Strong Regularity Assumptions Research Articles

Wild solutions to scalar Euler-Lagrange equations

We study very weak solutions to scalar Euler-Lagrange equations associated with quadratic convex functionals. We investigate whether W 1 , 1 W^{1,1} solutions are necessarily W loc 1 , 2 W^{1,2}_{\operatorname {loc}} , which would make the theories by De Giorgi-Nash and Schauder applicable. We answer this question positively for a suitable class of functionals. This is an extension of Weyl’s classical lemma for the Laplace equation to a wider class of equations under stronger regularity assumptions. Conversely, using convex integration, we show that outside this class of functionals, there exist W 1 , 1 W^{1,1} solutions of locally infinite energy to scalar Euler-Lagrange equations. In addition, we prove an intermediate result which permits the regularity of a W 1 , 1 W^{1,1} solution to be improved to W loc 1 , 2 W^{1,2}_{\operatorname {loc}} under suitable assumptions on the functional and solution.

Topologies on unparameterised path space

The signature of a path, introduced by K.T. Chen [10] in 1954, has been extensively studied in recent years. The fundamental 2010 paper [20] of Hambly and Lyons showed that the signature is an injective function on the space of continuous, finite-variation paths up to a general notion of reparameterisation called tree-like equivalence. This result has been extended to geometric rough paths by Boedihardjo et al. [5]. More recently, the approximation theory of the signature has been widely used in the literature in applications. The archetypal instance of these results, see e.g. [24], guarantees uniform approximation, on compact sets, of a continuous function by a linear functional on the (extended) tensor algebra acting on the signature.In this paper we study in detail, and for the first time, the properties of three natural candidate topologies on the set of unparameterised paths, i.e. the tree-like equivalence classes. These are obtained by privileging different properties of the signature and are: (1) the product topology, obtained by equipping the range of the signature with the (subspace topology of the) product topology in the extended tensor algebra and then requiring S to be an embedding, (2) the quotient topology derived from the 1-variation topology on the underlying path space, and (3) the metric topology associated to d([γ],[σ]):=||γ⁎−σ⁎||1 using the (constant-speed) tree-reduced representatives γ⁎ and σ⁎ of the respective equivalence classes. We evaluate these spaces from the point of view of their suitability when it comes to studying (probability) measures on them. We prove that the respective collections of open sets are ordered by strict inclusion, (1) being the weakest and (3) the strongest. Our other conclusions can be summarised as follows. All three topological spaces are separable and Hausdorff, (1) being both metrisable and σ-compact, but not a Baire space and hence being neither Polish nor locally compact. The completion of (1), in any metric inducing the product topology, is the subspace G⁎ of group-like elements. The quotient topology (2) is not metrisable and the metric d is not complete. We also discuss some open problems related to these spaces.We consider finally the implications of the selection of the topology for uniform approximation results involving the signature. A stereotypical model for a continuous function on (unparameterised) path space is the solution of a controlled differential equation. We thus prove, for a broad class of these equations, well-definedness and measurability of the (fixed-time) solution map with respect to the Borel sigma-algebra of each topology. Under stronger regularity assumptions, we further show continuity of this same map on explicit compact subsets of the product topology (1). We relate these results to the expected signature model of [24].

A General Iterative Procedure for Solving Nonsmooth Constrained Generalized Equations

In this paper, we concentrate on an abstract iterative procedure for solving nonsmooth constrained generalized equations. This procedure employs both the property of weak point-based approximation and the approach of searching for a feasible inexact projection on the constrained set. Utilizing the contraction mapping principle, we establish higher order local convergence of the proposed method under the assumption of metric regularity property which ensures that the iterative procedure generates a sequence converging to a solution of the constrained generalized equation. Under strong metric regularity assumptions, we obtain that each sequence generated by this procedure converges to a solution. Furthermore, a restricted version of the proposed method is considered, for which we establish the desired convergence for each iterative sequence without a strong metric subregularity condition. The obtained results are new even for generalized equations without a constraint set.

