ǫ −m , with arbitrary m (even of order exp ǫ −δ with δ > 0 in [ 4 , 68 , 15 ]). These results
are obtained under the assumption that the frequencies are completely resonant or highly non-resonant (Diophantine-type), by using the normal form techniques near an equilibrium (this is the reason for which they only apply to small amplitude so- lutions). See [ 6 ] and references therein for general theory of normal form for PDEs. In diﬀerence with the mentioned works, the research in this paper is based on the classical averaging method for ﬁnite dimensional systems, characterizing by the exis- tence of slow-fast variables. It deals with arbitrary solution of **equation** ( 3.1.2 ) with suﬃciently smooth initial data. Also note that the non-resonance assumption ( 3.1.4 ) is signiﬁcantly weaker than those in the mentioned works.

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1. Introduction
We are interested in finite elements approximations for Cauchy problems for stochastic parabolic PDEs of the form of **equation** (2.1) below. Such kind of equations arise in various fields of sciences and engineering, for example in **nonlinear** filtering of partially observed diffusion processes. Therefore these equations have been intensively studied in the litera- ture, and theories for their solvability and numerical methods for approximations of their solutions have been developed. Since the computational effort to get reasonably accurate numerical solutions grow rapidly with the dimension d of the state space, it is important to investigate the possibility of accelerating the convergence of spatial discretisations by Richardson extrapolation. About a century ago Lewis Fry Richardson had the idea in [18] that the speed of convergence of numerical approximations, which depend on some parameter h converging to zero, can be increased if one takes appropriate linear combina- tions of approximations corresponding to different parameters. This method to accelerate the convergence, called Richardson extrapolation, works when the approximations admit a power series expansion in h at h = 0 with a remainder term, which can be estimated by a higher power of h. In such cases, taking appropriate mixtures of approximations with different parameters, one can eliminate all other terms but the zero order term and the remainder in the expansion. In this way, the order of accuracy of the mixtures is the exponent k + 1 of the power h k+1 , that estimates the remainder. For various numerical methods applied to solving deterministic **partial** **differential** equations (PDEs) it has been proved that such expansions exist and that Richardson extrapolations can spectacularly increase the speed of convergence of the methods, see, e.g., [16], [17] and [20]. Richard- son’s idea has also been applied to numerical solutions of stochastic equations. It was shown first in [21] that by Richardson extrapolation one can accelerate the weak conver- gence of Euler approximations of stochastic **differential** equations. Further results in this direction can be found in [14], [15] and the references therein. For stochastic PDEs the first result on accelerated finite difference schemes appears in [7], where it is shown that

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5.1 Introduction
Burgers’ **equation** is one of the most elaborated parabolic **partial** **differential** equations (PDEs), which involves the effects of both **nonlinear** propagation and diffusion. This PDE was originally developed for modeling a one-dimensional turbulence and has been applied to differ- ent problems arising in physics, engineering, mathematical biology, etc. (see, e.g., [98, 135]). The broad range of application of Burgers’ equations motivated extensive investigations on the control of this type of PDEs in the literature and many solutions have been developed for different problems, such as linear boundary feedback control [36, 44], backstepping con- trol [80, 92], optimal control [71], and adaptive control [93]. It should be noted that the exponential stability (locally or globally) of PDEs is an essential requirement for asymptotic tracking control [28]. However, as Burgers’ **equation** is a **nonlinear** PDE, a linear feedback control can usually achieve only a local stability, which may be a performance restriction. In [36, 41], the local exponential stability for Burgers’ **equation** is obtained by using the classical energy method under some assumptions on initial data and the nonhomogeneous terms. A **nonlinear** boundary feedback control is introduced in [78], which can achieve a global exponential stability of the Burgers’ **equation**.

