200 research outputs found
Optimal space-time adaptive wavelet methods for degenerate parabolic PDEs
We analyze parabolic PDEs with certain type of weakly singular or degenerate time-dependent coefficients and prove existence and uniqueness of weak solutions in an appropriate sense. A localization of the PDEs to a bounded spatial domain is justified. For the numerical solution a space-time wavelet discretization is employed. An optimality result for the iterative solution of the arising systems can be obtained. Finally, applications to fractional Brownian motion models in option pricing are presente
A multiresolution space-time adaptive scheme for the bidomain model in electrocardiology
This work deals with the numerical solution of the monodomain and bidomain
models of electrical activity of myocardial tissue. The bidomain model is a
system consisting of a possibly degenerate parabolic PDE coupled with an
elliptic PDE for the transmembrane and extracellular potentials, respectively.
This system of two scalar PDEs is supplemented by a time-dependent ODE modeling
the evolution of the so-called gating variable. In the simpler sub-case of the
monodomain model, the elliptic PDE reduces to an algebraic equation. Two simple
models for the membrane and ionic currents are considered, the
Mitchell-Schaeffer model and the simpler FitzHugh-Nagumo model. Since typical
solutions of the bidomain and monodomain models exhibit wavefronts with steep
gradients, we propose a finite volume scheme enriched by a fully adaptive
multiresolution method, whose basic purpose is to concentrate computational
effort on zones of strong variation of the solution. Time adaptivity is
achieved by two alternative devices, namely locally varying time stepping and a
Runge-Kutta-Fehlberg-type adaptive time integration. A series of numerical
examples demonstrates thatthese methods are efficient and sufficiently accurate
to simulate the electrical activity in myocardial tissue with affordable
effort. In addition, an optimalthreshold for discarding non-significant
information in the multiresolution representation of the solution is derived,
and the numerical efficiency and accuracy of the method is measured in terms of
CPU time speed-up, memory compression, and errors in different norms.Comment: 25 pages, 41 figure
Some numerical methods for solving stochastic impulse control in natural gas storage facilities
The valuation of gas storage facilities is characterized as a stochastic impulse control problem with finite horizon resulting in Hamilton-Jacobi-Bellman (HJB) equations for the value function. In this context the two catagories of solving schemes for optimal switching are discussed in a stochastic control framework. We reviewed some numerical methods which include approaches related to partial differential equations (PDEs), Markov chain approximation, nonparametric regression, quantization method and some practitioners’ methods. This paper considers optimal switching problem arising in valuation of gas storage contracts for leasing the storage facilities, and investigates the recent developments as well as their advantages and disadvantages of each scheme based on dynamic programming principle (DPP
Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients
We consider elliptic partial differential equations with diffusion
coefficients that depend affinely on countably many parameters. We study the
summability properties of polynomial expansions of the function mapping
parameter values to solutions of the PDE, considering both Taylor and Legendre
series. Our results considerably improve on previously known estimates of this
type, in particular taking into account structural features of the affine
parametrization of the coefficient. Moreover, the results carry over to more
general Jacobi polynomial expansions. We demonstrate that the new bounds are
sharp in certain model cases and we illustrate them by numerical experiments
Error analysis of truncated expansion solutions to high-dimensional parabolic PDEs
We study an expansion method for high-dimensional parabolic PDEs which
constructs accurate approximate solutions by decomposition into solutions to
lower-dimensional PDEs, and which is particularly effective if there are a low
number of dominant principal components. The focus of the present article is
the derivation of sharp error bounds for the constant coefficient case and a
first and second order approximation. We give a precise characterisation when
these bounds hold for (non-smooth) option pricing applications and provide
numerical results demonstrating that the practically observed convergence speed
is in agreement with the theoretical predictions
Adaptive Multiresolution Methods for the Simulation ofWaves in Excitable Media
We present fully adaptive multiresolution methods for a class of spatially two-dimensional reaction-diffusion systems which describe excitable media and often give rise to the formation of spiral waves. A novel model ingredient is a strongly degenerate diffusion term that controls the degree of spatial coherence and serves as a mechanism for obtaining sharper wave fronts. The multiresolution method is formulated on the basis of two alternative reference schemes, namely a classical finite volume method, and Barkley's approach (Barkley in Phys. D 49:61-70, 1991), which consists in separating the computation of the nonlinear reaction terms from that of the piecewise linear diffusion. The proposed methods are enhanced with local time stepping to attain local adaptivity both in space and time. The computational efficiency and the numerical precision of our methods are assessed. Results illustrate that the fully adaptive methods provide stable approximations and substantial savings in memory storage and CPU time while preserving the accuracy of the discretizations on the corresponding finest uniform gri
Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux
A fully adaptive finite volume multiresolution scheme for one-dimensional
strongly degenerate parabolic equations with discontinuous flux is presented.
The numerical scheme is based on a finite volume discretization using the
Engquist--Osher approximation for the flux and explicit time--stepping. An
adaptivemultiresolution scheme with cell averages is then used to speed up CPU
time and meet memory requirements. A particular feature of our scheme is the
storage of the multiresolution representation of the solution in a dynamic
graded tree, for the sake of data compression and to facilitate navigation.
Applications to traffic flow with driver reaction and a clarifier--thickener
model illustrate the efficiency of this method
Innovative Approaches to the Numerical Approximation of PDEs
This workshop was about the numerical solution of PDEs for which classical
approaches,
such as the finite element method, are not well suited or need further
(theoretical) underpinnings.
A prominent example of PDEs for which classical methods are not well
suited are PDEs posed in high space dimensions.
New results on low rank tensor approximation for those problems were
presented.
Other presentations dealt with regularity of PDEs, the numerical solution
of PDEs on surfaces,
PDEs of fractional order, numerical solvers for PDEs that converge with
exponential rates, and
the application of deep neural networks for solving PDEs
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