Convexity Properties Research Articles

Semismoothness for Solution Operators of Obstacle-Type Variational Inequalities with Applications in Optimal Control

We prove that solution operators of elliptic obstacle-type variational inequalities (or, more generally, locally Lipschitz continuous functions possessing certain pointwise-a.e. convexity properties) are Newton differentiable when considered as maps between suitable Lebesgue spaces and equipped with the strong-weak Bouligand differential as a generalized set-valued derivative. It is shown that this Newton differentiability allows to solve optimal control problems with -cost terms and one-sided pointwise control constraints by means of a semismooth Newton method. The superlinear convergence of the resulting algorithm is proved in the infinite-dimensional setting, and its mesh independence is demonstrated in numerical experiments. We expect that the findings of this paper are also helpful for the design of numerical solution procedures for quasi-variational inequalities and the optimal control of obstacle-type variational problems.

A Study on the New Class of Inequalities of Midpoint-Type and Trapezoidal-Type Based on Twice Differentiable Functions with Conformable Operators

This paper derives some equalities via twice differentiable functions and conformable fractional integrals. With the help of the obtained identities, we present new trapezoid-type and midpoint-type inequalities via convex functions in the context of the conformable fractional integrals. New inequalities are obtained by taking advantage of the convexity property, power mean inequality, and Hölder’s inequality. We show that this new family of inequalities generalizes some previous research studies by special choices. Furthermore, new other relevant results with trapezoid-type and midpoint-type inequalities are obtained.

Geometry of infinite dimensional unitary groups: Convexity and fixed points

In this article we study convexity properties of distance functions in infinite dimensional Finsler unitary groups, such as the full unitary group, the unitary Schatten perturbations of the identity and unitary groups of finite von Neumann algebras. The Finsler structures are defined by translation of different norms on the tangent space at the identity. We first prove a convexity result for the metric derived from the operator norm on the full unitary group. We also prove strong convexity results for the squared metrics in Hilbert-Schmidt unitary groups and unitary groups of finite von Neumann algebras. In both cases the tangent spaces are endowed with an inner product defined with a trace. These results are applied to fixed point properties and to quantitative metric bounds in certain rigidity problems. Radius bounds for all convexity and fixed point results are shown to be optimal.

Truncated nonsmooth Newton multigrid for phase-field brittle-fracture problems, with analysis

AbstractWe propose the truncated nonsmooth Newton multigrid method (TNNMG) as a solver for the spatial problems of the small-strain brittle-fracture phase-field equations. TNNMG is a nonsmooth multigrid method that can solve biconvex, block-separably nonsmooth minimization problems with linear time complexity. It exploits the variational structure inherent in the problem, and handles the pointwise irreversibility constraint on the damage variable directly, without regularization or the introduction of a local history field. In the paper we introduce the method and show how it can be applied to several established models of phase-field brittle fracture. We then prove convergence of the solver to a solution of the nonsmooth Euler–Lagrange equations of the spatial problem for any load and initial iterate. On the way, we show several crucial convexity and regularity properties of the models considered here. Numerical comparisons to an operator-splitting algorithm show a considerable speed increase, without loss of robustness.

Coordinated scheduling for in-clinic and virtual medicine patients in a multi-station network

In this article, we study a coordinated scheduling problem with both Virtual Medicine patients (VM patients) and In-Clinic patients (IC patients) in a multi-disciplinary setting. The problem was motivated by appointment scheduling requirements in a multi-disciplinary clinic called an Integrated Practice Unit (IPU), which incorporates differing priorities, heterogeneous service time distributions, distinct cost structures and unique care paths in a multi-station network. We establish priority for IC patients and introduce time windows for VM patients to create flexibility. Recursion expressions are derived for a performance measure of interest, which balances revenue against clinic overtime and patient waiting time costs. We develop an approach where IC patients are scheduled first. To do so, we establish discrete convexity for a special tree-type directed network structure and generate near-optimal IC patient schedules for a more general acyclic directed network. Conditioned on IC patients’ schedule, we show that the VM patient scheduling problem has a discrete convexity property even in the presence of non-linear costs. Through numerical examples based on IPUs being implemented by the Dell Medical School at the University of Texas at Austin, we find that the introduction of VM patients can substantially improve system performance and patient access without adding resources.

