Dimensional Minimization Problem Research Articles

Sharp estimates of lowest positive Neumann eigenvalue for general indefinite Sturm-Liouville problems

Given two measures μ,ν and their total variations, we study the minimization of Neumann eigenvalues for measure differential equationdy•=y(t)dμ(t)+λydν(t). By solving the infinitely dimensional minimization problem of the lowest positive Neumann eigenvalue for the measure differential equation, we obtain the optimal lower bound of the lowest positive Neumann eigenvalue for the Sturm-Liouville problemy″=q(t)y+λm(t)y, where q(t) is a nonnegative potential function and the other potential function m(t) admits to change sign.

Asymptotic linear convergence of fully-corrective generalized conditional gradient methods

AbstractWe proposea fully-corrective generalized conditional gradient method(FC-GCG) for the minimization of the sum of a smooth, convex loss function and a convex one-homogeneous regularizer over a Banach space. The algorithm relies on the mutual update of a finite set $$\mathcal {A}_k$$Akof extremal points of the unit ball of the regularizer and of an iterate $$u_k \in {\text {cone}}(\mathcal {A}_k)$$uk∈cone(Ak). Each iteration requires the solution of one linear problem to update $$\mathcal {A}_k$$Akand of one finite dimensional convex minimization problem to update the iterate. Under standard hypotheses on the minimization problem we show that the algorithm converges sublinearly to a solution. Subsequently, imposing additional assumptions on the associated dual variables, this is improved to a linear rate of convergence. The proof of both results relies on two key observations: First, we prove the equivalence of the considered problem to the minimization of a lifted functional over a particular space of Radon measures using Choquet’s theorem. Second, the FC-GCG algorithm is connected to aPrimal-Dual-Active-Point methodon the lifted problem for which we finally derive the desired convergence rates.

Convergence of the forward-backward algorithm: beyond the worst-case with the help of geometry

We provide a comprehensive study of the convergence of the forward-backward algorithm under suitable geometric conditions, such as conditioning or Łojasiewicz properties. These geometrical notions are usually local by nature, and may fail to describe the fine geometry of objective functions relevant in inverse problems and signal processing, that have a nice behaviour on manifolds, or sets open with respect to a weak topology. Motivated by this observation, we revisit those geometric notions over arbitrary sets. In turn, this allows us to present several new results as well as collect in a unified view a variety of results scattered in the literature. Our contributions include the analysis of infinite dimensional convex minimization problems, showing the first Łojasiewicz inequality for a quadratic function associated to a compact operator, and the derivation of new linear rates for problems arising from inverse problems with low-complexity priors. Our approach allows to establish unexpected connections between geometry and a priori conditions in inverse problems, such as source conditions, or restricted isometry properties.

Minimizing spectral risk measures applied to Markov decision processes

We study the minimization of a spectral risk measure of the total discounted cost generated by a Markov Decision Process (MDP) over a finite or infinite planning horizon. The MDP is assumed to have Borel state and action spaces and the cost function may be unbounded above. The optimization problem is split into two minimization problems using an infimum representation for spectral risk measures. We show that the inner minimization problem can be solved as an ordinary MDP on an extended state space and give sufficient conditions under which an optimal policy exists. Regarding the infinite dimensional outer minimization problem, we prove the existence of a solution and derive an algorithm for its numerical approximation. Our results include the findings in Bäuerle and Ott (Math Methods Oper Res 74(3):361–379, 2011) in the special case that the risk measure is Expected Shortfall. As an application, we present a dynamic extension of the classical static optimal reinsurance problem, where an insurance company minimizes its cost of capital.

Numerical Comparison of some Crossover in Genetic Algorithm for Two Dimensional Global Minimization Problem using C++

In this paper we simulate the comparison of numerical results of the different crossover method in genetics algorithm for solving global minimization problem. In this paper, the objective function is for 2 two variables function. The genetics algorithm is run using C++ program and implemented to several benchmark test functions of global optimization. The performance of each crossover method is analyse according to the numerical results, which show that certain crossover method is better for certain benchmark test function.

