where each individual cell is represented in the model. This implies that the computational mesh has to have a typical resolution of a fewµm leading to huge computational challenges. In this paper we use a certain operator splitting of the coupled equations and showthat this leads to systems that can be solved in parallel. This opens up for the possibility of simulating large numbers of coupled cardiac cells.
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As described in the previous chapter, the human body consists of billions of cells, which may be connected by various coupling mechanisms depending on the type of tissue under consideration. When constructing mathematical models for... more
As described in the previous chapter, the human body consists of billions of cells, which may be connected by various coupling mechanisms depending on the type of tissue under consideration. When constructing mathematical models for electrical activity in the tissue, one possible approach would be to model each cell as a separate unit, and couple them together using mathematical models
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The physical relevance of computations based on the model problems arising from the electrical activity in the heart depends on high accuracy of the solution. High accuracy requires the solution of large linear or nonlinear systems of... more
The physical relevance of computations based on the model problems arising from the electrical activity in the heart depends on high accuracy of the solution. High accuracy requires the solution of large linear or nonlinear systems of ODEs and PDEs. This chapter deals with solution algorithms for the discretization of (linear) PDEs, which is a huge research field around the
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The operator splitting algorithms introduced in Chapter 3 reduced the solution of the bidomain equations to solving linear PDE systems and nonlinear systems of ODEs. Techniques for discretizing the PDE system were presented in Chapter 3,... more
The operator splitting algorithms introduced in Chapter 3 reduced the solution of the bidomain equations to solving linear PDE systems and nonlinear systems of ODEs. Techniques for discretizing the PDE system were presented in Chapter 3, while techniques for solving the resulting linear systems were discussed in Chapter 4. What remains to have a complete computational method for the bidomain
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The mathematical models derived in the previous chapter give a quantitative description of the electrical activity in the heart, from the level of electrochemical reactions in the cells to body surface potentials that may be recorded as... more
The mathematical models derived in the previous chapter give a quantitative description of the electrical activity in the heart, from the level of electrochemical reactions in the cells to body surface potentials that may be recorded as ECGs. However, the models are formulated as systems of nonlinear partial and ordinary differential equations, for which analytical solutions are not available. To be of any practical use, the equations of the models must therefore be solved with numerical methods. The choice of numerical methods that may be applied to the equations is large, see e.g. [83], but we have chosen to focus entirely on finite element methods (FEM). One reason for this is that the geometries of the heart and the body are irregular, and this is more conveniently handled by FEM than, for instance, by finite difference methods.
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Our knowledge about the heart dates back more than two millenia. Already in the days of Aristotle (350 b.c.) the importance of the heart was recognized, and it was, in fact, considered to be the most important organ in the body. Other... more
Our knowledge about the heart dates back more than two millenia. Already in the days of Aristotle (350 b.c.) the importance of the heart was recognized, and it was, in fact, considered to be the most important organ in the body. Other vital organs, such as the brain and lungs, were thought to exist merely to cool the blood. Over
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Cardiomyocytes are the functional building blocks of the heart—yet most models developed to simulate cardiac mechanics do not represent the individual cells and their surrounding matrix. Instead, they work on a homogenized tissue level,... more
Cardiomyocytes are the functional building blocks of the heart—yet most models developed to simulate cardiac mechanics do not represent the individual cells and their surrounding matrix. Instead, they work on a homogenized tissue level, assuming that cellular and subcellular structures and processes scale uniformly. Here we present a mathematical and numerical framework for exploring tissue-level cardiac mechanics on a microscale given an explicit three-dimensional geometrical representation of cells embedded in a matrix. We defined a mathematical model over such a geometry and parametrized our model using publicly available data from tissue stretching and shearing experiments. We then used the model to explore mechanical differences between the extracellular and the intracellular space. Through sensitivity analysis, we found the stiffness in the extracellular matrix to be most important for the intracellular stress values under contraction. Strain and stress values were observed to...
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In the preceding chapters, we have discussed various numerical techniques for solving the different parts of our mathematical model problem. Now it is time to turn our attention to simulating the complete mathematical model. First, we... more
In the preceding chapters, we have discussed various numerical techniques for solving the different parts of our mathematical model problem. Now it is time to turn our attention to simulating the complete mathematical model. First, we will explain the diverse computational tasks that constitute an electrocardiac simulator. Then, we will estimate the computational resources needed to carry out high-resolution simulations.
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IntroductionFor low-cost PC clusters that are made up of only o-the-shelf components, a commonnotion is that they can not be real competitors for conventional supercomputersin respect of delivering sustained computing power. We aim to... more
IntroductionFor low-cost PC clusters that are made up of only o-the-shelf components, a commonnotion is that they can not be real competitors for conventional supercomputersin respect of delivering sustained computing power. We aim to show in the presentpaper that this needs not be the case, especially when PC clusters are used to solvepartial dierential equations (PDEs). More specically, we will
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We explain how domain decomposition methods combined with objectoriented programming techniques can simplify the process of developing parallel simulation codes in computational mechanics. More speci cally, we present a generic software... more
We explain how domain decomposition methods combined with objectoriented programming techniques can simplify the process of developing parallel simulation codes in computational mechanics. More speci cally, we present a generic software framework for implementing overlapping Schwarz methods that possess high numerical eÆciency and inherent parallelism. The actual parallelization work in this framework reduces to deriving a couple of small C++ subclasses. We also examine a concrete example of parallelizing a nonlinear water wave simulator in the program system Di pack, with a detailed report on the speed-up results obtained on a Beowulf PC cluster.