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Let us recall some previous well-known results of the literature. The first examples of **nonlinear** feedbacks were only concerning feedback functions having a polynomial growth in a neighborhood of 0 (see e.g. [ 36 , 24 ] and the references therein). As far as we know, the first paper considering the case of arbitrary growing feedbacks (in a neighborhood of 0) is [ 26 ]. In this paper, the analysis is based on the existence (always true) of a concave function h satisfying h(sρ(s)) > s 2 + ρ 2 (s) for all |s| 6 N (see (1.3) in [ 26 ]). The paper is very interesting but provides only two examples of construction of such function h in Corollary 2, namely the linear and polynomial growing feedbacks. The results use only the Jensen’s inequality (not the Young’s inequality), and allow the authors to compare the decay of the energy with the decay of the solution of an ordinary **differential** **equation** S ′ (t) + q(S(t)) = 0 where q(x) = x − (I + p) −1 (x) and p(x) = (cI + h(Cx)) −1 (Kx) where c, C, K are non explicit constants and f −1 stands for the inverse function of f . In the general case, these results do not give the ways to build an explicite concave function satisfying h(sρ(s)) > s 2 + ρ 2 (s). No general energy decay rates are given in an explicit, simple and general formula, which besides this, could be shown to be "optimal". Due to this lack of explicit examples of decay rates for arbitrary growing feedbacks in other situations than the linear or polynomial cases, other results were obtained, also based on convexity arguments but through other constructions in [ 32 , 33 ] (see also [ 37 ]) through linear energy integral inequalities and in [ 30 ] through the comparison with a dissipative ordinary **differential** inequality. In both cases, optimality is not guaranteed. In particular, [ 32 , 33 ] do not allow to recover the well-known expected "optimal" energy decay rates in the case of polynomially growing feedbacks. Optimality can be shown in particular geometrical situations, in one dimension when the feedback is very weak (as for ρ(s) = e −1/s for s > 0 close to 0 for instance), see e.g. [ 42 , 5 ]. Hence the challenging questions are not only to derive energy decay rates for arbitrary growing feedbacks, but to determine whether if these decay rates are optimal, at least in finite dimensions and in some situations in the infinite dimensional case, and also to derive one-step, simple and semi-explicite formula which are valid in the general case. This is the main contribution of [ 5 , 8 ] for direct methods and of the present paper for indirect methods for the continuous as well as the discretized settings (see also [ 9 ] for the continuous setting). Note also that the direct method is valid for bounded as well as unbounded feedback operators.

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|g ′ | ≤ γ |f ′ |, with 0 ≤ γ < 1, (3a) lim
y→±∞ f (y) = ±∞. (3b)
**Equation** (1) is typically issued from hyperbolic **partial** **differential** equations with **nonlinear** boundary conditions [11, 7]. A closely related system of neu- tral equations has been derived, for instance, in the case of elastic wave propagation across two **nonlinear** cracks [14]: y is then the dilatation of the crack, the shift 1 is the normalized travel time between the cracks, f and g denote the **nonlinear** contact law, and s is the T -periodic excitation.

Probabilistic representations of **nonlinear** **Partial** **Differential** Equations (PDEs) are interesting in several aspects. From a theoretical point of view, such representations allow for probabilistic tools to study the analytic properties of the **equation** (existence and/or uniqueness of a solution, regularity,. . . ). They also have their own interest typically when they provide a microscopic interpretation of physical phenomena macroscopically drawn by a **nonlinear** PDE. Similarly, stochastic control problems are a way of interpreting non-linear PDEs through Hamilton-Jacobi-Bellman **equation** that have their own theoretical and practical interests (see [16]). Besides, from a numerical point of view, such representations allow for new approxi- mation schemes potentially less sensitive to the dimension of the state space thanks to their probabilistic nature involving Monte Carlo based methods.

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1. Motivation
Hyperbolic PDEs such as the wave **equation** are known to propagate singularities, unlike parabolic (or elliptic) PDEs, whose solutions are more regular than the corresponding data. Besides, in the context of hyperbolic PDEs, nonlinearities are responsible for the build-up of finite time singularities in the form of shock waves. Therefore, the notion of a “**nonlinear** regularizing effect” for hyperbolic PDEs may seem somewhat of a paradox.

First, we suppose the hedging instruments are modeled by a Stochastic **Differential** **Equation** (SDE) with drift µ and diffusion σ. We also consider contingent claims of
the form H T = h(X T ). Second, we suppose that the contingent claim is evaluated
exogenously by a valuation process V t = v(t, X t ) for some function v. For instance, v is

operator and H ( p ) = jpj and for convex initial sets were studied using dierent methods
by Yip [Y].
(iii) Asymptotic problems in phase transitions
We present here an example of an asymptotic problem arising in phase transitions | see [BCESS] and [BS] for an extended discussion of such problems in the deterministic setting and [LS2,3] for more general problems in random environments. The problem is about a modied Allen-Cahn **equation** of the form

Solutions of certain **nonlinear** dispersive equations present a remarkable behavior: they can be decomposed in a radiative component and a soliton component. The amplitude of the radiative component decays in time polynomially: this is due to the dispersive effect. The soliton component instead is the effect of a delicate balance between the **nonlinear** and dispersive effects, which generates one or more stable waves, called solitons. These waves maintain a constant shape while propagating, and have constant amplitude and velocity. In other words, they characterize the long time behavior of the solution. Even from such a short introduction, the importance of solitons in applications should be clear. We shall focus on the effects of noise on this special class of solutions, studying in particular their stability with respect to noise perturbations of the initial condition and the possibility of creating solitons from random initial conditions. This is done for two important equations, which are analyzed and compared: the **nonlinear** Schrödinger (NLS) **equation** and the Korteweg - de Vries (KdV) **equation**. Both equations have a central role in literature for historical reasons and for their general character. Indeed, they are used in a wide range of very different physical models, mentioned in the introduction of chapter 2. Here we only observe that a main application of the **nonlinear** Schrödinger **equation** is the modelling of the propagation of short light pulses in optical fibers, and the Korteweg - de Vries **equation** is used for example to model the propagation of water waves in a channel, and is the first **equation** that has been used to study the dynamic of solitons.