Well-posedness of a Navier–Stokes–Cahn–Hilliard system for incompressible two-phase flows with surfactant

We investigate a diffuse-interface model that describes the dynamics of viscous incompressible two-phase flows with surfactant. The resulting system of partial differential equations consists of a sixth-order Cahn–Hilliard equation for the difference of local concentrations of the binary fluid mixture coupled with a fourth-order Cahn–Hilliard equation for the local concentration of the surfactant. The former has a smooth potential, while the latter has a singular potential. Both equations are coupled with a Navier–Stokes system for the (volume averaged) fluid velocity. The evolution system is endowed with suitable initial conditions, a no-slip boundary condition for the velocity field and homogeneous Neumann boundary conditions for the phase functions as well as for the chemical potentials. We first prove the existence of a global weak solution, which turns out to be unique in two dimensions. Stronger regularity assumptions on the initial data allow us to prove the existence of a unique global (respectively, local) strong solution in two (respectively, three) dimensions. In the two-dimensional case, we derive a continuous dependence estimate with respect to the norms controlled by the total energy. Moreover, we establish instantaneous regularization properties of global weak solutions for [Formula: see text]. In particular, we show that the surfactant concentration stays uniformly away from the pure states 0 and 1 after some positive time.

On criticality theory for elliptic mixed boundary value problems in divergence form

The paper is devoted to the study of positive solutions of a second-order linear elliptic equation in divergence form in a domain [Formula: see text] that satisfy an oblique boundary condition on a portion of [Formula: see text]. First, we study weak solutions for the degenerate mixed boundary value problem [Formula: see text] where [Formula: see text] is a bounded Lipschitz domain, [Formula: see text] is a relatively open portion of [Formula: see text], and [Formula: see text] is an oblique (Robin) boundary operator defined on [Formula: see text] in a weak sense. In particular, we discuss the unique solvability of the above problem, the existence of a principal eigenvalue, and the existence of a minimal positive Green function. Then we establish a criticality theory for positive weak solutions of the operator [Formula: see text] in a general domain [Formula: see text] with no boundary condition on [Formula: see text] and no growth condition at infinity. The paper extends results obtained by Pinchover and Saadon for classical solutions of such a problem, where stronger regularity assumptions on the coefficients of [Formula: see text], and the boundary [Formula: see text] are assumed.

Multiscale modeling of fiber reinforced materials via non-matching immersed methods

Fiber reinforced materials (FRMs) can be modeled as bi-phasic materials, where different constitutive behaviors are associated with different phases. The numerical study of FRMs through a full geometrical resolution of the two phases is often computationally infeasible, and therefore most works on the subject resort to homogenization theory, and exploit strong regularity assumptions on the fibers distribution. Both approaches fall short in intermediate regimes where lack of regularity does not justify a homogenized approach, and when the fiber geometry or their numerosity render the fully resolved problem numerically intractable.In this paper, we propose a distributed Lagrange multiplier approach, where the effect of the fibers is superimposed on a background isotropic material through an independent description of the fibers. The two phases are coupled through a constraint condition, opening the way for intricate fiber-bulk couplings as well as allowing complex geometries with no alignment requirements between the discretisation of the background elastic matrix and the fibers.We analyze both a full order coupling, where the elastic matrix is coupled with fibers that have a finite thickness, as well as a reduced order model, where the position of their centerline uniquely determines the fibers. Well posedness, existence, and uniquess of solutions are shown both for the continuous models, and for the finite element discretizations. We validate our approach against the models derived by the rule of mixtures, and by the Halpin-Tsai formulation.

A dynamic model for viscoelastic materials with prescribed growing cracks

In this paper we prove the existence of solutions for a class of viscoelastic dynamic systems on time--dependent cracked domains, with possibly degenerate viscosity coefficients. Under stronger regularity assumptions we also show a uniqueness result. Finally, we exhibit an example where the energy--dissipation balance is not satisfied, showing there is an additional dissipation due to the crack growth.

New asymptotic estimates of solutions for generalized Rosenau equations

This paper presents new methods widely applicable to expand solutions for wave equations with damping terms such as Rosenau‐type equations. Some of them have the diffusion structure that appears strongly in the low‐frequency region, and some detailed analysis on diffusion waves is seen in this report. In the high‐frequency region, difficulties arising from the regularity‐loss type are overcome by a new discovery of suitable asymptotic profiles and expanding techniques of solutions even if regularity assumptions on the initial data are not imposed. It is also shown that stronger regularity assumptions on the initial data give better asymptotic estimates.

