e-mail: Raphaele.Herbin@cmi.univ-mrs.fr - Web page: http://www.cmi.univ-mrs.fr/∼herbin
Key words: porous media, reactive transport, code coupling, **nonlinear** **conjugate** gra- dient method, preconditioning
Abstract. In the framework of the evaluation of nuclear waste disposal safety, the French Atomic Energy Commission (CEA) is interested in modelling the reactive transport in porous media. At a given time step, the equation system of reactive-transport can be writ- ten as a system of **nonlinear** coupled equations F(x) = 0. In the computational code which is presently used, this system is solved using classical sequential iterative algorithms (SIA) [2]. We are currently investigating **nonlinear** **conjugate** **gradient** **methods** to improve the resolution of the system F(x) = 0 where x is the discrete unknown. Indeed, the handling of the coupling is improved by numerical derivation along the descent direction. The orig- inal feature of this method is the use of an explicit formula for the descent parameter. We choose an approach involving two distinct codes, that is one code for the chemistry and one code for the transport equations.

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The difficulty of solving the system depends on the discretization scheme used. For localized basis schemes like the Finite Element Method, H is sparse, and efficient direct **methods** for large scale sparse matrices can be used [39]. For the Fourier pseudo-spectral scheme, which we use in this paper, H is not sparse, and only matrix-vector products are available efficiently through FFTs (a matrix-free problem). This means that the system has to be solved using an iterative method. Since it is a symmetric but indefinite problem, the solver of choice is MINRES [18], although the solver BICGSTAB has been used [7, 9]. The number of iterations of this solver will then grow when the grid spacing tends to zero, which shows that BE also has a CFL-like limitation. However, as is well-known, Krylov **methods** [39] only depend on the square root of the condition number for their convergence, as opposed to the condition number itself for fixed-point type **methods** [15, 46]. This explains why BE with a Krylov solver is preferred to FE in practice [7, 9].

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When a differentiable regularizer, such as a Tikhonov penalty term, is considered, one can address (2) by means of a **nonlinear** **conjugate** **gradient** method [5], which is particularly suited for least squares problems. However, the computation of the line search parameter at each iteration may require several evaluations of both the function and its **gradient** in order to ensure convergence [4], which significantly increases computational time when such evaluations are expensive, as is the case of the DIC functional. Furthermore, a **conjugate** **gradient** method can not handle the presence of a non differentiable regularizer, such as the total variational functional.

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In this paper we focus on **Conjugate** **Gradient** (CG) [16], a Krylov projection method for symmetric (Hermitian) positive definite matrices. We discuss two new versions of **Conjugate** **Gradient** (section 3). The first method, multiple search direction with orthogonalization CG (MSDO-CG), is an adapted version of MSD-CG [14] with the A-orthonormalization of the search directions to obtain a projection method that guarentees convergence at least as fast as CG. The second projection method that we propose here, long recurrence enlarged CG (LRE-CG), is similar to GMRES in that we build an orthonormal basis for the enlarged Krylov subspace rather than finding search directions. Then, we use the whole basis to update the solution and the residual. Both **methods** converge faster than CG in terms of iterations, but LRE-CG converges faster than MSDO-CG since it uses the whole basis to update the solution rather than only t search directions. And the more subdomains are introduced or the larger t is, the faster is the convergence of both **methods** with respect to CG in terms of iterations. For example, for t “ 64 the MSDO-CG and LRE-CG **methods** converge in 75% up to 98% less iteration with respect to CG for the different test matrices. But increasing t also means increasing the memory requirements. Thus, in practice, t should be relatively small, depending on the available memory, on the size of the matrix, and on the number of iterations needed for convergence, as explained in section 4. We also present the parallel algorithms along with their expected performance based on the estimated run times, and the preconditioned versions with their convergence behavior.

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Figure 1. Star-graph with N = 6 edges
attractive Dirac type interaction at the vertex, Adami, Cacciapuoti, Finco and Noja [4, 5] established under a mass condition and for sub-critical nonlinearities the existence of a (local or global) minimizer of the energy at fixed mass, with an explicit formula for the minimizer (see Section 2 for more details and explanations). For more general **nonlinear** quantum graphs, Adami, Serra and Tilli [9, 10, 11] have focused on the case of Kirchhoff-Neumann boundary conditions for non-compact connected metric graphs with a finite number of edges and vertices. In particular, they obtained a topological condition (see Assumption 2.3 (H)) under which no ground state exists. On the other hand, in some cases metric, properties of the graph and the value of the mass constraint influence the existence or non-existence of the ground state [10, 11].

