Inverse problems for multi-valued quasi variational inequalities and noncoercive variational inequalities with noisy data

ABSTRACT

We study the inverse problem of identifying a variable parameter in variational and quasi-variational inequalities. We consider a quasi-variational inequality involving a multi-valued monotone map and give a new existence result. We then formulate the inverse problem as an optimization problem and prove its solvability. We also conduct a thorough study of the inverse problem of parameter identification in noncoercive variational inequalities which appear commonly in applied models. We study the inverse problem by posing optimization problems using the output least-squares and the modified output least-squares. Using regularization, penalization, and smoothing, we obtain a single-valued parameter-to-selection map and study its differentiability. We consider optimization problems using the output least-squares and the modified output least-squares for the regularized, penalized and smoothened variational inequality. We give existence results, convergence analysis, and optimality conditions. We provide applications and numerical examples to justify the proposed framework.     

Stationary Transonic Solutions of a One–Dimensional Hydrodynamic Model for Semiconductors

We study a hydrodynamic model for semiconductors where the energy equation is replaced by a pressure-density relationship. We construct artificial viscosity solutions, prove BV estimates independent of the viscosity coefficient and study the transonic weak limit. We also study the behavior of the limiting solution at the boundary for subsonic data. We find that a boundary layer can be formed on each side of the boundary and has a condition that determines the possible range of discontinuities for the density.     

Dirichlet forms on fractals: Poincaré constant and resistance

Summary We study Dirichlet forms associated with random walks on fractal-like finite grahs. We consider related Poincaré constants and resistance, and study their asymptotic behaviour. We construct a Markov semi-group on fractals as a subsequence of random walks, and study its properties. Finally we construct self-similar diffusion processes on fractals which have a certain recurrence property and plenty of symmetries.Partly supported by the JSPS Program     

Computation of the Delta in Multidimensional Jump-Diffusion Setting with Applications to Stochastic Volatility Models

We study the robustness of options prices to model variation in a multidimensional jump-diffusion framework. In particular, we consider price dynamics in which small variations are modeled either by a Poisson random measure with infinite activity or by a Brownian motion. We consider both European and Exotic options and we study their deltas using two approaches: the Malliavin method and the Fourier method. We prove robustness of the deltas to model variation. We apply these results to the study of stochastic volatility models for the underlying and the corresponding options.     

On the mathematical controllability in a simple growth tumors model by the internal localized action of inhibitors

We study a model of growth of tumors with a free boundary delaying the tumor region. We take into account the presence of inhibitors and its interaction with the nutrients. We study the approximate controllability of the internal distribution of density of cells, that is proportional to concentration of nutrients, injecting inhibitor in a small inner region.     

Optimal growth with investment enhancing labor

We study a non-convex optimal growth problem with investment enhancing labor. We prove that there exists an optimal growth path, that all optimal paths are interior and we provide a condition under which at least one of them is monotonic. We also study the existence and uniqueness of the steady state. We show in particular that a rise in the efficiency of the investment enhancing labor does not necessarily lead to an increase in the steady state value of this labor. Furthermore we provide a complete study of the dynamics of the optimal solution in the special case of a logarithmic utility function and a Cobb–Douglas production function.     

2-D singular Cauchy problems for quasilinear equations

We consider 2-dimensional quasilinear Cauchy problems for singular initial values in a complex domain. We study the singularities of the solution, in terms of monoidal transformation. We study whether the singularities propagate toward characteristic directions, and whether the singularities branch.     

Behavior of the solution of a random semilinear heat equation

We consider a semilinear heat equation in one space dimension, with a random source at the origin. We study the solution, which describes the equilibrium of this system, and prove that, as the space variable tends to infinity, the solution becomes a.s. asymptotic to a steady state. We also study the fluctuations of the solution around the steady state.     

Periodic perturbations of a nonlinear oscillator

We study small time-periodic perturbations of an oscillator with a power-law odd restoring force with exponent exceeding unity. We study two problems, one on the stability of the equilibrium and the other on the bifurcation of an invariant two-dimensional torus from the equilibrium. We construct a focal quantity and a bifurcation equation that find the character of stability and branching of the equilibrium.     

The dynamical behavior of the discontinuous Galerkin method and related difference schemes

We study the dynamical behavior of the discontinuous Galerkin finite element method for initial value problems in ordinary differential equations. We make two different assumptions which guarantee that the continuous problem defines a dissipative dynamical system. We show that, under certain conditions, the discontinuous Galerkin approximation also defines a dissipative dynamical system and we study the approximation properties of the associated discrete dynamical system. We also study the behavior of difference schemes obtained by applying a quadrature formula to the integrals defining the discontinuous Galerkin approximation and construct two kinds of discrete finite element approximations that share the dissipativity properties of the original method.

     

On the existence of the biharmonic Green kernels and the adjoint biharmonic functions

We study the existence and the regularity of the biharmonic Green kernel in a Brelot biharmonic space whose associated harmonic spaces have Green kernels. We show by some examples that this kernel does not always exist. We then introduce and study the adjoint of the given biharmonic space. This study was initiated by Smyrnelis, however, it seems that several results were incomplete and we clarify them here.     