Bayesian Model Comparison with the Hyvärinen Score: Computation and Consistency

ABSTRACT The Bayes factor is a widely used criterion in model comparison and its logarithm is a difference of out-of-sample predictive scores under the logarithmic scoring rule. However, when some of the candidate models involve vague priors on their parameters, the log-Bayes factor features an arbitrary additive constant that hinders its interpretation. As an alternative, we consider model comparison using the Hyvärinen score. We propose a method to consistently estimate this score for parametric models, using sequential Monte Carlo methods. We show that this score can be estimated for models with tractable likelihoods as well as nonlinear non-Gaussian state-space models with intractable likelihoods. We prove the asymptotic consistency of this new model selection criterion under strong regularity assumptions in the case of nonnested models, and we provide qualitative insights for the nested case. We also use existing characterizations of proper scoring rules on discrete spaces to extend the Hyvärinen score to discrete observations. Our numerical illustrations include Lévy-driven stochastic volatility models and diffusion models for population dynamics. Supplementary materials for this article are available online.

Dirac concentrations in a chemostat model of adaptive evolution

This paper deals with a non-local parabolic equation of Lotka-Volterra type that describes the evolution of phenotypically structured populations. Nonlinearities appear in these systems to model interactions and competition phenomena leading to selection. In this paper, the equation on the structured population is coupled with a differential equation on the nutrient concentration that changes as the total population varies. Different methods aimed at showing the convergence of the solutions to a moving Dirac mass are reviewed. Using either weak or strong regularity assumptions, the authors study the concentration of the solution. To this end, BV estimates in time on appropriate quantities are stated, and a constrained Hamilton-Jacobi equation to identify where the solutions concentrates as Dirac masses is derived.

A Predictor-Corrector Path-Following Algorithm for Dual-Degenerate Parametric Optimization Problems

Most path-following algorithms for tracing a solution path of a parametric nonlinear optimization problem are only certifiably convergent under strong regularity assumptions about the problem functions. In particular, linear independence of the constraint gradients at the solutions is typically assumed, which implies unique multipliers. In this paper we propose a procedure designed to solve problems satisfying a weaker set of conditions, allowing for nonunique (but bounded) multipliers. Each iteration along the path consists of three parts: (1) a Newton corrector step for the primal and dual variables, which is obtained by solving a linear system of equations, (2) a predictor step for the primal and dual variables, which is found as the solution of a quadratic programming problem, and (3) a jump step for the dual variables, which is found as the solution of a linear programming problem. We present a convergence proof and demonstrate the successful solution tracking of the algorithm numerically on a couple...

Globally stable quasistatic evolution for strain gradient plasticity coupled with damage

We consider evolutions for a material model which couples scalar damage with strain gradient plasticity, in small strain assumptions. For strain gradient plasticity, we follow the Gurtin–Anand formulation (J Mech Phys Solids 53:1624–1649, 2005). The aim of the present model is to account for different phenomena: On the one hand, the elastic stiffness reduces and the plastic yield surface shrinks due to material’s degradation, on the other hand the dislocation density affects the damage growth. The main result of this paper is the existence of a globally stable quasistatic evolution (in the so-called energetic formulation). Furthermore, we study the limit model as the strain gradient terms tend to zero. Under stronger regularity assumptions, we show that the evolutions converge to the ones for the coupled elastoplastic damage model studied in Crismale (ESAIM Control Optim Calc Var 22:883-912, 2016).

Interpolation via weighted ℓ1 minimization

Functions of interest are often smooth and sparse in some sense, and both priors should be taken into account when interpolating sampled data. Classical linear interpolation methods are effective under strong regularity assumptions, but cannot incorporate nonlinear sparsity structure. At the same time, nonlinear methods such as ℓ1 minimization can reconstruct sparse functions from very few samples, but do not necessarily encourage smoothness. Here we show that weightedℓ1 minimization effectively merges the two approaches, promoting both sparsity and smoothness in reconstruction. More precisely, we provide specific choices of weights in the ℓ1 objective to achieve approximation rates for functions with coefficient sequences in weighted ℓp spaces with p≤1. We consider implications of these results for spherical harmonic and polynomial interpolation, in the univariate and multivariate setting. Along the way, we extend concepts from compressive sensing such as the restricted isometry property and null space property to accommodate weighted sparse expansions; these developments should be of independent interest in the study of structured sparse approximations and continuous-time compressive sensing problems.

