An explicit algorithm for the minimization of an ℓ 1 penalized least squares functional, with non-separable ℓ 1 term, is proposed. Each step in the iterative algorithm requires four matrix vector multiplications and a single simple projection on a convex set (or equivalently thresholding). Convergence is proven and a 1/N convergence rate is derived for the functional. In the special case where the matrix in the ℓ 1 term is the identity (or orthogonal), the algorithm reduces to the traditional iterative soft-thresholding algorithm. In the special case where the matrix in the quadratic term is the identity (or orthogonal), the algorithm reduces to a gradient projection algorithm for the dual problem.By replacing the projection with a simple proximity operator, other convex nonseparable penalties than those based on an ℓ 1 -norm can be handled as well.

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We consider the cubic and quartic Hénon-Heiles Hamiltonians with additional inverse square terms, which pass the Painlevé test for only seven sets of coefficients. For all the not yet integrated cases we prove the singlevaluedness of the general solution. The seven Hamiltonians enjoy two properties: meromorphy of the general solution, which is hyperelliptic with genus two and completeness in the Painlevé sense (impossibility to add any term to the Hamiltonian without destroying the Painlevé property).

The quartic Hénon-Heiles Hamiltonian passes the Painlevé test for only four sets of values of the constants. Only one of these, identical to the traveling-wave reduction of the Manakov system, has been explicitly integrated (Wojciechowski, 1985), while the other three have not yet been integrated in the general case (α, β, γ) = (0, 0, 0). We integrate them by building a birational transformation to two fourth-order firstdegree equations in the Cosgrove classification of polynomial equations that have the Painlevé property. This transformation involves the stationary reduction of various partial differential equations. The result is the same as for the three cubic Hénon-Heiles Hamiltonians, namely, a general solution that is meromorphic and hyperelliptic with genus two in all four quartic cases. As a consequence, no additional autonomous term can be added to either the cubic or the quartic Hamiltonians without destroying the Painlevé integrability (the completeness property).

The generalized Hénon-Heiles Hamiltonian H = 1/2(P 2 X + P 2 Y + c 1 X 2 + c 2 Y 2 ) + aXY 2 − bX 3 /3 with an additional nonpolynomial term µY −2 is known to be Liouville integrable for three sets of values of (b/a, c 1 , c 2 ). It has been previously integrated by genus two theta functions only in one of these cases. Defining the separating variables of the Hamilton-Jacobi equations, we succeed here, in the two other cases, to integrate the equations of motion with hyperelliptic functions. 1

We consider an integrable scalar partial differential equation (PDE) that is second order in time. By rewriting it as a system and applying the Wahlquist-Estabrook prolongation algebra method, we obtain the zero curvature representation of the equation, which leads to a Lax representation in terms of an energy-dependent Schrödinger spectral problem of the type studied by Antonowicz and Fordy. The solutions of this PDE system, and of its associated hierarchy of commuting flows, display weak Painlevé behaviour, i.e. they have algebraic branching. By considering the travelling wave solutions of the next flow in the hierarchy, we find an integrable perturbation of the case (ii) Hénon-Heiles system which has the weak Painlevé property. We perform separation of variables for this generalized Hénon-Heiles system, and describe the corresponding solutions of the PDE.

A qualitative comparison of total variation like penalties (total variation, Huber variant of total variation, total generalized variation, . . . ) is made in the context of global seismic tomography. Both penalized and constrained formulations of seismic recovery problems are treated. A number of simple iterative recovery algorithms applicable to these problems are described. The convergence speed of these algorithms is compared numerically in this setting. For the constrained formulation a new algorithm is proposed and its convergence is proven.