![Results for the computation of the expected value for the solution, E[u].](https://www.researchgate.net/profile/Raul-Tempone/publication/220179808/figure/fig1/AS:340393892827136@1458167636273/Results-for-the-computation-of-the-expected-value-for-the-solution-Eu_Q320.jpg)
![Results for the computation of the variance of the solution, V ar[u].](https://www.researchgate.net/profile/Raul-Tempone/publication/220179808/figure/fig2/AS:340393892827139@1458167636478/Results-for-the-computation-of-the-variance-of-the-solution-V-aru_Q320.jpg)
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A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
Abstract and Figures
We investigate the convergence rate of approximations by finite sums of rank-1 tensors of solutions of multiparametric elliptic PDEs. Such PDEs arise, for example, in the parametric, deterministic reformulation of elliptic PDEs with random field inputs, based, for example, on the M-term truncated Karhunen–Lo`eve expansion. Our approach could be regarded as either a class of compressed approximations of these solutions or as a new class of iterative elliptic problem solvers for high-dimensional, parametric, elliptic PDEs providing linear scaling complexity in the dimension M of the parameter space. It is based on rank-reduced, tensor-formatted separable approximations of the high-dimensional tensors and matrices involved in the iterative process, combined with the use of spectrally equivalent low-rank tensor-structured preconditioners to the parametric matrices resulting from a finite element discretization of the high-dimensional parametric, deterministic problems. Numerical illustrations for the M-dimensional parametric elliptic PDEs resulting from sPDEs on parameter spaces of dimensions M ≤ 100 indicate the advantages of employing low-rank tensor-structured matrix formats in the numerical solution of such problems.
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... The representation (5.7) is called a lognormal parameterization, if the parameters σ = (σ j ) Ns j=1 are independently and identically distributed (i.i.d.) standard normal random variables, that is (σ j ) Ns j=1 ∼ P := Ns j=1 N (0, 1), see, e.g., [2]. Parameterizations of this form have origins in Karhunen-Loève expansions of lognormal random fields, see, e.g., [22]. is assumed, and we set Q = √ 10 · P F , where P F is the orthogonal projection in H onto span(F ), where F = 1 cos(πx) cos(2πx) ⊤ , and P = 1 H in (2.1), where 1 H denotes the identity operator in H, as well as the initial condition y • (x) = sin(2πx) − 1. ...
... Similarly, in the case with convection b = 0.1, the feedback (5.2) tracks the target better than the feedback (5.3): while (5.3) clearly fails to track the target for σ (2) test , σ ...
Optimal control problems of tracking type for a class of linear systems with uncertain parameters in the dynamics are investigated. An affine tracking feedback control input is obtained by considering the minimization of an energy-like functional depending on a finite ensemble of training/sample parameters. It is computed from the nonnegative definite solution of an associated differential Riccati equation. Simulations are presented showing the tracking performance of the computed input for trained as well as untrained parameters.
This work is concerned with the use of Gaussian surrogate models for Bayesian inverse problems associated with linear partial differential equations. A particular focus is on the regime where only a small amount of training data is available. In this regime the type of Gaussian prior used is of critical importance with respect to how well the surrogate model will perform in terms of Bayesian inversion. We extend the framework of Raissi et. al. (2017) to construct PDE-informed Gaussian priors that we then use to construct different approximate posteriors. A number of different numerical experiments illustrate the superiority of the PDE-informed Gaussian priors over more traditional priors.
Uncertainty quantification (UQ) is rapidly becoming a sine qua non for all forms of computational science out of which actionable outcomes are anticipated. Much of the microscopic world of atoms and molecules has remained immune to these developments but due to the fundamental problems of reproducibility and reliability, it is essential that practitioners pay attention to the issues concerned. Here a UQ study is undertaken of classical molecular dynamics with a particular focus on uncertainties in the high-dimensional force-field parameters, which affect key quantities of interest, including material properties and binding free energy predictions in drug discovery and personalized medicine. Using scalable UQ methods based on active subspaces that invoke machine learning and Gaussian processes, the sensitivity of the input parameters is ranked. Our analyses reveal that the prediction uncertainty is dominated by a small number of the hundreds of interaction potential parameters within the force fields employed. This ranking highlights what forms of interaction control the prediction uncertainty and enables systematic improvements to be made in future optimizations of such parameters.