A randomized orthogonal array-based procedure for the estimation of first- and second-order Sobol' indices

In variance-based sensitivity analysis, the method of Sobol' [Sensitivity analysis for nonlinear mathematical models. Math Model Comput Exp. 1993;1:407–414] allows one to compute Sobol' indices (SI) using the Monte Carlo integration. One of the main drawbacks of this approach is that estimating SI requires a number of simulations which is dependent on the dimension of the model of interest. For example, estimating all the first- or second-order SI of a d-dimensional function basically requires or independent input vectors, respectively. Some interesting combinatorial results have been introduced to weaken this defect, in particular by Saltelli [Making best use of model evaluations to compute sensitivity indices. Comput Phys Commun. 2002;145:280–297] and more recently by Owen [Variance components and generalized Sobol' indices. SIAM/ASA J Uncertain Quantification. 2013;1:19–41], but the quantities they estimate still depend linearly on the dimension d. In this paper, we introduce a new approach to estimate all the first- and second-order SI by using only two input vectors. We establish theoretical properties of such a method for the estimation of first-order SI and discuss the generalization to higher order indices. In particular, we prove on numerical examples that this procedure is tractable and competitive for the estimation of all first- and second-order SI. As an illustration, we propose to apply this new approach to a marine ecosystem model of the Ligurian sea (northwestern Mediterranean) in order to study the relative importance of its several parameters. The calibration process of this kind of chemical simulators is well known to be quite intricate, and a rigorous and robust – that is, valid without strong regularity assumptions – sensitivity analysis, as the method of Sobol' provides, could be of great help. This article has supplementary material online.

On Quadratic Expansions of Log-Likelihoods and a General Asymptotic Linearity Result

Irrespective of the statistical model under study, the derivation of limits, in the Le Cam sense, of sequences of local experiments (see [7]-[10]) often follows along very similar lines, essentially involving differentiability in quadratic mean of square roots of (conditional) densities. This chapter establishes two abstract and very general results providing sufficient and nearly necessary conditions for (i) the existence of a quadratic expansion, and (ii) the asymptotic linearity of local log-likelihood ratios (asymptotic linearity is needed whenever unspecified model parameters are to be replaced, in some statistic of interest, with some preliminary estimator). Such results have been established, for locally asymptotically normal (LAN) models involving independent and identically distributed observations, by, e.g., [1], [11] and [12]. Similar results are provided here for models exhibiting serial dependencies which, so far, have been treated on a case-by-case basis (see [4] and [5] for typical examples) and, in general, under stronger regularity assumptions. Unlike their i.i.d. counterparts, our results moreover extend beyond the context of LAN experiments, so that non-stationary unit-root time series and cointegration models, for instance, also can be handled (see, e.g., [6]).

Optimal multiple stopping time problem

We study the optimal multiple stopping time problem defined for each stopping time $S$ by $v(S)=\operatorname {ess}\sup_{\tau_1,...,\tau_d\geq S}E[\psi(\tau_1,...,\tau_d)|\mathcal{F}_S]$. The key point is the construction of a new reward $\phi$ such that the value function $v(S)$ also satisfies $v(S)=\operatorname {ess}\sup_{\theta\geq S}E[\phi(\theta)|\mathcal{F}_S]$. This new reward $\phi$ is not a right-continuous adapted process as in the classical case, but a family of random variables. For such a reward, we prove a new existence result for optimal stopping times under weaker assumptions than in the classical case. This result is used to prove the existence of optimal multiple stopping times for $v(S)$ by a constructive method. Moreover, under strong regularity assumptions on $\psi$, we show that the new reward $\phi$ can be aggregated by a progressive process. This leads to new applications, particularly in finance (applications to American options with multiple exercise times).