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[7] M. J. Esteban., P. L. Felmer and A. Quaas, Superlinear elliptic equations for fully **nonlinear** operators without growth restrictions for the data, Proc. Edinb. Math. Soc. 53 (2010), 125–141
[8] G. Galise and A. Vitolo, Viscosity Solutions of Uniformly Elliptic Equations with- out Boundary and Growth Conditions at Infinity, Int. J. Differ. Equations, vol. 2011, Article ID 453727, 18 pages, 2011. doi:10.1155/2011/453727

A detailed review of these results as well as announcement of new results can be found in [ 1 ]. The minimizer is in fact explicitly known, and we use its explicit form in Section 5.1 to compare the outcome of our numerical experiences with the theoretical ground states.
2.2.2. General non-compact graphs with Kirchhoff condition. The existence of ground states with prescribed mass for the focusing **nonlinear** Schr¨ odinger equation on non-compact graphs G equipped with Kirchhoff bound- ary conditions is linked to the topology of the graph. Actually, a topological hypothesis, usually referred to as Assumption (H) can prevent a graph from having ground states for every value of the mass (see [ 12 ] for a review). For the sake of clarity, we recall that a trail in a graph is a path made of adjacent edges, in which every edge is run through exactly once. In a trail, vertices can be run through more than once. The Assumption (H) has many formulations (see [ 12 ]) but we give here only the following one.

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Optimization algorithms are at the heart of neural network design in deep learning. One of the most studied procedures is the fixed step Stochastic **Gradient** Descent (SGD) [1]; although very robust, SGD may converge too slow for small learning rates or become unstable if the learning rate is too large. Each problem having its own optimal learning rate, there is no general recipe to adapt it automatically; to address this issue, several approaches have been put forward among which the use of momentum [14], Adam [4], RMSprop [16] and so on. On the other hand, recent research efforts have been directed towards finding, heuristically or theoretically, the best learning rate [11, 13, 12] with [17], which uses an estimate of the Lipschitz constant, being a very recent example.

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1 Introduction
Modern control-command software for industrial systems are becoming more and more complex to design. On the one side, the description of the physical system that must be controlled, a power plant for instance, is frequently done using partial differential equations or **nonlinear** ordinary differential equations, whose number can grow very fast when one tries to have a precise model. On the other side, the complexity of the controller also increases when one wants it to be precise and efficient. In particular, adaptive controllers (which embed information on the plant dynamics) are more and more used: for such systems, the controller may need to compute approximations of the plant evolution using a look-up table or a simple approximation scheme as in [5]. As an extreme example, consider a controller for an air conditioning device in a car. In order to correctly and pleasantly regulate the temperature in the car, the controller takes information from the temperature of the engine but also from the outside temperature and the sunshine on the car. Based on these data, it acts on a cooling device, which is usually made of a hot and a cold fluid circuit, and is thus described using usual equations in fluid dynamics, which are given by high dimensional **nonlinear** differential equations.

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. (A.25)
A.2.5 Algorithms for linearly modulated signals
Pattern recognition algorithms depend on features that can be extracted from modulated re- ceived signals. For linear modulation recognition, features extracted from the instantaneous amplitude and phase of the received signal were exploited . The variance of the absolute value of the normalized-centered instantaneous amplitude was used to differentiate between 2ASK and 4ASK [ AN96a ; AN96b ; NA97 ; NA98 ; WN01 ]. For PSK signals, the phase PDF is multimodal, thus the number of modes provides information for PSK order identifica- tion [ YS91a ; SH92 ; YS91b ; YS95 ; YS97 ; YL98 ]. In the high-SNR region, MPSK exhibits M distinct modes. However with the decrease of SNR or the increase of M , the peaks smear off and the PDF converges to a uniform PDF [ SH92 ]. An approximation using the Tikhonov PDF and a Fourier series expansion of the phase PDF with a log-likelihood ratio test were employed [ YS91a ; YS97 ; YL98 ]. By using these **methods** to compute the phase PDF, closed-form expressions for the phase statistical moments were derived, and pdfs of the sample estimates of the moments were used for decision making [ YS91a ; SH92 ; YS95 ]. The histogram of phase difference between two adjacent symbols was compared against a particular pattern, for PSK signal identification in [ Lie84 ; HS89 ; HS90 ]. The DFT was ap- plied to the phase histogram of the received symbols to analyze the periodic components of the phase PDF or the empirical characteristic function of the phase [ SMH95 ; SR97 ]. By ex- ploiting an additional information about the magnitude of the received signal, the algorithm in [ SR97 ] was extended to QAM signal classification. Other features such as the kurtosis of the amplitude extracted from the instantaneous amplitude and phase were investigated for PSK and QAM identification in [ TM99 ; DBG91 ; UIK00 ]. Different PSK signals give rise to different sets of peak values in the magnitude of the Haar wavelet transform, and hence, the input was classified as PSK of order M if the histogram of the peak magnitudes had M/2 to M-1 distinct modes [ HPC95 ; HPC00 ].

