475 research outputs found
Approximation of Stochastic Partial Differential Equations by a Kernel-based Collocation Method
In this paper we present the theoretical framework needed to justify the use
of a kernel-based collocation method (meshfree approximation method) to
estimate the solution of high-dimensional stochastic partial differential
equations (SPDEs). Using an implicit time stepping scheme, we transform
stochastic parabolic equations into stochastic elliptic equations. Our main
attention is concentrated on the numerical solution of the elliptic equations
at each time step. The estimator of the solution of the elliptic equations is
given as a linear combination of reproducing kernels derived from the
differential and boundary operators of the SPDE centered at collocation points
to be chosen by the user. The random expansion coefficients are computed by
solving a random system of linear equations. Numerical experiments demonstrate
the feasibility of the method.Comment: Updated Version in International Journal of Computer Mathematics,
Closed to Ye's Doctoral Thesi
An Improved Solver for the M/EEG Forward Problem
Noninvasive investigation of the brain activity via
electroencephalography (EEG) and magnetoencephalography
(MEG) involves a typical inverse problem whose solution process
requires an accurate and fast forward solver. We propose the
Method of Fundamental Solutions (MFS) as a truly meshfree
alternative to the Boundary Element Method (BEM) for solving
the M/EEG forward problem. The solution of the forward
problem is obtained, via the Method of Particular Solutions
(MPS), by numerically solving a set of coupled boundary value
problems for the 3D Laplace equation. Numerical accuracy and
computational load are investigated for spherical geometries and
comparisons with a state-of-the-art BEM solver shows that the
proposed method is competitive
IL METODO DELLE SOLUZIONI FONDAMENTALI PER LA SOLUZIONE DEL PROBLEMA DIRETTO M/EEG
The research already started on the mesh-free solution of the M / EEG direct problem has led to the development of a solver based on the method of fundamental solutions (MFS, method of fundamental solutions) able to manage the physical-geometric complexity of realistic models of the head more efficiently than traditional
ADVANCED BIO-ELECTROMAGNETIC NUMERICAL MODELLING AND ICT FOR HUMAN BRAIN RESEARCH
Functional imaging is used in the research area
neurological, neurophysiology and cognitive psychology, for the diagnosis of diseases
metabolic and for the detection of thin / squamous lesions (eg Alzheimer's disease) and for
the development of neural interfaces (brain-computer interfaces - BCI)
STIMA DEL POTENZIALE ELETTRICO IN tDCS CON APPROCCIO MESHLESS INNOVATIVO
Transcranial DC stimulation (transcranial Direct Current Stimulation,
tDCS) is a non-invasive technique aimed at modifying neuronal activity for the purpose
therapeutic and / or for the improvement of mental performance. A continuous current of entity
modest (below the threshold of perception) is injected into the brain via electrodes placed on the
scalp surface to produce changes in long-term cortical activity.
Despite the increasing use of this and other similar techniques, and the relevant ones
applications - for example in the field of neuropsychological rehabilitation - their impact
on neuronal activity is not yet fully known, mainly due to the difficulty of
predict the spatial distribution of the current within the brain, and to determine the
optimal position and size of the electrodes
Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Distributional Operators
In this paper we introduce a generalized Sobolev space by defining a
semi-inner product formulated in terms of a vector distributional operator
consisting of finitely or countably many distributional operators
, which are defined on the dual space of the Schwartz space. The types of
operators we consider include not only differential operators, but also more
general distributional operators such as pseudo-differential operators. We
deduce that a certain appropriate full-space Green function with respect to
now becomes a conditionally positive
definite function. In order to support this claim we ensure that the
distributional adjoint operator of is
well-defined in the distributional sense. Under sufficient conditions, the
native space (reproducing-kernel Hilbert space) associated with the Green
function can be isometrically embedded into or even be isometrically
equivalent to a generalized Sobolev space. As an application, we take linear
combinations of translates of the Green function with possibly added polynomial
terms and construct a multivariate minimum-norm interpolant to data
values sampled from an unknown generalized Sobolev function at data sites
located in some set . We provide several examples, such
as Mat\'ern kernels or Gaussian kernels, that illustrate how many
reproducing-kernel Hilbert spaces of well-known reproducing kernels are
isometrically equivalent to a generalized Sobolev space. These examples further
illustrate how we can rescale the Sobolev spaces by the vector distributional
operator . Introducing the notion of scale as part of the
definition of a generalized Sobolev space may help us to choose the "best"
kernel function for kernel-based approximation methods.Comment: Update version of the publish at Num. Math. closed to Qi Ye's Ph.D.
thesis (\url{http://mypages.iit.edu/~qye3/PhdThesis-2012-AMS-QiYe-IIT.pdf}
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