Single Snapshot Super-Resolution DOA Estimation for Arbitrary Array Geometries

We address the problem of search-free direction of arrival (DOA) estimation for sensor arrays of arbitrary geometry under the challenging conditions of a single snapshot and coherent sources. We extend a method of searchfree super-resolution beamforming, originally applicable only for uniform linear arrays, to arrays of arbitrary geometry. The infinite dimensional primal atomic norm minimization problem in continuous angle domain is converted to a dual problem. By exploiting periodicity, the dual function is then represented with a trigonometric polynomial using a truncated Fourier series. A linear rule of thumb is derived for selecting the minimum number of Fourier coefficients required for accurate polynomial representation, based on the distance of the farthest sensor from a reference point. The dual problem is then expressed as a semidefinite program and solved efficiently. Finally, the searchfree DOA estimates are obtained through polynomial rooting, and source amplitudes are recovered through least squares. Simulations using circular and random planar arrays show perfect DOA estimation in noise-free cases.

Homogenization of kinetic laminates in linearized elasticity

The homogenization of kinetic laminates in the framework of time‐dependent linearized elasticity is studied from a variational point of view through the Γ‐convergence of the associated energies. The characterization of the effective coefficients is achieved by means of a finite dimensional minimization problem.

Non-orthogonal joint block diagonalization based on the LU or QR factorizations for convolutive blind source separation

This article addresses the problem of blind source separation, in which the source signals are most often of the convolutive mixtures, and moreover, the source signals cannot satisfy independent identical distribution generally. One kind of prevailing and representative approaches for overcoming these difficulties is joint block diagonalization (JBD) method. To improve present JBD methods, we present a class of simple Jacobi-type JBD algorithms based on the LU or QR factorizations. Using Jacobi-type matrices we can replace high dimensional minimization problems with a sequence of simple one-dimensional problems. The novel methods are more general i.e. the orthogonal, positive definite or symmetric matrices and a preliminary whitening stage is no more compulsorily required, and further, the convergence is also guaranteed. The performance of the proposed algorithms, compared with the existing state-of-the-art JBD algorithms, is evaluated with computer simulations and vibration experimental. The results of numerical examples demonstrate that the robustness and effectiveness of the two novel algorithms provide a significant improvement i.e., yield less convergence time, higher precision of convergence, better success rate of block diagonalization. And the proposed algorithms are effective in separating the vibration signals of convolutive mixtures.

Lithosphere rigidity by adjoint-based inversion of interseismic GPS data, application to the Western United States

While vertical motion induced by long-term geological loads is often used to estimate the flexural rigidity of the lithosphere, we intend to evaluate the shear rigidity of the lithosphere using horizontal motion. Our approach considers that the rigidity of the lithosphere may be defined as its resistance to horizontal tectonic lateral forces. In this case, a spatial distribution of an effective shear rigidity can be estimated from the analysis of the interseismic velocity fields. We consider the Western United States zone where weakly strained areas (e.g., the Sierra Nevada) are connected with areas of large strain rate (e.g. San Andreas Fault system). By inverting interseismic strain distribution measured by geodetic methods, we infer the effective shear rigidity of the lithosphere. The forward problem is defined using the equations of linear elasticity. The inversion relies on the minimization of the sum of a quadratic measure of the differences between measured and modelled velocity fields. The functional also includes regularization terms for the parameters of the model. The gradient of the functional with respect to the minimization parameters is computed using an adjoint formulation. This permits the treatment of large dimensional minimization problems. Finally, a measure of the uncertainty of our inversion is illustrated through the covariance matrix of the parameters at the optimum. The optimization chart is validated on two synthetic velocity distributions. Then, the effective shear rigidity variations of the Western United States are estimated using the CMM3 interseismic velocities. The inversion displays low effective rigidities along the San Andreas Fault system, the Mojave Desert and in the Eastern California Shear Zone, while rigid areas are found in the Sierra Nevada and in the South Basin and Range. Finally, we discuss the differences between our strain rate and rigidity maps with previously published results for the Western United States.

A Regularized Newton Method for Computing Ground States of Bose–Einstein Condensates

In this paper, we propose a regularized Newton method for computing ground states of Bose-Einstein condensates (BECs), which can be formulated as an energy minimization problem with a spherical constraint. The energy functional and constraint are discretized by either the finite difference, or sine or Fourier pseudospectral discretization schemes and thus the original infinite dimensional nonconvex minimization problem is approximated by a finite dimensional constrained nonconvex minimization problem. Then an initial solution is first constructed by using a feasible gradient type method, which is an explicit scheme and maintains the spherical constraint automatically. To accelerate the convergence of the gradient type method, we approximate the energy functional by its second-order Taylor expansion with a regularized term at each Newton iteration and adopt a cascadic multigrid technique for selecting initial data. It leads to a standard trust-region subproblem and we solve it again by the feasible gradient type method. The convergence of the regularized Newton method is established by adjusting the regularization parameter as the standard trust-region strategy. Extensive numerical experiments on challenging examples, including a BEC in three dimensions with an optical lattice potential and rotating BECs in two dimensions with rapid rotation and strongly repulsive interaction, show that our method is efficient, accurate and robust.

