Abstract
Approximation of **elliptic** **PDEs** **with** **random** diffusion **coefficients** typically re- quires a representation of the diffusion field in terms of a sequence y = (y j ) j≥1
of scalar **random** variables. One may then apply high-dimensional approximation methods to the solution map y 7→ u(y). Although Karhunen-Lo` eve representa- tions are commonly used, it was recently shown, in the relevant case of lognormal diffusion fields, that they do not generally yield optimal approximation rates. Mo- tivated by these results, we construct wavelet-type representations of stationary Gaussian **random** fields defined on bounded domains. The size and localization properties of these wavelets are studied, and used to obtain polynomial approxima- tion results for the related **elliptic** PDE which outperform those achievable when using Karhunen-Lo` eve representations. Our construction is based on a periodic extension of the **random** field, and the expansion on the domain is then obtained by simple restriction. This makes the approach easily applicable even when the computational domain of the PDE has a complicated geometry. In particular, we apply this construction to the class of Gaussian processes defined by the family of Mat´ ern covariances.

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key words: **Elliptic** **PDEs** **with** **random** coeﬃcients, Stochastic collo- cation method, Anisotropic sparse grid, Adaptation method.
1 Introduction
The Monte Carlo method [9] is the most classical approach used to compute statistical quantities of interest depending on the solution of partial differen- tial equation **with** stochastic inputs. It consists of solving M deterministic problems, where M is the number of independent and identically distributed simulations (iid) of the parameters. The main disadvantage of this approach is its slow convergence given by the order O( √ 1

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is defined from the parameter domain U to the solution space V .
Equations of this type arise both in stochastic and deterministic modeling, depending on the nature of the parameters y j which may either be **random** or deterministic variables. In both settings,
one main computational challenge is to approximate the entire solution map y 7→ u(y) up to a prescribed accuracy, **with** reasonable computational cost. This task has been intensively studied since the 1990s, see in particular [14, 15, 20, 22] for general treatments. It becomes very challenging when the number of parameters J is large due to the curse of dimensionality. Ideally, one would like to design numerical methods that are immune to the growth of J , which in principle amounts to treat the case of countably many variables, that is,

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From the above estimate, a reasonable strategy is to choose the truncation level J of the same order as n, up to logarithmic factors.
4 Examples of **random** fields
The analysis in §2 and §3 shows that for both Galerkin and weighted least-squares methods, the total error of approximation is split into two terms, resulting from the spatial and parametric discretization, respectively. The spatial error term is controlled by the H¨ olderian regularity of the sample path of b, while the parametric error term is controlled by the size properties of the functions ψ j in the representation of the Gaussian **random** field b.

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SECOND ORDER **PDES** **WITH** DIRICHLET WHITE NOISE BOUNDARY CONDITIONS
ZDZISLAW BRZE´ ZNIAK, BEN GOLDYS, SZYMON PESZAT, AND FRANCESCO RUSSO
Abstract. In this paper we study the Poisson and heat equations on bounded and unbounded domains **with** smooth boundary **with** **random** Dirichlet boundary conditions. The main novelty of this work is a convenient framework for the analysis of such equations excited by the white in time and/or space noise on the boundary. Our approach allows us to show the existence and uniqueness of weak solutions in the space of distributions. Then we prove that the solutions can be identified as smooth functions inside the domain, and finally the rate of their blow up at the boundary is estimated. A large class of noises including Wiener and fractional Wiener space time white noise, homogeneous noise and L´evy noise is considered.

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In this work we consider the **random** discrete L 2 projection on polynomial spaces (hereafter RDP) for the approximation of scalar Quantities of Interest (QOIs) related to the solution of a Partial Differential Equation model **with** **random** input parameters. In the RDP technique the QOI is first computed for independent samples of the **random** input parameters, as in a standard Monte Carlo approach, and then the QOI is approximated by a multivariate polynomial function of the input parameters using a discrete least squares approach. We consider several examples including the Darcy equations **with** **random** permeability; the linear elasticity equations **with** **random** elastic coefficient; the Navier–Stokes equations in **random** geometries and **with** **random** fluid viscosity. We show that the RDP technique is well suited to QOIs that depend smoothly on a moderate number of **random** parameters. Our numerical tests confirm the theoretical findings in [MNvST11], which have shown that, in the case of a single uniformly distributed **random** parameter, the RDP technique is stable and optimally convergent if the number of sampling points is proportional to the square of the dimension of the polynomial space. Here optimality means that the weighted L 2 norm of the RDP error is bounded from above by the best L ∞ error achievable in the given polynomial space, up to logarithmic factors. In the case of several **random** input parameters, the numerical evidence indicates that the condition on quadratic growth of the number of sampling points could be relaxed to a linear growth and still achieve stable and optimal convergence. This makes the RDP technique very promising for moderately high dimensional uncertainty quantification.