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When Λ = 0, PDEs of the type (1.1) are non-linear generalizations of the Fokker-Planck **equation**. In that case, solutions v of (1.1) are in general conservative in the sense that R R v(t, x)dx is constant in t, so equal to 1
if the initial condition is a probability measure. In particular when Φ and g do not depend on v, then previ- ous **equation** is a classical (time-dependent) Fokker-Planck type **equation**. Under reasonnable conditions on Φ and g (for instance if they are Lipschitz with linear growth or bounded continuous), then according to The- orem 5.1.1 and Corollary 6.4.4 of [17], there is a process Y which is a solution, at least in law (for any initial condition) to a SDE with diffusion (resp. drift) coefficient equal to Φ (resp. g). Indeed that solution does not explode. So Itô’s formula applied to ϕ(Y ), where ϕ is a test function, allows to show that the function ν de- fined on [0, T ] with values in the space of finite measures such that ν t is the marginal law of Y t , is a solution

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• Article [40] deals with he specific study of the Schrödinger **equation** without using transmutation techniques, in the case of a cascade system of two equations with one control force, using Carleman estimates.
• Article [4] treats the case of some linear systems of two periodic and one-dimensional non- conservative transport equations with the same speed of propagation, space-time varying cou- pling matrix and one control are also analyzed, together with some **nonlinear** variants, thanks to the fictitious control method.

In this paper, we introduce a method to get necessary and sufficient stability conditions for systems governed by 1-D **nonlinear** hyperbolic **partial**-**differential** equations with closed-loop in- tegral controllers, when the linear frequency analysis cannot be used anymore. We study the stability of a general **nonlinear** transport **equation** where the control input and the measured out- put are both located on the boundaries. The principle of the method is to extract the limiting part of the stability from the solution using a projector on a finite-dimensional space and then use a Lyapunov approach. This paper improves a result of Trinh, Andrieu and Xu, and gives an optimal condition for the design of the controller. The results are illustrated with numerical simulations where the predicted stable and unstable regions can be clearly identified.

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[28] Dovetta S., Tentarelli L., L 2 -critical NLS on noncompact metric graphs with localized nonlinearity: topological and metric features, Calc. Var. **Partial** **Differential** Equations 58 (2019), no. 3, art. num. 108, 26 pp.
[29] Dovetta S., Tentarelli L., Ground states of the L 2 -critical NLS **equation** with localized nonlinearity on a tadpole graph, accepted by Oper. Theory Adv. Appl., arXiv:1803.09246 [math.AP] (2018). [30] Duca A., Global exact controllability of the bilinear Schr¨ odinger potential type models on quantum

MULTIPOINT PROBLEM WITH EQUIDISTANT NODES FOR **PARTIAL** **DIFFERENTIAL** EQUATIONS
IRYNA KLYUS AND INNA KUDZINOVS’KA
Abstract. Correctness of the problem with multipoint conditions in time variable and frequency of the spatial coordinates for **partial** **differential** equations with shifts is investi- gated. The conditions of existence and uniqueness of the problem solution, metric theorems on lower bounds of small denominators arising in the construction of the solution of the problem are proved.

In the last decade a new geostatistical modelling paradigm based on these considerations (either explic- itly or implicitly) has been developed. It is called the SPDE Approach. It has arisen from the needs of the statistical community and not from the probabilist community. It consists in interpreting the studied variable as the realisation of a Random Function which satisfies some SPDE. Although this kind of modelling has always been done in Stochastic Analysis, it has not necessarily been grounded on the need of conveniently fitting a stochastic model to a data-set, nor by the need of interpreting statistical techniques in an analyst way. This approach has allowed many theoretical and practical developments. From the practical point of view, it allows the analysis of geostatistical models through the use of numerical tools used in the analysis of **Partial** **Differential** Equations (PDEs). PDE numerical solvers such as the Finite Element Method (FEM) or spec- tral methods can now be used to inspire new simulation and statistical inference methods of geostatistical models. All the imaginable benefits of the world of Numerical Analysis are then applicable in Geostatistics. In particular, the computing time for simulations and inference methods has been notably reduced thanks to the fast computing performance of PDE numerical solvers in some contexts. From the theoretical view- point, this approach has allowed the introduction of new geostatistical models related to SPDEs which can be added to the already known valid covariance models. In some cases, these models can present a tradi- tional physical meaning, and hence, the parameters of classical geostatistical covariance models can carry a traditional physical interpretation. A classical geostatistical parameter such as the scale, which describes roughly the spatial or spatio-temporal range, defined as the distance below which the correlation is signifi- cant enough, can be interpreted as a damping parameter. Other parameters, now considered as parameters of the associated SPDE rather than of the covariance model itself, can be also physically interpreted. This is the case for example of a velocity vector, a diffusivity coefficient or an anisotropic diffusivity matrix, a curvature coefficient, or a wave propagation velocity.

