10 research outputs found
Sparse Compression of Expected Solution Operators
We show that the expected solution operator of prototypical linear elliptic
partial differential operators with random coefficients is well approximated by
a computable sparse matrix. This result is based on a random localized
orthogonal multiresolution decomposition of the solution space that allows both
the sparse approximate inversion of the random operator represented in this
basis as well as its stochastic averaging. The approximate expected solution
operator can be interpreted in terms of classical Haar wavelets. When combined
with a suitable sampling approach for the expectation, this construction leads
to an efficient method for computing a sparse representation of the expected
solution operator
Sparse compression of expected solution operators
We show that the expected solution operator of prototypical linear elliptic partial differential operators with random coefficients is well approximated by a computable sparse matrix. This result is based on a random localized orthogonal multiresolution decomposition of the solution space that allows both the sparse approximate inversion of the random operator represented in this basis as well as its stochastic averaging. The approximate expected solution operator can be interpreted in terms of classical Haar wavelets. When combined with a suitable sampling approach for the expectation, this construction leads to an efficient method for computing a sparse representation of the expected solution operator
A probabilistic finite element method based on random meshes: Error estimators and Bayesian inverse problems
We present a novel probabilistic finite element method (FEM) for the solution
and uncertainty quantification of elliptic partial differential equations based
on random meshes, which we call random mesh FEM (RM-FEM). Our methodology
allows to introduce a probability measure on standard piecewise linear FEM. We
present a posteriori error estimators based uniquely on probabilistic
information. A series of numerical experiments illustrates the potential of the
RM-FEM for error estimation and validates our analysis. We furthermore
demonstrate how employing the RM-FEM enhances the quality of the solution of
Bayesian inverse problems, thus allowing a better quantification of numerical
errors in pipelines of computations
Probabilistic Gradients for Fast Calibration of Differential Equation Models
Calibration of large-scale differential equation models to observational or
experimental data is a widespread challenge throughout applied sciences and
engineering. A crucial bottleneck in state-of-the art calibration methods is
the calculation of local sensitivities, i.e. derivatives of the loss function
with respect to the estimated parameters, which often necessitates several
numerical solves of the underlying system of partial or ordinary differential
equations. In this paper we present a new probabilistic approach to computing
local sensitivities. The proposed method has several advantages over classical
methods. Firstly, it operates within a constrained computational budget and
provides a probabilistic quantification of uncertainty incurred in the
sensitivities from this constraint. Secondly, information from previous
sensitivity estimates can be recycled in subsequent computations, reducing the
overall computational effort for iterative gradient-based calibration methods.
The methodology presented is applied to two challenging test problems and
compared against classical methods
Sparse operator compression of higher-order elliptic operators with rough coefficients
We introduce the sparse operator compression to compress a self-adjoint
higher-order elliptic operator with rough coefficients and various boundary
conditions. The operator compression is achieved by using localized basis
functions, which are energy-minimizing functions on local patches. On a regular
mesh with mesh size , the localized basis functions have supports of
diameter and give optimal compression rate of the solution
operator. We show that by using localized basis functions with supports of
diameter , our method achieves the optimal compression rate of
the solution operator. From the perspective of the generalized finite element
method to solve elliptic equations, the localized basis functions have the
optimal convergence rate for a th-order elliptic problem in the
energy norm. From the perspective of the sparse PCA, our results show that a
large set of Mat\'{e}rn covariance functions can be approximated by a rank-
operator with a localized basis and with the optimal accuracy
Universal Scalable Robust Solvers from Computational Information Games and fast eigenspace adapted Multiresolution Analysis
We show how the discovery of robust scalable numerical solvers for arbitrary
bounded linear operators can be automated as a Game Theory problem by
reformulating the process of computing with partial information and limited
resources as that of playing underlying hierarchies of adversarial information
games. When the solution space is a Banach space endowed with a quadratic
norm , the optimal measure (mixed strategy) for such games (e.g. the
adversarial recovery of , given partial measurements with
, using relative error in -norm as a loss) is a
centered Gaussian field solely determined by the norm , whose
conditioning (on measurements) produces optimal bets. When measurements are
hierarchical, the process of conditioning this Gaussian field produces a
hierarchy of elementary bets (gamblets). These gamblets generalize the notion
of Wavelets and Wannier functions in the sense that they are adapted to the
norm and induce a multi-resolution decomposition of that is
adapted to the eigensubspaces of the operator defining the norm .
When the operator is localized, we show that the resulting gamblets are
localized both in space and frequency and introduce the Fast Gamblet Transform
(FGT) with rigorous accuracy and (near-linear) complexity estimates. As the FFT
can be used to solve and diagonalize arbitrary PDEs with constant coefficients,
the FGT can be used to decompose a wide range of continuous linear operators
(including arbitrary continuous linear bijections from to or
to ) into a sequence of independent linear systems with uniformly bounded
condition numbers and leads to
solvers and eigenspace adapted Multiresolution Analysis (resulting in near
linear complexity approximation of all eigensubspaces).Comment: 142 pages. 14 Figures. Presented at AFOSR (Aug 2016), DARPA (Sep
2016), IPAM (Apr 3, 2017), Hausdorff (April 13, 2017) and ICERM (June 5,
2017