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amplitudes become more and more sensitive to the particular geometry of the surface. Possible explanations might be numerical instabilities or energy losses due to downward scattering.
Recently, identical calculations [28] have been made, also on a 3 m diameter circular sinusoidal profile but **with** a structural wavelength of 20 cm and amplitude of 6 cm, so only differing slightly from our reference surface. Kosaka and Sakuma studied the effect of the receiver mesh radius, the discretisation of the source and receiver mesh, the minimal number of structural periods for the sample and the influence of the shape of the sample. They came to the conclusion that, for calculations up to 2000 Hz, the radius of the receiver mesh should be at least one diameter of the sample and the elevation step size for the receiver mesh should be at most 3°. They further found a minimum number of 10 structural periods to obtain scattering **coefficients** representative for infinite periodical surfaces. They finally applied the same procedure on a 3 x 3 m² square sinusoidal sample **with** the same structural wavelength and amplitude as the circular sinusoidal sample. They obtained practically identical

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nilpotent orbit theorem [4], [24] states that it degenerates along Y M ∗ into a variation of M HS **with** weight filtration W M = W (Σ i∈M N i ) shifted by m, however this result doesn’t lead directly
to a structure of mixed Hodge complex since what happens at the intersection of Y M and Y K
for two subsets M and K of I couldn’t be explained until the discovery of perverse sheaves. The M HS we are looking for cannot be obtained from Hodge complexes defined by smooth and proper varieties, so it was only after the purity theorem [2], [5], [24] and the work on perverse sheaves [2], that the weight filtration W(N ) on Ψ f (L) has been defined by the logarithm of the

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If u vanishes on a larger set, one may still hope to conclude, under some weaker condition on a, that u ≡ 0 in Ω. Such a result was obtained by B´ enilan and Brezis [6, Appendix D] (**with** a contribution by R. Jensen) in the case where a ∈ L 1 (Ω) and supp u is a compact subset of Ω. Their maximum principle has been further extended by Ancona [4], who proved Theorem 2.1 below.

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were logarithmic sweeps. **With** this choice the smallest possible time variance effect is expected. **With** MLS signals these effects would be larger [21].
To study this (short time) effect, the average of 6 impulse responses was measured for a stationary turntable but for different time intervals between measurements **with** and without door openings [14]. The reverberation times as obtained from these averaged impulse responses were normalised to the reverberation time calculated from one impulse response and are plotted in Figure 16. A small change in reverberation time can be seen due to temperature and humidity drifts when the door stays closed. However, when opening and closing the door between the measurements, the late parts of the impulse responses are more incoherent and result in a lower reverberation time after averaging.

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problem of a (representative) agent, using a control α based on her/his current private state X t at time t, and of the information brought by the common noise F t 0 , typically the condi-
tional mean E[X t |W 0 ], which represents, in the large population equilibrium interpretation,
the limit of the empirical mean of the state of all the players when their number tend to infinity from the propagation of chaos. In other words, the control α may be viewed as a semi closed-loop control, i.e., closed-sloop w.r.t. the state process, and open-loop w.r.t. the common noise W 0 , or alternatively as a F 0 -progressively measurable **random** field control

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AND APPLICATIONS TO DISCRETIZATION OF DEGENERATE PARABOLIC-**ELLIPTIC** **PDES**
B. ANDREIANOV, C. CANC` ES, AND A. MOUSSA
Abstract. We propose a discrete functional analysis result suitable for prov- ing compactness in the framework of fully discrete approximations of strongly degenerate parabolic problems. It is based on the original exploitation of a result related to compensated compactness rather than on a classical estimate on the space and time translates in the spirit of Simon (Ann. Mat. Pura Appl. 1987). Our approach allows to handle various numerical discretizations both in the space variables and in the time variable. In particular, we can cope quite easily **with** variable time steps and **with** multistep time differentiation methods like, e.g., the backward differentiation formula of order 2 (BDF2) scheme. We illustrate our approach by proving the convergence of a two-point flux Finite Volume in space and BDF2 in time approximation of the porous medium equation.

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In this correspondence we mainly consider the asymptotic distribution of the estimator of circularity **coefficients** of scalar and multidimensional complex **random** variables (RV). A particular attention is paid to rectilinear RV. After deriving new properties of the circularity **coefficients**, the maximum likelihood estimate of the circularity **coefficients** in the Gaussian case and asymptotic distribution of this estimate for arbitrary distributions are given. Finally, an illustrative example is presented in order to strengthen the obtained theoretical results.

