291 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
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
A Meshfree Solver for the MEG Forward Problem
Noninvasive estimation of brain activity via magnetoencephalography (MEG) involves an inverse problem whose solution requires an accurate and fast forward solver. To this end, we propose the Method of Fundamental Solutions (MFS) as a meshfree alternative to the Boundary Element Method (BEM). The solution of the MEG forward problem is obtained, via the Method of Particular
Solutions (MPS), by numerically solving a boundary value problem for the electric scalar potential, derived from the quasi-stationary approximation of Maxwell’s equations. The magnetic field is then computed by the Biot-Savart law. Numerical experiments have been carried out in a realistic single-shell head geometry. The proposed solver is compared with a state-of-the-art BEM solver. A good agreement and a reduced computational load show the attractiveness of the meshfree approach
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}
The Challenge of Machine Learning in Space Weather Nowcasting and Forecasting
The numerous recent breakthroughs in machine learning (ML) make imperative to
carefully ponder how the scientific community can benefit from a technology
that, although not necessarily new, is today living its golden age. This Grand
Challenge review paper is focused on the present and future role of machine
learning in space weather. The purpose is twofold. On one hand, we will discuss
previous works that use ML for space weather forecasting, focusing in
particular on the few areas that have seen most activity: the forecasting of
geomagnetic indices, of relativistic electrons at geosynchronous orbits, of
solar flares occurrence, of coronal mass ejection propagation time, and of
solar wind speed. On the other hand, this paper serves as a gentle introduction
to the field of machine learning tailored to the space weather community and as
a pointer to a number of open challenges that we believe the community should
undertake in the next decade. The recurring themes throughout the review are
the need to shift our forecasting paradigm to a probabilistic approach focused
on the reliable assessment of uncertainties, and the combination of
physics-based and machine learning approaches, known as gray-box.Comment: under revie
Bayesian Probabilistic Numerical Methods in Time-Dependent State Estimation for Industrial Hydrocyclone Equipment
The use of high-power industrial equipment, such as large-scale mixing equipment or a hydrocyclone for separation of particles in liquid suspension, demands careful monitoring to ensure correct operation. The fundamental task of state-estimation for the liquid suspension can be posed as a time-evolving inverse problem and solved with Bayesian statistical methods. In this article, we extend Bayesian methods to incorporate statistical models for the error that is incurred in the numerical solution of the physical governing equations. This enables full uncertainty quantification within a principled computation-precision trade-off, in contrast to the over-confident inferences that are obtained when all sources of numerical error are ignored. The method is cast within a sequential Monte Carlo framework and an optimized implementation is provided in Python
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