there are doubly active indices, the Jacobian used in ( 1.3 ), determined by the choice ( 1.2 ), is an element of the Clarke generalized Jacobian [ 4 ] of the function in ( 1.4 ). This description makes it natural to call Newton-min the algorithm that updates x by the formulas ( 1.2 )-( 1.3 ).
The algorithm sketched above and that we further explore in this paper can be traced back at least to the work of Aganagi´c [ 1 , 1984], who proposed a Newton-type algorithm for solving LC(M, q) when M is a hidden Z-matrix, which is a matrix that can be written M X = Y where X and Y are Z-matrices having particular properties (an M -matrix is a particular instance of hidden Z-matrix). In his approach, the nonsmooth equation ( 1.4 ) to solve is then expressed in terms of a Z-matrix associated with M ; a convergence result is established. The algorithm was then rediscovered in the form ( 1.2 )–( 1.3 ) by Bergounioux, Ito, and Kunisch [ 2 , 1999] for solving quadratic optimal control problems under the name of primal-dual active set strategy (see also [ 12 , 2008]). Hinterm¨ uller [ 10 , 2003] uses the same algorithm for solving constrained optimal control problems, shows that the method does not cycle in the case of bilateral constraints, and proves convergence in finite and infinite dimension. It is shown in [ 11 , 2003] that the algorithm can be viewed as a nonsmooth Newton method for solving ( 1.4 ), which motivates the name of the algorithm given in this note, and that it converges locally superlinearly if M is a P -matrix. Kanzow [ 13 , 2004] proved its convergence in at most n iterations when M is an M -matrix.
The last approaches are based on the non-smooth modelling and simulation framework. The two last approaches only differ in their time-integration techniques. For the modelling part, the goal is to design some ideal electrical models by means of formulations extensively used in the mathematical programming theory. To list a few of them, the models that are based on variational inequalities (VI) [ 24 ], com- plementarity problems (CP) [ 22 ], inclusions into a normal cone to a convex set [ 46 ] and non-smooth and convex optimisation problems [ 33 ] are very good candidates. The main difference with the hybrid approach is the intrinsic functional description (possibly multi-valued) of the component rather than a discrete/modal description. In Fig. 14.1 e, a complementarity modelling of the diode with possible residual cur- rent b and voltage a is depicted. It can be defined by the following complementarity condition, or inclusion into a normal cone, as
In this paper, we propose two algorithms for solving lin- ear inverse problems when the observations are corrupted by noise. A proper data fidelity term (log-likelihood) is in- troduced to reflect the statistics of the noise (e.g. Gaus- sian, Poisson). On the other hand, as a prior, the images to restore are assumed to be positive and sparsely repre- sented in a dictionary of waveforms. Piecing together the data fidelity and the prior terms, the solution to the in- verse problem is cast as the minimization of a non-smooth convex functional. We establish the well-posedness of the optimization problem, characterize the corresponding min- imizers, and solve it by means of primal and primal-dual proximal splitting algorithms originating from the field of non-smooth convex optimization theory. Experimental re- sults on deconvolution, inpainting and denoising with some comparison to prior methods are also reported.
J. Behrndt and M. Langer, Boundary value problems for elliptic partial differential operators on bounded domains, J. Funct. Anal. 243 (2007), 536-565.
P.L. Butzer, J.J. Koliha, The a-Drazin inverse and ergodic behaviour of semigroups and cosine operator functions, J. Operator theory 62; 2(2009), 297-326.
N. Khaldi, M. Benharrat, B. Messirdi, On the Spectral Boundary Value Problems and Boundary Approximate Controllability of Linear Systems. Rend. Circ. Mat. Palermo, 63 (2014) 141-153.
problem subject to linear constraints.
Recovering measures Recovering a measure given its moments is another question of interest. Typically one is given some set of moments up to order d, and the goal is to recover an atomic measure that is consistent with these moments. In the limited information setting, we wish to recover a measure given a small set of linear combinations of the moments. Such a question is also of interest in system identification. The set A in this setting is the moment curve, and its convex hull goes by several names including the Caratheodory orbitope . Discretized versions of this problem correspond to the set A being a finite number of points on the moment curve; the convex hull C( A ) is then a cyclic polytope .
5 Linear arboricity
The linear arboricity of a graph G is the least number k such that the edges of G can be partitionned into k classes, each of them being a forest of paths (the disjoints union of paths – trees of maximal degree 2). The corresponding LP is very similar to the one giving edge-disjoint spanning trees
of LLS problems, using the Frobenius norm to measure the perturbations of A. Since then many results on normwise LLS condition numbers have been published (see e.g. [2, 6, 11, 15, 16]).
It was observed in  that normwise condition numbers can lead to a loss of information since they consolidate all sensitivity information into a single number. Indeed in some cases this sensitivity can vary significantly among the different solution components (some examples for LLS are presented in [2, 22]). To overcome this issue, it was proposed the notion of “componentwise” condition numbers or condition numbers for the solution components . Note that this approach must be distinguished from the componentwise metric also applied to LLS for instance in [4, 10]. This approach was generalized by the notion of partial or subspace condition numbers where we study the conditioning of L T x with L ∈ R n×k , k
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This last benchmark problem also deals with linear elastic stability. The corresponding buckling analysis is based on the eigenvalue equation ( 12 ), which allows the formula- tion of the geometric stiffness matrix K σ to be checked once again. The test consists of a portion of a submarine hull subjected to external pressure, as illustrated in Fig. 6 . This test was previously used in [ 40 ], while in [ 41 ] the submarine was modeled in its entirety. The current analysis only considers a single ring, as described in Fig. 6 , and the geometric and material data are reported in Table 9 .
Figure 3.2: Performance comparison of dense v.s. sparse Cholesky factorization.
3.4 Mixed precision Additive Schwarz preconditioner
Motivated by accuracy reasons, many large-scale scientific applications and industrial numerical simulation codes are fully implemented in 64-bit floating-point arithmetic. On the other hand, many recent processor architectures exhibit 32-bit computational power that is significantly higher than that for 64-bit. One recent and significant example is the IBM CELL multiprocessor that is projected to have a peak performance near 256 GFlops in 32-bit and “only” 26 GFlops in 64-bit computation. More extreme and common examples are the processors that possess a SSE (streaming SIMD exten- sion) execution unit can perform either two 64-bit instructions or four 32-bit instructions in the same time. This class of chip includes for instance the IBM PowerPC G5, the AMD Opteron and the Intel Pentium. For illustration purpose, are displayed below the time and the ratio of the time to perform a 32-bit operation over the time to perform the corresponding 64-bit one on some of the basic dense kernels involved in our hybrid solver implementation. In Table 3.4 are displayed the performance of B LAS -2 (_GEMV) and B LAS -3 (_GEMM) routines for various problems sizes. The comparison