Complex Cauchy Problems with Intersecting Singularities

We consider linear Cauchy problems of order two in a complex domain. We assume that the initial values have singularities along a family of hypersurfaces, which cross pairwise transversally along a single intersection. We study the propagation of the singularities of the solution. We show that the solution may have anomalous singularities, and study the monodromy of the solution.     

Pseudodeformations

We study the deformation functor of a reducible pseudocharacter. We show that there is a natural filtration (the complexity filtration) on such a functor, and that this filtration induces a filtration on the tangent space, whose graded pieces can be described in terms of the extension spaces between the irreducible components of the pseudocharacter. We also study the obstruction theory of that deformation problem.     

Semi-Slant Submanifolds of a Sasakian Manifold

We define and study both bi-slant and semi-slant submanifolds of an almost contact metric manifold and, in particular, of a Sasakian manifold. We prove a characterization theorem for semi-slant submanifolds and we obtain integrability conditions for the distributions which are involved in the definition of such submanifolds. We also study an interesting particular class of semi-slant submanifolds.     

增长网络的形成机理和度分布计算

关于增长网络的形成机理,着重介绍由线性增长与择优连接组成的BA模型, 以及加速增长模型.此外,我们提出了一个含反择优概率删除旧连线的模型,这个模型能自组织演化成scale-free(SF)网络.关于计算SF网络的度分布,简要介绍文献上常用的基于连续性理论的动力学方法(包括平均场和率方程)和基于概率理论的主方程方法.另外,我们基于马尔可夫链理论还首次尝试了数值计算方法.这一方法避免了复杂方程的求解困难,所以较有普适性,因此可用于研究更为复杂的网络模型.我们用这种数值计算方法研究了一个具有对数增长的加速增长模型,这个模型也能自组织演化成SF网络.     

Nonlocal Interaction Equations in Environments with Heterogeneities and Boundaries

We study well-posedness of a class of nonlocal interaction equations with spatially dependent mobility. We also allow for the presence of boundaries and external potentials. Such systems lead to the study of nonlocal interaction equations on subsets ? of ? ^d endowed with a Riemannian metric g. We obtain conditions, relating the interaction potential and the geometry, which imply existence, uniqueness and stability of solutions. We study the equations in the setting of gradient flows in the space of probability measures on ? endowed with Riemannian 2-Wasserstein metric.     

Asymptotic Solution of the Autoresonance Problem

We study the problem of forced oscillations near a stable equilibrium of a two-dimensional nonlinear Hamiltonian system of equations. A given exciting force is represented as rapid oscillations with a small amplitude and a slowly varying frequency. We study the conditions under which such a perturbation makes the phase trajectory of the system recede from the original equilibrium point to a distance of the order of unity. To study the problem, we construct asymptotic solutions using a small amplitude parameter. We present the solution for not-too-small values of time outside the original boundary layer.     

Main eigenvalues and automorphisms of a graph

We study some aspects of the relationship between algebras associated with graphs and automorphism groups. We study an algebra generated by the adjacent matrix of a graph and the all ones matrix, and derive a lower bound for the rank of the automorphism group of a graph. If a graph attains the equality in the above bound, it is calledextremal. We also describe some properties and examples of extremal graphs.     

Application and Implementation of Incorporating Local Boundary Conditions into Nonlocal Problems

We study nonlocal equations from the area of peridynamics, an instance of nonlocal wave equation, and nonlocal diffusion on bounded domains whose governing equations contain a convolution operator based on integrals. We generalize the notion of convolution to accommodate local boundary conditions. On a bounded domain, the classical operator with local boundary conditions has a purely discrete spectrum, and hence, provides a Hilbert basis. We define an abstract convolution operator using this Hilbert basis, thereby automatically satisfying local boundary conditions. The main goal in this paper is twofold: apply the concept of abstract convolution operator to nonlocal problems and carry out a numerical study of the resulting operators. We study the corresponding initial value problems with prominent boundary conditions such as periodic, antiperiodic, Neumann, and Dirichlet. To connect to the standard convolution, we give an integral representation of the abstract convolution operator. For discretization, we use a weak formulation based on a Galerkin projection and use piecewise polynomials on each element which allows discontinuities of the approximate solution at the element borders. We study convergence order of solutions with respect to polynomial order and observe optimal convergence. We depict the solutions for each boundary condition.     

Algebraic Levi-Flat Hypervarieties in Complex Projective Space

We study singular real-analytic Levi-flat hypersurfaces in complex projective space. We define the rank of an algebraic Levi-flat hypersurface and study the connections between rank, degree, and the type and size of the singularity. In particular, we study degenerate singularities of algebraic Levi-flat hypersurfaces. We then give necessary and sufficient conditions for a Levi-flat hypersurface to be a pullback of a real-analytic curve in ℂ via a meromorphic function. Among other examples, we construct a nonalgebraic semianalytic Levi-flat hypersurface with compact leaves that is a perturbation of an algebraic Levi-flat variety.