443,837 research outputs found
Matrix-free weighted quadrature for a computationally efficient isogeometric -method
The -method is the isogeometric method based on splines (or NURBS, etc.)
with maximum regularity. When implemented following the paradigms of classical
finite element methods, the computational resources required by the method
are prohibitive even for moderate degree. In order to address this issue, we
propose a matrix-free strategy combined with weighted quadrature, which is an
ad-hoc strategy to compute the integrals of the Galerkin system. Matrix-free
weighted quadrature (MF-WQ) speeds up matrix operations, and, perhaps even more
important, greatly reduces memory consumption. Our strategy also requires an
efficient preconditioner for the linear system iterative solver. In this work
we deal with an elliptic model problem, and adopt a preconditioner based on the
Fast Diagonalization method, an old idea to solve Sylvester-like equations. Our
numerical tests show that the isogeometric solver based on MF-WQ is faster than
standard approaches (where the main cost is the matrix formation by standard
Gaussian quadrature) even for low degree. But the main achievement is that,
with MF-WQ, the -method gets orders of magnitude faster by increasing the
degree, given a target accuracy. Therefore, we are able to show the
superiority, in terms of computational efficiency, of the high-degree
-method with respect to low-degree isogeometric discretizations. What we
present here is applicable to more complex and realistic differential problems,
but its effectiveness will depend on the preconditioner stage, which is as
always problem-dependent. This situation is typical of modern high-order
methods: the overall performance is mainly related to the quality of the
preconditioner
Accelerating the Fourier split operator method via graphics processing units
Current generations of graphics processing units have turned into highly
parallel devices with general computing capabilities. Thus, graphics processing
units may be utilized, for example, to solve time dependent partial
differential equations by the Fourier split operator method. In this
contribution, we demonstrate that graphics processing units are capable to
calculate fast Fourier transforms much more efficiently than traditional
central processing units. Thus, graphics processing units render efficient
implementations of the Fourier split operator method possible. Performance
gains of more than an order of magnitude as compared to implementations for
traditional central processing units are reached in the solution of the time
dependent Schr\"odinger equation and the time dependent Dirac equation
Michaelis-Menten dynamics in protein subnetworks
To understand the behaviour of complex systems it is often necessary to use
models that describe the dynamics of subnetworks. It has previously been
established using projection methods that such subnetwork dynamics generically
involves memory of the past, and that the memory functions can be calculated
explicitly for biochemical reaction networks made up of unary and binary
reactions. However, many established network models involve also
Michaelis-Menten kinetics, to describe e.g. enzymatic reactions. We show that
the projection approach to subnetwork dynamics can be extended to such
networks, thus significantly broadening its range of applicability. To derive
the extension we construct a larger network that represents enzymes and enzyme
complexes explicitly, obtain the projected equations, and finally take the
limit of fast enzyme reactions that gives back Michaelis-Menten kinetics. The
crucial point is that this limit can be taken in closed form. The outcome is a
simple procedure that allows one to obtain a description of subnetwork
dynamics, including memory functions, starting directly from any given network
of unary, binary and Michaelis-Menten reactions. Numerical tests show that this
closed form enzyme elimination gives a much more accurate description of the
subnetwork dynamics than the simpler method that represents enzymes explicitly,
and is also more efficient computationally
GPU acceleration of time-domain fluorescence lifetime imaging
Fluorescence lifetime imaging microscopy (FLIM) plays a significant role in biological sciences, chemistry, and medical research. We propose a Graphic Processing Units (GPUs) based FLIM analysis tool suitable for high-speed and flexible time-domain FLIM applications. With a large number of parallel processors, GPUs can significantly speed up lifetime calculations compared to CPU-OpenMP (parallel computing with multiple CPU cores) based analysis. We demonstrate how to implement and optimize FLIM algorithms on GPUs for both iterative and non-iterative FLIM analysis algorithms. The implemented algorithms have been tested on both synthesized and experimental FLIM data. The results show that at the same precision the GPU analysis can be up to 24-fold faster than its CPU-OpenMP counterpart. This means that even for high precision but time-consuming iterative FLIM algorithms, GPUs enable fast or even real-time analysis
Efficient Explicit Time Stepping of High Order Discontinuous Galerkin Schemes for Waves
This work presents algorithms for the efficient implementation of
discontinuous Galerkin methods with explicit time stepping for acoustic wave
propagation on unstructured meshes of quadrilaterals or hexahedra. A crucial
step towards efficiency is to evaluate operators in a matrix-free way with
sum-factorization kernels. The method allows for general curved geometries and
variable coefficients. Temporal discretization is carried out by low-storage
explicit Runge-Kutta schemes and the arbitrary derivative (ADER) method. For
ADER, we propose a flexible basis change approach that combines cheap face
integrals with cell evaluation using collocated nodes and quadrature points.
Additionally, a degree reduction for the optimized cell evaluation is presented
to decrease the computational cost when evaluating higher order spatial
derivatives as required in ADER time stepping. We analyze and compare the
performance of state-of-the-art Runge-Kutta schemes and ADER time stepping with
the proposed optimizations. ADER involves fewer operations and additionally
reaches higher throughput by higher arithmetic intensities and hence decreases
the required computational time significantly. Comparison of Runge-Kutta and
ADER at their respective CFL stability limit renders ADER especially beneficial
for higher orders when the Butcher barrier implies an overproportional amount
of stages. Moreover, vector updates in explicit Runge--Kutta schemes are shown
to take a substantial amount of the computational time due to their memory
intensity
Derivation of the bacterial run-and-tumble kinetic equation from a model with biochemical pathway
Kinetic-transport equations are, by now, standard models to describe the
dynamics of populations of bacteria moving by run-and-tumble. Experimental
observations show that bacteria increase their run duration when encountering
an increasing gradient of chemotactic molecules. This led to a first class of
models which heuristically include tumbling frequencies depending on the
path-wise gradient of chemotactic signal.
More recently, the biochemical pathways regulating the flagellar motors were
uncovered. This knowledge gave rise to a second class of kinetic-transport
equations, that takes into account an intra-cellular molecular content and
which relates the tumbling frequency to this information. It turns out that the
tumbling frequency depends on the chemotactic signal, and not on its gradient.
For these two classes of models, macroscopic equations of Keller-Segel type,
have been derived using diffusion or hyperbolic rescaling. We complete this
program by showing how the first class of equations can be derived from the
second class with molecular content after appropriate rescaling. The main
difficulty is to explain why the path-wise gradient of chemotactic signal can
arise in this asymptotic process.
Randomness of receptor methylation events can be included, and our approach
can be used to compute the tumbling frequency in presence of such a noise
A Supervisor for Control of Mode-switch Process
Many processes operate only around a limited number of operation points. In order to have adequate control around each operation point, and adaptive controller could be used. When the operation point changes often, a large number of parameters would have to be adapted over and over again. This makes application of conventional adaptive control unattractive, which is more suited for processes with slowly changing parameters. Furthermore, continuous adaptation is not always needed or desired. An extension of adaptive control is presented, in which for each operation point the process behaviour can be stored in a memory, retrieved from it and evaluated. These functions are co-ordinated by a ¿supervisor¿. This concept is referred to as a supervisor for control of mode-switch processes. It leads to an adaptive control structure which quickly adjusts the controller parameters based on retrieval of old information, without the need to fully relearn each time. This approach has been tested on experimental set-ups of a flexible beam and of a flexible two-link robot arm, but it is directly applicable to other processes, for instance, in the (petro) chemical industry
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