We present a non-intrusive adaptive stochastic collocation method coupled with a data-consistent inference framework to optimize stochastic inverse problems solve in CFD. The purpose of the proposed data-consistent method is, given a model and some observed output probability density function (pdf), to build a new model input pdf which is consistent with both the model and the data. Solving stochastic inverse problems in CFD is however very costly, which is why we use a surrogate or metamodel in the data-consistent inference method. This surrogate model is built using an adaptive stochastic collocation approach based on a stochastic error estimator and simplex elements in the parameters space. The efficiency of the proposed method is evaluated on analytical test cases and two CFD configurations. The metamodel inference results are shown to be as accurate as crude Monte Carlo inferences while performing 103 less deterministic computations for smooth and discontinuous response surfaces. Moreover, the proposed method is shown to be able to reconstruct both an observed pdf on the data and a data-generating distribution in the uncertain parameter space under certain conditions.
In this paper, we apply the stochastic perturbation technique to solve the optimal control problem governed by elliptic partial differential equation with small uncertainty in the random input. We first use finite-dimensional noise assumption and perturbation technique to establish the first-order and second-order deterministic optimality systems, and then discretize the two systems by standard finite-element method. Furthermore, we derive a posteriori error estimators for the finite-element approximation of the state, co-state and control in two different norms, respectively. These error estimators are then used to build our adaptive algorithm. Finally, some numerical examples are presented to verify the effectiveness of the derived estimators.
We now return to the possible solutions X
t
(ω) of the stochastic differential equation$$\frac{{d{X_t}}}{{dt}} = b(t,{\kern 1pt} {X_t}) + \sigma (t,{\kern 1pt} {X_t}){W_t},{\kern 1pt} b(t,{\kern 1pt} x) \in R,\sigma (t,{\kern 1pt} x) \in R$$ (5.1)
where W
t
is 1-dimensional “white noise”. As discussed in Chapter III the Ito interpretation of (5.1) is that X
t
satisfies the stochastic integral equation$${X_t} = {X_0} + \int\limits_0^t {b(s,{\kern 1pt} {X_s})} ds + \int\limits_0^t {\sigma (s,{\kern 1pt} {X_s})} d{B_s}$$
or in differential form$$d{X_t} = b(t,{\kern 1pt} {X_t})dt + \sigma (t,{\kern 1pt} {X_t})d{B_t}.$$ (5.2)
Review from zbMath: Let {ξn} be a sequence on n points from [−1,+1] varying with n; let Ln(x) denote the sequence of Lagrange polynomials coinciding with a given R integrable function f(x) at the points ξn. The authors are interested in the mean convergence
limn→∞∫+1−1|f(x)−Ln(x)|pdx=0(*)
for p=2 and p=1. Let ξn be the zeros of the orthogonal polynomial pn(x) of degree n corresponding to the weight function w(x)≥μ>0. Then (*) holds with p=2. The same is true if we choose for ξn the zeros of pn(x)+Anpn−1(x)+Bnpn−2(x), where An arbitrary real, Bn≤0. If ∫+1−1w(x)dx and ∫+1−1w(x)−1dx exist and ξn is defined by the zeros of the linear combination mentioned, (*) holds with p=1. Finally the existence of a continuous function f(x) is proved for which (*) with p=2 does not hold provided that ∑nk=1∫+1−1lk(x)2dx is unbounded; here lk(x) are the fundamental polynomials of the Lagrange interpolation corresponding to the set ξn.
URL: http://www.jstor.org/stable/1968516
We introduce a stochastic model of flow through highly heterogeneous,
composite porous media that greatly improves estimates of pressure head
statistics. Composite porous media consist of disjoint blocks of
permeable materials, each block comprising a single material type.
Within a composite medium, hydraulic conductivity can be represented
through a pair of random processes: (1) a boundary process that
determines block arrangement and extent and (2) a stationary process
that defines conductivity within a given block. We obtain second-order
statistics for hydraulic conductivity in the composite model and then
contrast them with statistics obtained from a standard univariate model
that ignores the boundary process and treats a composite medium as if it
were statistically homogeneous. Next, we develop perturbation expansions
for the first two moments of head and contrast them with expansions
based on the homogeneous approximation. In most cases the bivariate
model leads to much sharper perturbation approximations than does the
usual model of flow through an undifferentiated material when both are
applied to highly heterogeneous media. We make this statement precise.
We illustrate the composite model with examples of one-dimensional flows
which are interesting in their own right and which allow us to compare
the accuracy of perturbation approximations of head statistics to exact
analytical solutions. We also show the boundary process of our bivariate
model is equivalent to the indicator functions often used to represent
composite media in Monte Carlo simulations.
We investigate the concept of dual-weighted residuals for measuring model errors in the numerical solution of nonlinear partial differential equations. The method is first derived in the case where only model errors arise and then extended to handle simultaneously model and discretization errors. We next present an adaptive model/mesh refinement procedure where both sources of error are equilibrated. Various test cases involving Poisson equations and convection diffusion-reaction equations with complex diffusion models (oscillating diffusion coefficient, nonlinear diffusion, multicomponent diffusion matrix) confirm the reliability of the analysis and the efficiency of the proposed methodology.