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The second-order Adams-Bashforth-Moulton scheme is a predictor-corrector approach to solving initial-value problems that involve fractional derivatives. The number of required floating-point operations grows as O(N^2), where N is the... more
The second-order Adams-Bashforth-Moulton scheme is a predictor-corrector approach to solving initial-value problems that involve fractional derivatives. The number of required floating-point operations grows as O(N^2), where N is the number of time steps. This potentially huge amount of computation, together with the large memory requirement of order O(N), calls for efficient software implementations. In this context, we will demonstrate the importance of code optimization techniques like loop fusion, loop unrolling and alternating directions for loop traversal. On the basis of efficient serial implementations, high-performance parallel implementations, via MPI and OpenMP programming, can be easily developed for modern multi-core processors. The particular Bagley-Torvik equation, which can be formulated as a system of four coupled fractional differential equations, is used as a test bed for studying the obtainable computing speeds.
This paper studies the CUDA programming challenges with using multiple GPUs inside a single machine to carry out plane-by-plane updates in parallel 3D sweeping algorithms. In particular, care must be taken to mask the overhead of various... more
This paper studies the CUDA programming challenges with using multiple GPUs inside a single machine to carry out plane-by-plane updates in parallel 3D sweeping algorithms. In particular, care must be taken to mask the overhead of various data movements between the GPUs. Multiple OpenMP threads on the CPU side should be combined multiple CUDA streams per GPU to hide the data transfer cost related to the halo computation on each 2D plane. Moreover, the technique of peer-to-peer data motion can be used to reduce the impact of 3D volumetric data shuffles that have to be done between mandatory changes of the grid partitioning. We have investigated the performance improvement of 2- and 4-GPU implementations that are applicable to 3D anisotropic front propagation computations related to geological folding. In comparison with a straightforward multi-GPU implementation, the overall performance improvement due to masking of data movements on four GPUs of the Fermi architecture was 23%. The co...
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Computational Fluid Dynamics (CFD) is an important research field with a broad spectrum, requiring a close interplay between mathematical modeling, numerical methods, software development, and high performance computing. Programming CFD... more
Computational Fluid Dynamics (CFD) is an important research field with a broad spectrum, requiring a close interplay between mathematical modeling, numerical methods, software development, and high performance computing. Programming CFD applications is an inherently difficult task due to many factors related to numerics and software. For modern CFD applications, use of parallel computers is a must, which makes the CFD code development even more challenging. Rapid and flexible programming of a parallel CFD application calls for re-usable serial and parallel software components plus a modular overall parallel framework that allows an easy coupling between different components. Moreover, more attention needs to be put on securing the performance of the software components. This minisymposium thus attempts to shed some light into the recent developments in respect of CFD-related numerical methods, programming techniques and software libraries.
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If you know the characteristics of something today, and you know the laws of change, then you can figure out what the characteristicswill be tomorrow. This is the basic idea of modeling lots of natural processes. Since we know the weather... more
If you know the characteristics of something today, and you know the laws of change, then you can figure out what the characteristicswill be tomorrow. This is the basic idea of modeling lots of natural processes. Since we know the weather today and we know the equations modeling the changes of the weather, we can predict it some days ahead. We know the heat of an object now and we know the equations describing how heat changes; thus we can predict how warm an object will be later on. This is really at the heart of science and has been so for quite a while. But how do we express change? How do we make these vague statements precise and suitable for computer simulations? We do so by expressing the laws of change in terms of differential equations.
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In Chap. 7 we studied several aspects of the theory of diffusion processes. We saw how these equations arise in models of several physical phenomena and how they can be approximately solved by suitable numerical methods. The analysis of... more
In Chap. 7 we studied several aspects of the theory of diffusion processes. We saw how these equations arise in models of several physical phenomena and how they can be approximately solved by suitable numerical methods. The analysis of diffusion equations is a classic subject of applied mathematics and of scientific computing. Its impact on the field of partial differential
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In Oslo, there is a chain of small cafés called Bagel and Juice that serve fresh bagels and tasty juice. We know of such a café on Hegdehaugsveien, fairly close to the University of Oslo. The owner of this café, as well as all the other... more
In Oslo, there is a chain of small cafés called Bagel and Juice that serve fresh bagels and tasty juice. We know of such a café on Hegdehaugsveien, fairly close to the University of Oslo. The owner of this café, as well as all the other owners, faces one particular problem each night: She has to determine how many bagels