0 (Ω)| div(ς∇v) ∈ L 2 (Ω)} ⊂ P H 1+s (Ω) (algebraically and topologically). The exponent s > 0 depends both on the geometrical setting and on the value of κς. Moreover, it can be arbitrarily close to zero. For a FE approximation of degree `, one has for h small enough ε h ≤ C h min(s,`) .
Let us make some comments regarding this analysis. First, a direct application of the Aubin-Nitsche lemma allows one to derive results of approximation of the eigenfunctions in the L 2 (Ω)-norm. Using isoparametric quadrilateral FE, one can deal **with** curved interfaces. Recently, in [1], an alternative ap- proach, based on an optimisation method, has been proposed to consider the source term problem asso- ciated **with** (1). It has the advantage of requiring no geometrical assumption on the mesh. It would be interesting to investigate if it can be employed to deal **with** the eigenvalue problem (1).

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(u, v) ∈ D 1 ∩ R 2 + , H(x, u, v) = a(x)u α+1 v β+1 ,
∗ M athematics Subject Classifications : 35 J 20 , 35 J 25 , 35 J 60 , 35 J 65 , 35 J 70 .
Key words : **Elliptic** systems , p - Laplacian , variational methods , mountain - pass Lemma , Palais - Smale condition , potential function , Moser iterative method .

However, in practice, we may be faced **with** situations where, at the same time, **coefficients** in an optimisation problem are **random** variables and their realisations are not completely known. This is the case when the optimisation problem coeffi- cients cover a set of possible scenarios (expressing variability of situations where an optimal decision is to be made), each of which is imprecisely known (for instance, precision of measured values is limited). When **random** variables take values that are known through fuzzy intervals, it leads to the concept of fuzzy **random** vari- ables, first introduced by Kwakernaak (1978). Later, other authors like Kruse and Meyer (1987), Puri and Ralescu (1986), among others studied this concept. Puri and Ralescu consider a fuzzy **random** variable as a classical one taking values on a space of fuzzy sets understood as a metric space of membership functions. Kwakernaak, as well as Kruse and Meyer, consider a fuzzy **random** variable as a function from a probability space to a set of fuzzy intervals, where the latter restrict the actual values of standard **random** variables. This is the view adopted here. Recently Couso and Dubois (2009), Couso and Sánchez (2011) proposed yet another interpretation of this concept as a conditional possibility measure dom- inating a set of conditional probabilities, and they compare it to the two other views.

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We prove in Section 3, Theorem 3.10, a uniqueness result for strong-viscosity solutions to path-dependent **PDEs** proceeding as in the finite dimensional Markovian case, i.e., by means of probabilistic methods based on the theory of backward stochastic differential equations. We also prove an existence result (Theorem 3.12) for strong-viscosity solutions in a more restrictive frame- work, which is based on the idea that a candidate solution to the path-dependent PDE is deduced from the corresponding backward stochastic differential equation. The existence proof consists in building a sequence of strict solutions (we prefer to use the term strict in place of classical, because even the notion of smooth solution can not be considered classical for path-dependent partial differential equations; indeed, all the theory is very recent) to perturbed path-dependent **PDEs** converging to our strong-viscosity solution. This regularization procedure is performed hav- ing in mind the following simple property: when the **coefficients** of the path-dependent partial differential equation are smooth enough the solution is smooth as well, i.e., the solution is strict. In the path-dependent case, smooth **coefficients** means cylindrical **coefficients**, i.e., smooth maps of integrals of regular functions **with** respect to the path, as in the statement of Theorem 3.6.

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firstname.lastname@inria.fr
Abstract—Machine learning from brain images is a central tool for image-based diagnosis and diseases characterization. Predicting behavior from functional imaging, brain decoding, analyzes brain activity in terms of the behavior that it implies. While these multivariate techniques are becoming standard brain mapping tools, like mass-univariate analysis, they entail much larger computational costs. In an time of growing data sizes, **with** larger cohorts and higher-resolutions imaging, this cost is increasingly a burden. Here we consider the use of **random** sampling and projections as fast data approximation techniques for brain images. We evaluate their prediction accuracy and computation time on various datasets and discrimination tasks. We show that the weight maps obtained after **random** sampling are highly consistent **with** those obtained **with** the whole feature space, while having a fair prediction performance. Altogether, we present the practical advantage of **random** sampling methods in neuroimaging, showing a simple way to embed back the reduced **coefficients**, **with** only a small loss of information.

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[4] H. Brezis and P.L. Lions, A note on isolated singularities for linear **elliptic** equations, Mathematical analysis and applications, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York–London, 1981, 263–266.
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