Towards a complete characterization of the effective elasticity tensors of mixtures of an elastic phase and an almost rigid phase

The set $GU_f$ of possible effective elastic tensors of composites built from two materials with positive definite elasticity tensors $\BC_1$ and $\BC_2=\Gd\BC_0$ comprising the set $U=\{\BC_1,\Gd\BC_0\}$ and mixed in proportions $f$ and $1-f$ is partly characterized in the limit $\Gd\to\infty$. The material with tensor $\BC_2$ corresponds to a material which (for technical reasons) is almost rigid in the limit $\Gd\to \infty$. The paper, and the underlying microgeometries, have many aspects in common with the companion paper "On the possible effective elasticity tensors of 2-dimensional printed materials". The chief difference is that one has a different algebraic problem to solve: determining the subspaces of stress fields for which the thin walled structures can be rigid, rather than determining, as in the companion paper, the subspaces of strain fields for which the thin walled structure is compliant. Recalling that $GU_f$ is completely characterized through minimums of sums of energies, involving a set of applied strains, and complementary energies, involving a set of applied stresses, we provide descriptions of microgeometries that in appropriate limits achieve the minimums in many cases. In these cases the calculation of the minimum is reduced to a finite dimensional minimization problem that can be done numerically. Each microgeometry consists of a union of walls in appropriate directions, where the material in the wall is an appropriate $p$-mode material, that is almost rigid to $6-p\leq 5$ independent applied stresses, yet is compliant to any strain in the orthogonal space. Thus the walls, by themselves, can support stress with almost no deformation. The region outside the walls contains "Avellaneda material" that is a hierarchical laminate which minimizes an appropriate sum of elastic energies.

On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials

The set $GU_f$ of possible effective elastic tensors of composites built from two materials with elasticity tensors $\BC_1>0$ and $\BC_2=0$ comprising the set $U=\{\BC_1,\BC_2\}$ and mixed in proportions $f$ and $1-f$ is partly characterized. The material with tensor $\BC_2=0$ corresponds to a material which is void. (For technical reasons $\BC_2$ is actually taken to be nonzero and we take the limit $\BC_2\to 0$). Specifically, recalling that $GU_f$ is completely characterized through minimums of sums of energies, involving a set of applied strains, and complementary energies, involving a set of applied stresses, we provide descriptions of microgeometries that in appropriate limits achieve the minimums in many cases. In these cases the calculation of the minimum is reduced to a finite dimensional minimization problem that can be done numerically. Each microgeometry consists of a union of walls in appropriate directions, where the material in the wall is an appropriate $p$-mode material, that is easily compliant to $p\leq 5$ independent applied strains, yet supports any stress in the orthogonal space. Thus the material can easily slip in certain directions along the walls. The region outside the walls contains "complementary Avellaneda material" which is a hierarchical laminate which minimizes the sum of complementary energies.

Γ-convergence of multiscale periodic functionals depending on the time-derivative and the curl operator

We study the Γ-convergence of a family of multiscale periodic quadratic integral functionals defined in a product space, whose densities depend on the time-derivative and on the curl of solenoidal fields, through the multiscale convergence in time–space and the multiscale Young measures in time–space associated with relevant sequences of pairs. An explicit representation of the Γ-limit density is given by means of an infinite dimensional minimization problem.

Variational justification of the dimensional-scaling method in chemical physics: the H-atom

The dimensional scaling (D-scaling) method first originated from quantum chromodynamics by using the spatial dimension D as an order parameter. It later has found many useful applications in chemical physics and other fields. It enables, e.g., the calculation of the energies of the Schrodinger equation with Coulomb potentials without having to solve the partial differential equation (PDE). This is done by imbedding the PDE in a D-dimensional space and by letting D tend to infinity. One can avoid the partial derivatives and then solve instead a reduced-order finite dimensional minimization problem. Nevertheless, mathematical proofs for the D-scaling method remain to be rigorously established. In this paper, we will establish this by examining the D-scaling procedures from the variational point of view. We show how the ground state energy of the hydrogen atom model can be calculated by justifying the singular perturbation procedures. In the process, we see in a more clear and mathematical way confirming (Herschbach J Chem Phys 85:838, 1986 Sect. II.A) how the D-dimensional electron wave function “condenses into a particle,” the Dirac delta function, located at the unit Bohr radius.