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Abstract
This dissertation presents theoretical advances in the application of the Stochastic **Partial** **Differential** Equa- tion (SPDE) approach in Geostatistics. This recently developed approach consists in interpreting a region- alised data-set as a realisation of a Random Field satisfying a SPDE. Within the theoretical framework of Generalized Random Fields with a mean-square analysis, we are able to describe with a great generality the influence of a linear SPDE over the covariance structure of its potential solutions. A criterion of existence and uniqueness of stationary solutions for a wide-class of conveniently defined linear SPDEs has been ob- tained, together with an expression for the related spectral measures. This result allows to encompass a great variety of already known relationships between stationary covariance models and SPDEs. It also allows us to obtain new stationary covariance models that are easily related to SPDEs, and to propose SPDEs for some already known covariance models such as the Stein model and the J Bessel model. We apply these results to construct spatio-temporal covariance models having non-trivial properties. By analysing evolution equations presenting an arbitrary fractional temporal derivative order, we have been able to develop non- separable models with controllable non-symmetric conditions and separate regularity over space and time. We present results concerning stationary solutions for physically inspired SPDEs such as the advection- diffusion **equation**, the Heat **equation**, some Langevin equations and the Wave **equation**. We also present developments on the resolution of a first order evolution **equation** with initial condition. We then study a method of non-conditional simulation of stationary models within the SPDE approach, following the reso- lution of the associated SPDE through a convenient PDE numerical solver. This simulation method, whose practical applications are already present in the literature, can be catalogued as a spectral method. It consists in obtaining an approximation of the Fourier Transform of the stationary Random Field, using a procedure related to the classical development on Fourier basis, and for which the computations can be efficiently ob- tained through the use of the Fast Fourier Transform. We have theoretically proved the convergence of this method in suitable weak and strong senses. We show how to apply it to numerically solve SPDEs relating the stationary models developed in this work, and we present a qualitative error analysis in the case of the Matérn model. Illustrations of models presenting non-trivial properties and related to physically driven equations are then given.

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29. P. Rapha¨ el, On the blow up phenomenon for the L 2 critical non linear Schr¨ odinger **equation**,
Lectures on **nonlinear** dispersive equations, GAKUTO Internat. Ser. Math. Sci. Appl., vol. 27, Gakk¯ otosho, Tokyo, 2006, pp. 9–61.
30. A. V. Rybin, G. G. Varzugin, M. Lindberg, J. Timonen, and R. K. Bullough, Similarity solutions and collapse in the attractive Gross-Pitaevskii **equation**, Phys. Rev. E (3) 62 (2000), no. 5, part A, 6224–6228.

RESIDUAL EQUILIBRIUM SCHEMES FOR TIME DEPENDENT **PARTIAL** **DIFFERENTIAL** EQUATIONS
LORENZO PARESCHI ∗ AND THOMAS REY †
Abstract. Many applications involve **partial** **differential** equations which admits nontrivial steady state solutions. The design of schemes which are able to describe correctly these equilibrium states may be challenging for numerical methods, in particular for high order ones. In this paper, inspired by micro-macro decomposition methods for kinetic equations, we present a class of schemes which are capable to preserve the steady state solution and achieve high order accuracy for a class of time dependent **partial** **differential** equations including **nonlinear** diffusion equations and kinetic equations. Extension to systems of conservation laws with source terms are also discussed.

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Modeling complex biological phenomenons with ordinary **differential** equations (ODEs) some- times involves a large number of variables and parameters, which makes the analysis of these models cumbersome. Modeling such systems with delay **differential** equations (DDEs) helps to describe the important dynamics with fewer variables and parameters [44]. Another advantage of using a DDE model over an ODE model is that the parameters in the DDE model usually have a direct biological interpretation [4]. Thus, providing Monolix and MlxPlore with a DDE solver is of a major importance.