Nonparametric estimation of the anisotropic probability density of mixed variables

The problem of nonparametric estimation of the joint probability density of a vector of continuous and ordinal/nominal categorical random variables with bounded support is considered. There are numerous publications devoted to the cases of either continuous or categorical variables, and the curse of dimensionality and strong regularity assumptions are the two familiar issues in the literature. Mixed variables occur in practically all applications of the statistical science and, nonetheless, the literature devoted to the joint density estimation is practically next to none. This paper develops the theory of estimation of the density of mixed variables which is on par with results known for simpler settings. Specifically, a data-driven estimator is developed that adapts to unknown anisotropic smoothness of the joint density and, whenever the density depends on a smaller number of variables, performs a dimension reduction that implies the corresponding optimal rate of the mean integrated squared error (MISE) convergence. The results hold without traditional, in the density estimation literature, minimal regularity assumptions like differentiability or continuity of the density. The procedure of estimation is based on mimicking an oracle-estimator that knows the underlying density, and the main theoretical result is the oracle inequality which relates the MISEs of the estimator and the oracle-estimator. The proof is based on a new exponential inequality for Sobolev statistics which is of interest on its own merits.

An overview on convergence rates for Tikhonov regularization methods for non-linear operators

There exists a vast literature on convergence rates results for Tikhonov regularized minimizers. The first convergence rates results for non-linear problems have been developed by Engl, Kunisch and Neubauer in 1989 [Inverse Problems 5: 523–540, 1989]. While these results apply for operator equations formulated in Hilbert spaces, the results of Burger and Osher from 2004 [Inverse Problems 20: 1411–1421, 2004], more generally, apply to operators formulated in Banach spaces. Recently, Resmerita and Scherzer [Inverse Problems 22: 801–814, 2006] presented a modification of the convergence rates result of Burger and Osher which turns out a complete generalization of the result of Engl et. al. In all these papers relatively strong regularity assumptions are made. However, it has been observed numerically, that violations of the smoothness assumptions on the operator do not necessarily affect the convergence rate negatively. We have taken this observation and weakened the smoothness assumptions on the operator and have proved a novel convergence rate result published in [Hofmann, Kaltenbacher, Pöschl, Scherzer, Inverse Problems 23: 987–1010, 2007]. The most significant difference of this result to the previous ones is that the source condition is formulated as a variational inequality and not as an equation as before, which is necessary due to the lack of smoothness assumptions on F .

Generalized Cauchy difference equations. II

The main result is an improvement of previous results on the equation\[f(x)+f(y)−f(x+y)=g[ϕ(x)+ϕ(y)−ϕ(x+y)]f(x)+f(y)-f(x+y)=g[\phi (x)+\phi (y)-\phi (x+y)]\]for a given functionϕ\phi. We find its general solution assuming only continuous differentiability and local nonlinearity ofϕ\phi. We also provide new results about the more general equation\[f(x)+f(y)−f(x+y)=g(H(x,y))f(x)+f(y)-f(x+y)=g(H(x,y))\]for a given functionHH. Previous uniqueness results required strong regularity assumptions on a particular solutionf0,g0f_{0},g_{0}. Here we weaken the assumptions onf0,g0f_{0},g_{0}considerably and find all solutions under slightly stronger regularity assumptions onHH.

Sparse adaptive finite elements for radiative transfer

The linear radiative transfer equation, a partial differential equation for the radiation intensity u ( x , s ) , with independent variables x ∈ D ⊂ R n in the physical domain D of dimension n = 2 , 3 , and angular variable s ∈ S 2 : = { y ∈ R 3 : | y | = 1 } , is solved in the n + 2 -dimensional computational domain D × S 2 . We propose an adaptive multilevel Galerkin finite element method (FEM) for its numerical solution. Our approach is based on (a) a stabilized variational formulation of the transport operator, (b) on so-called sparse tensor products of two hierarchic families of finite element spaces in H 1 ( D ) and in L 2 ( S 2 ) , respectively, and (c) on wavelet thresholding techniques to adapt the discretization to the underlying problem. An a priori error analysis shows, under strong regularity assumptions on the solution, that the sparse tensor product method is clearly superior to a discrete ordinates method, as it converges with essentially optimal asymptotic rates while its complexity grows essentially only as that for a linear transport problem in R n . Numerical experiments for n = 2 on a set of example problems agree with the convergence and complexity analysis of the method and show that introducing adaptivity can improve performance in terms of accuracy vs. number of degrees even further.