Diffeomorphic Matching and Dynamic Deformable Surfaces in 3d Medical Imaging

AbstractWe consider optimal matching of submanifolds such as curves and surfaces by a variational approach based on Hilbert spaces of diffeomorphic transformations. In an abstract setting, the optimal matching is formulated as a minimization problem involving actions of diffeomorphisms on regular Borel measures considered as supporting measures of the reference and the target submanifolds. The objective functional consists of two parts measuring the elastic energy of the dynamically deformed surfaces and the quality of the matching. To make the problem computationally accessible, we use reproducing kernel Hilbert spaces with radial kernels and weighted sums of Dirac measures which gives rise to diffeomorphic point matching and amounts to the solution of a finite dimensional minimization problem. We present a matching algorithm based on the first order necessary optimality conditions which include an initial-value problem for a dynamical system in the trajectories describing the deformation of the surfaces and a final-time problem associated with the adjoint equations. The performance of the algorithm is illustrated by numerical results for examples from medical image analysis.

A finite dimensional linear programming approximation of Mather's variational problem

We provide an approximation of Mather variational problem by finite dimensional minimization problems in the framework of Γ-convergence. By a linear programming interpretation as done in [Evans and Gomes, ESAIM: COCV 8 (2002) 693–702] we state a duality theorem for the Mather problem, as well a finite dimensional approximation for the dual problem.

Optimal control of cancer chemotherapy model using the coupled Adomian/Alienor methods

PurposeThis paper deals with the use of the combined Adomian/Alienor methods for solving the problem of optimal cancer chemotherapy model.Design/methodology/approachThe combination of the Adomian decomposition method and the Alienor reducing transformation method allows us to solve the control problem as if it were a classical one dimensional minimization problem. The mathematical model used here describes specific model based on cell‐cycle kinetics.FindingsIt was found that the goal is to maintain the number of the cancerous cells around a desired value while keeping the toxicity to the normal tissues acceptable.Practical implicationsSimulation results are given for illustration.Originality/valueNew combined approach to optimal control of cancer chemotherapy using Adomian/Alienor Methods.

A survey of the spectral gradient method

Finite dimensional unconstrained minimization problems are generally solved by iterative methods like Newton, quasi-Newton, steepest descent, conjugate gradient methods and its variations. All of them share the same property: they are descent methods. In 1988 Barzilai and Borwein introduce the spectral gradient method which have been an

Large Deviation Bounds for Single Class Queueing Networks and Their Calculation

We investigate large deviations for the behavior of single class queueing networks. The starting point is a sample large deviation principle on the path-space of network primitives describing the cumulative external arrivals, service time requirements and routing decisions. The behavior of the network, capturing the cumulative total arrivals, idle times and queue lengths, is characterized by a path-space fixed point equation containing the network primitives. The mapping from the network primitives to the set of fixed points is partially upper semicontinuous. This set-valued continuity allows us to derive large deviation bounds for the network behavior in the form of variational problems. The analysis is carried out on the doubly-infinite time axis R and can directly capture stationary and non-Markovian situations. By relaxing the fixed point equation the upper bounds and minimizing paths can be approximated with piecewise linear paths. For a class of typical rate functions we specify sequences of finite dimensional minimization problems which permit the calculation of large deviation rates and minimizing paths for the tail probabilities of queue lengths. We illustrate the approach with an example.

Multi-atomic Young measure and artificial boundary in approximation of micromagnetics

Some micromagnetic phenomena can be modelled by a minimization problem of a nonconvex energy. A numerical method to compute the micromagnetic field, which gives rise to a finite dimensional unconstrained minimization problem, is given and analyzed. In our method, the Maxwell's equation defined on the whole space is solved by a finite element method using artificial boundary, and the highly oscillatory magnetization structure is approximated by an element-wise constant Young measure supported on a finite number of unknown points on the unit sphere. Numerical experiments on some uniaxial and cubic anisotropic energy densities show that the method is efficient.