483 research outputs found
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New Discretization Methods for the Numerical Approximation of PDEs
The construction and mathematical analysis of numerical methods for PDEs is a fundamental area of modern applied mathematics. Among the various techniques that have been proposed in the past, some – in particular, finite element methods, – have been exceptionally successful in a range of applications. There are however a number of important challenges that remain, including the optimal adaptive finite element approximation of solutions to transport-dominated diffusion problems, the efficient numerical approximation of parametrized families of PDEs, and the efficient numerical approximation of high-dimensional partial differential equations (that arise from stochastic analysis and statistical physics, for example, in the form of a backward Kolmogorov equation, which, unlike its formal adjoint, the forward Kolmogorov equation, is not in divergence form, and therefore not directly amenable to finite element approximation, even when the spatial dimension is low). In recent years several original and conceptionally new ideas have emerged in order to tackle these open problems.
The goal of this workshop was to discuss and compare a number of novel approaches, to study their potential and applicability, and to formulate the strategic goals and directions of research in this field for the next five years
Sensitivity analysis for models with dynamic inputs: a case study to control the heat consumption of a real passive house
International audienceIn this communication, we perform the sensitivity analysis of a building energy model. The aim is to assess the impact of the weather data on the performance of a model of a passive house, in order to better control it. The weather data are uncertain dynamic inputs to the model. To evaluate their impact, the problem of generating coherent weather data arises. To solve it, we carry out the Karhunen-Loève decomposition of the uncertain dynamic inputs. We then propose an approach for the sensitivity analysis of this kind of models. The originality for sensitivity analysis purpose is to separate the random variable of the dynamic inputs, propagated to the model response, from the deterministic spatio/temporal function. This analysis highlights the role of the solar gain on a high-insulated passive building, during winter time
Numerical Methods for the Chemical Master Equation
The dynamics of biochemical networks can be described by a Markov jump process on a high-dimensional state space, with the corresponding probability distribution being the solution of the Chemical Master Equation (CME). In this thesis, adaptive wavelet methods for the time-dependent and stationary CME, as well as for the approximation of committor probabilities are devised. The methods are illustrated on multi-dimensional models with metastable solutions and large state spaces
Numerical investigation of Differential Biological-Models via GA-Kansa Method Inclusive Genetic Strategy
In this paper, we use Kansa method for solving the system of differential
equations in the area of biology. One of the challenges in Kansa method is
picking out an optimum value for Shape parameter in Radial Basis Function to
achieve the best result of the method because there are not any available
analytical approaches for obtaining optimum Shape parameter. For this reason,
we design a genetic algorithm to detect a close optimum Shape parameter. The
experimental results show that this strategy is efficient in the systems of
differential models in biology such as HIV and Influenza. Furthermore, we prove
that using Pseudo-Combination formula for crossover in genetic strategy leads
to convergence in the nearly best selection of Shape parameter.Comment: 42 figures, 23 page
Numerical solution for anti-persistent process based stochastic integral equations
In this article, we propose the shifted Legendre polynomial solutions for anti-persistent process based stochastic integral equations. The operational matrices for stochastic integration and fractional stochastic integration are efficiently generated using the properties of shifted Legendre polynomials. In addition, the original problem can be reduced to a system of simultaneous equations with (N + 1) unknowns in the function approximation. By solving the given stochastic integral equations, we obtain numerical solutions. The proposed method’s convergence is derived in terms of the error function’s expectation, and the upper bound of the error in L² norm is also discussed in detail. The applicability of this methodology is demonstrated using numerical examples and the solution’s quality is statistically validated by comparing it with the exact solution.Publisher's Versio
Uncertainty quantification for problems in radionuclide transport
The field of radionuclide transport has long recognised the stochastic nature of the problems
encountered. Many parameters that are used in computational models are very difficult,
if not impossible, to measure with any great degree of confidence. For example,
bedrock properties can only be measured at a few discrete points, the properties between
these points may be inferred or estimated using experiments but it is difficult to achieve
any high levels of confidence.
This is a major problem when many countries around the world are considering deep
geologic repositories as a disposal option for long-lived nuclear waste but require a high
degree of confidence that any release of radioactive material will not pose a risk to future
populations.
In this thesis we apply Polynomial Chaos methods to a model of the biosphere that is
similar to those used to assess exposure pathways for humans and associated dose rates
by many countries worldwide.
We also apply the Spectral-Stochastic Finite Element Method to the problem of contaminated
fluid flow in a porous medium. For this problem we use the Multi-Element
generalized Polynomial Chaos method to discretise the random dimensions in a manner
similar to the well known Finite Element Method. The stochastic discretisation is then
refined adaptively to mitigate the build up errors over the solution times.
It was found that these methods have the potential to provide much improved estimates
for radionuclide transport problems. However, further development is needed in order to
obtain the necessary efficiency that would be required to solve industrial problems
JDNN: Jacobi Deep Neural Network for Solving Telegraph Equation
In this article, a new deep learning architecture, named JDNN, has been
proposed to approximate a numerical solution to Partial Differential Equations
(PDEs). The JDNN is capable of solving high-dimensional equations. Here, Jacobi
Deep Neural Network (JDNN) has demonstrated various types of telegraph
equations. This model utilizes the orthogonal Jacobi polynomials as the
activation function to increase the accuracy and stability of the method for
solving partial differential equations. The finite difference time
discretization technique is used to overcome the computational complexity of
the given equation. The proposed scheme utilizes a Graphics Processing Unit
(GPU) to accelerate the learning process by taking advantage of the neural
network platforms. Comparing the existing methods, the numerical experiments
show that the proposed approach can efficiently learn the dynamics of the
physical problem
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Computation and Learning in High Dimensions (hybrid meeting)
The most challenging problems in science often involve the learning and
accurate computation of high dimensional functions.
High-dimensionality is a typical feature for a multitude of problems
in various areas of science.
The so-called curse of dimensionality typically negates the use of
traditional numerical techniques for the solution of
high-dimensional problems. Instead, novel theoretical and
computational approaches need to be developed to make them tractable
and to capture fine resolutions and relevant features. Paradoxically,
increasing computational power may even serve to heighten this demand,
since the wealth of new computational data itself becomes a major
obstruction. Extracting essential information from complex
problem-inherent structures and developing rigorous models to quantify
the quality of information in a high-dimensional setting pose
challenging tasks from both theoretical and numerical perspective.
This has led to the emergence of several new computational methodologies,
accounting for the fact that by now well understood methods drawing on
spatial localization and mesh-refinement are in their original form no longer viable.
Common to these approaches is the nonlinearity of the solution method.
For certain problem classes, these methods have
drastically advanced the frontiers of computability.
The most visible of these new methods is deep learning. Although the use of deep neural
networks has been extremely successful in certain
application areas, their mathematical understanding is far from complete.
This workshop proposed to deepen the understanding of
the underlying mathematical concepts that drive this new evolution of
computational methods and to promote the exchange of ideas emerging in various
disciplines about how to treat multiscale and high-dimensional problems
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