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Abstract : The most important part of this work deals with the mathematical analysis of numerical **methods** for the homogenization of multiple integrals widely used in **nonlinear** elasticity. These **methods** couple, at the mesoscopic scale, a heterogeneous hyperelastic material or a network of interacting bonds with, at the macroscopic scale, a **nonlinear** elasticity model. The macroscopic constitutive law is obtained by solving mesoscopic problems, either continuous or discrete. In chapters 1, 2, and 3, we introduce the mechanical models and mathematical tools we use in the sequel. In chapters 5, 6, and 7, we present a direct method for the numerical solution of the homogenized behavior of a periodic composite material at ﬁnite strains, and a general framework to study numerical homogenization **methods**. We prove the convergence of such **methods** within general hypotheses and provide a numerical corrector convergence result. We also extend the analysis to cover the cases of oversampling and windowing. In chapters 8, 9, and 10, we consider a mesoscopic model based on discrete systems of bonds. We ﬁrst study a G-closure problem for a network of conductances. In the next chapter, we prove an integral representation result for a system of interacting spins. We then address the rigorous derivation of a continuous hyperelastic model starting from a stochastic network of interacting points. We apply this result to prove the convergence of discrete models for rubber developed in mechanics. In the last chapter, we introduce a new solution method for ﬂuid structure interaction problems with three dimensional shell elements to describe the structure.

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Research Report n° 8511 — March 2014 — 26 pages
Abstract: This work aims at formulating a shape optimization problem within a multiobjective optimization framework and approximating it by means of the so-called Multiple-**Gradient** Descent Algorithm (MGDA), a **gradient**-based strategy that extends classical Steepest-Descent Method to the case of the simultaneous optimization of several criteria. We describe several variants of MGDA and we apply them to a shape optimization problem in linear elasticity using a numerical solver based on IsoGeometric Analysis (IGA). In particular, we study a multiobjective **gradient**- based method that approximates the gradients of the functionals by means of the Finite Difference Method; kriging-assisted MGDA that couples a statistical model to predict the values of the objec- tive functionals rather than actually computing them; a variant of MGDA based on the analytical gradients of the functionals extracted from the NURBS -based parametrization of the IGA solver. Some numerical simulations for a test case in computational mechanics are carried on to validate the **methods** and a comparative analysis of the results is presented.

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IEEE Transactions on Signal Processing, vol. 54, no. 7, pp. 2678-2690, July 2006
[7] A. Salamech and N. Tayem, “**Conjugate** MUSIC for non-circular sources,” submitted to IEEE Trans.
on Antennas and Propagation, 2006.
[8] N. Tayem, A. Salamech, “Angle of arrival estimation for non-circular signals,” IEEE Sarnoff Sympo-

An important asset of the method is that it captures the **gradient** e↵ect, i.e., the dependence of the strain energy on the **gradient** of strain and not just on the strain. This makes it possible to derive higher-order reduced models o↵ering the following advantages: (i ) they feature faster convergence towards the solution of the full (non-reduced) problem and (ii ) they are well-suited to the analysis of localization in slender structures. Localization is ubiquitous in slender structures, from neck formation in polymer bars under traction (G’Sell et al., 1983), to beading in cylinders made up of soft gels (Matsuo and Tanaka, 1992; Mora et al., 2010), to bulges produced by the inflation of cylindrical party balloons (Kyriakides and Chang, 1990), and to kinks in bent tape springs (Se↵en and Pellegrino, 1999). Classical reduced models depending on strain only cannot resolve the sharp interfaces that result from localization, and are mathematically ill-posed. By contrast, higher-order models capturing the dependence on the **gradient** of strain allows the interfaces to be resolved and are well-posed in the context of localization. In prior work, asymptotic 1d strain-**gradient** models have been obtained as refinements over the standard theory for linearly elastic beams (Trabucho and Via˜ no, 1996; Buannic and Cartaud, 2000), inextensible ribbons (Sadowsky, 1930; Wunderlich, 1962) and thin-walled beams (Freddi et al., 2004). The possibility of using 1d models to analyze localization in slender structures easily and accurately has emerged recently in the context of necking in bars and bulging in balloons (Audoly and Hutchinson, 2016; Lestringant and Audoly, 2018). Several other localization phenomena could be better understood if 1d models were available.

