25,900 research outputs found
A continuous analogue of the tensor-train decomposition
We develop new approximation algorithms and data structures for representing
and computing with multivariate functions using the functional tensor-train
(FT), a continuous extension of the tensor-train (TT) decomposition. The FT
represents functions using a tensor-train ansatz by replacing the
three-dimensional TT cores with univariate matrix-valued functions. The main
contribution of this paper is a framework to compute the FT that employs
adaptive approximations of univariate fibers, and that is not tied to any
tensorized discretization. The algorithm can be coupled with any univariate
linear or nonlinear approximation procedure. We demonstrate that this approach
can generate multivariate function approximations that are several orders of
magnitude more accurate, for the same cost, than those based on the
conventional approach of compressing the coefficient tensor of a tensor-product
basis. Our approach is in the spirit of other continuous computation packages
such as Chebfun, and yields an algorithm which requires the computation of
"continuous" matrix factorizations such as the LU and QR decompositions of
vector-valued functions. To support these developments, we describe continuous
versions of an approximate maximum-volume cross approximation algorithm and of
a rounding algorithm that re-approximates an FT by one of lower ranks. We
demonstrate that our technique improves accuracy and robustness, compared to TT
and quantics-TT approaches with fixed parameterizations, of high-dimensional
integration, differentiation, and approximation of functions with local
features such as discontinuities and other nonlinearities
Profile extrema for visualizing and quantifying uncertainties on excursion regions. Application to coastal flooding
We consider the problem of describing excursion sets of a real-valued
function , i.e. the set of inputs where is above a fixed threshold. Such
regions are hard to visualize if the input space dimension, , is higher than
2. For a given projection matrix from the input space to a lower dimensional
(usually ) subspace, we introduce profile sup (inf) functions that
associate to each point in the projection's image the sup (inf) of the function
constrained over the pre-image of this point by the considered projection.
Plots of profile extrema functions convey a simple, although intrinsically
partial, visualization of the set. We consider expensive to evaluate functions
where only a very limited number of evaluations, , is available, e.g.
, and we surrogate with a posterior quantity of a Gaussian process
(GP) model. We first compute profile extrema functions for the posterior mean
given evaluations of . We quantify the uncertainty on such estimates by
studying the distribution of GP profile extrema with posterior
quasi-realizations obtained from an approximating process. We control such
approximation with a bound inherited from the Borell-TIS inequality. The
technique is applied to analytical functions () and to a -dimensional
coastal flooding test case for a site located on the Atlantic French coast.
Here is a numerical model returning the area of flooded surface in the
coastal region given some offshore conditions. Profile extrema functions
allowed us to better understand which offshore conditions impact large flooding
events
Efficient Localization of Discontinuities in Complex Computational Simulations
Surrogate models for computational simulations are input-output
approximations that allow computationally intensive analyses, such as
uncertainty propagation and inference, to be performed efficiently. When a
simulation output does not depend smoothly on its inputs, the error and
convergence rate of many approximation methods deteriorate substantially. This
paper details a method for efficiently localizing discontinuities in the input
parameter domain, so that the model output can be approximated as a piecewise
smooth function. The approach comprises an initialization phase, which uses
polynomial annihilation to assign function values to different regions and thus
seed an automated labeling procedure, followed by a refinement phase that
adaptively updates a kernel support vector machine representation of the
separating surface via active learning. The overall approach avoids structured
grids and exploits any available simplicity in the geometry of the separating
surface, thus reducing the number of model evaluations required to localize the
discontinuity. The method is illustrated on examples of up to eleven
dimensions, including algebraic models and ODE/PDE systems, and demonstrates
improved scaling and efficiency over other discontinuity localization
approaches
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