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1. Introduction
Inverse problems are common in applied physics. They consist of estimating the
parameters of a model from incomplete and noisy measurements. Examples are
tomography which is a key technology in the field of medical imaging, or identifying the targets using sonars and radars. Blind source separation, which is an active topic of research in audio-processing, also falls in this category. In this contribution, we target the localization of the brain regions whose neural activations produce electromagnetic fields measured by Magnetoencephalography (MEG) and Electroencephalography (EEG), which we will refer to collectively as M/EEG. The sources of M/EEG are current generators classically modeled by current dipoles. Given a limited number of noisy measurements of the electromagnetic fields associated to neural activity, the task is to estimate the positions and amplitudes of the sources that have generated the signals. By solving this problem, M/EEG become noninvasive **methods** for functional brain imaging with a high temporal resolution.

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1.2 Problem Statement
The goal of this work is to develop data-driven **methods** for statistical verification of **nonlinear** closed-loop systems. While analytical verification techniques provide prov- able guarantees, their restrictive modeling assumptions and conservativeness limit their utility and availability in complex **nonlinear** systems. For example, the large state and parameter spaces associated with industrial applications challenge analytical **methods** [29,30]. In many of these applications, simulation-based statistical **methods** are significantly easier and faster to perform than it is to compute an analytical so- lution, if that is even feasible [30, 31]. Some analytical **methods** are able to scale to arbitrarily complex systems; however, they typically require very restrictive or con- servative assumptions and abstractions to achieve that result. While they do produce a solution, the resulting guarantees may not reflect the full response of the actual system. Therefore, statistical verification is often a more suitable approach towards verification of arbitrary closed-loop systems with adaptive, **nonlinear**, or complex control systems.

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Then, in [9], we write a discretization scheme when the wave-field depends only on one space variable in the direction of propagation of the laser beam. The electro- magnetic field is discretized using the classical Yee scheme [3] with temporal and spatial staggered grids. To obtain a second-order scheme, the points of discretiza- tion of the polarization and the density matrix must be chosen carefully. The time- derivative of the polarization is computed with the Bloch equations and is replaced in the Maxwell equations by its expression as a function of the density matrix and the electric field. Thus, in order to compute the electric field, at each time step, we have to solve a bloc-diagonal linear system. The Bloch equations are solved using a splitting scheme: the Hamiltonian is divided into the free Hamiltonian and the Hamiltonian resulting from the interaction of the wave-field with the matter. With this scheme, we compare our model with two macroscopic models based on **nonlinear** Maxwell equations [2]. In the first one, the polarization is instantaneous, while it takes the linear and quadratic dispersions into account in the second one. We show that the Maxwell-Bloch model renders more physical effects than these macroscopic models. Indeed, with this model, we can see the saturation of the non- linearity or Raman scattering. . . Furthermore, the **nonlinear** polarization is not re- stricted to its quadratic part as it is in the macroscopic models. Every order of the nonlinearity is computed and takes the dispersion into account.

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V. G ENERALIZED C ONTINUOUS - TIME G RADIENT
P ROJECTION M ETHOD
As shown in [7, Appendix B], the continuous-time gradi- ent projection method can be obtained by taking the limit as the stepsize of the original **gradient** projection method [5], [6] is decreased to zero. This has been used successfully in the context of adaptive control to bound parameter estimates in some a priori known region in the parameter space [7, Sections 4.4, 8.4.2, and 8.5.5]. Ref. [11] presents another popular projection scheme used in adaptive control. While each has its merits, both of these **methods** are limited to projection with respect to a single constraint. This section presents a generalized continuous-time **gradient** projection method that can accommodate multiple **nonlinear** constraints. Let I n = {1, 2, . . . , n} where n is some positive integer.

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propriately adapted initial guess should outperform the currently proposed messenger-field approaches. We also note that better preconditioners have been already proposed in particular in the map-making context e.g. [6, 21]. This notwithstanding, the messenger-field approach may still be found of interest in some specific applications.
In the context of the PCG **methods**, we have found that the convergence may be sped up by an appropriate choice of the initial vector. While the gain is largely negligible for the cases with a low signal-to-noise solution, it can become significant if the solution is expected to have high signal content. We have found this e ffect particularly relevant for the map-making procedure, where we have showed that the choice of the simple binned map as the initial vector can result in a significant improvement of the map-making solver convergence.

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During the last decade, various numerical techniques for computing **nonlinear** normal modes (e.g., the harmonic balance method and the continuation of periodic solutions) have emerged to overcome the inherent limitations of analytical techniques. The development of such algorithms paves the way for the computation of NNMs of large-scale, real-life applications. However, most existing computational **methods** consider conservative systems. If the effect of weak damping can often be considered as purely parasitic, this no longer holds for systems with moderate or high damping or with **nonlinear** damping.

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