Correlation-based full-waveform shear wave elastography

Objective. With the ultimate goal of reconstructing 3D elasticity maps from ultrasound particle velocity measurements in a plane, we present in this paper a methodology of inverting for 2D elasticity maps from measurements on a single line. Approach. The inversion approach is based on gradient optimization where the elasticity map is iteratively modified until a good match is obtained between simulated and measured responses. Full-wave simulation is used as the underlying forward model to accurately capture the physics of shear wave propagation and scattering in heterogeneous soft tissue. A key aspect of the proposed inversion approach is a cost functional based on correlation between measured and simulated responses. Main results. We illustrate that the correlation-based functional has better convexity and convergence properties compared to the traditional least-squares functional, and is less sensitive to initial guess, robust against noisy measurements and other errors that are common in ultrasound elastography. Inversion with synthetic data illustrates the effectiveness of the method to characterize homogeneous inclusions as well as elasticity map of the entire region of interest. Significance. The proposed ideas lead to a new framework for shear wave elastography that shows promise in obtaining accurate maps of shear modulus using shear wave elastography data obtained from standard clinical scanners.

Measuring pants

We investigate the terms arising in an identity for hyperbolic surfaces proved by Luo and Tan, namely showing that they vary monotonically in terms of lengths and that they verify certain convexity properties. Using these properties, we deduce two results. As a first application, we show how to deduce a theorem of Thurston which states, in particular for closed hyperbolic surfaces, that if a simple length spectrum “dominates” another, then in fact the two surfaces are isometric. As a second application, we show how to find upper bounds on the number of pairs of pants of bounded length that only depend on the boundary length and the topology of the surface.

Extremum seeking control for real-time optimization of high temperature heat pump systems incorporating vapor injection

In order to minimize the power consumption of a vapor-injected high temperature heat pump system with R245fa as refrigerant, a control strategy, namely Extremum seeking control, is proposed. It can implement real-time model-free optimization control of plants with concave and convex properties. Modelica-based dynamic simulation model of the proposed system is developed. The Newton-based Extremum seeking control is used to optimize the intermediate pressure of the proposed system to obtain the minimum power consumption of the compressor at a fixed heating capacity (200 kW). The adopted ESC strategy requires four measured quantities, namely, injection pressure, condenser outlet water temperature, evaporator outlet superheat and compressor power consumption, to achieve optimal control of the system in real time. Two operation scenarios are simulated: stable and freely variable heat source temperature. For both scenarios, the condenser outlet water temperature was set to 140 °C. The simulation results show that at stable heat source temperature, the power consumption is reduced by 5.2% and the COP is increased by 7.7% compared to a fixed initial intermediate pressure. Under the free-variable heat source condition, the power consumption can be reduced by up to 5 kW compared to a constant input. A total of 283 kWh of power was saved during the 48 h simulation. Thus, the Newton-based extremum seeking control strategy can optimize the intermediate pressure to achieve the minimum power consumption while maintaining excellent transient performance, and demonstrate the superiority of the ESC strategy for real-time optimization..

Unique continuation for the Lamé system using stabilized finite element methods

We introduce an arbitrary order, stabilized finite element method for solving a unique continuation problem subject to the time-harmonic elastic wave equation with variable coefficients. Based on conditional stability estimates we prove convergence rates for the proposed method which take into account the noise level and the polynomial degree. A series of numerical experiments corroborates our theoretical results and explores additional aspects, e.g. how the quality of the reconstruction depends on the geometry of the involved domains. We find that certain convexity properties are crucial to obtain a good recovery of the wave displacement outside the data domain and that higher polynomial orders can be more efficient but also more sensitive to the ill-conditioned nature of the problem.

Properties of Convex Fuzzy-Number-Valued Functions on Harmonic Convex Set in the Second Sense and Related Inequalities via Up and Down Fuzzy Relation

In this paper, we provide different variants of the Hermite–Hadamard (H⋅H) inequality using the concept of a new class of convex mappings, which is referred to as up and down harmonically s-convex fuzzy-number-valued functions (UDH s-convex FNVM) in the second sense based on the up and down fuzzy inclusion relation. The findings are confirmed with certain numerical calculations that take a few appropriate examples into account. The results deal with various integrals of the 2ρσρ+σ type and are innovative in the setting of up and down harmonically s-convex fuzzy-number-valued functions. Moreover, we acquire classical and new exceptional cases that can be seen as applications of our main outcomes. In our opinion, this will make a significant contribution to encouraging more research.

Bilinear matrix inequalities and polynomials in several freely noncommuting variables

Matrix-valued polynomials in any finite number of freely noncommuting variables that enjoy certain canonical partial convexity properties are characterized, via an algebraic certificate, in terms of Linear Matrix Inequalities and Bilinear Matrix Inequalities.

Probabilistic and Analytical Aspects of the Symmetric and Generalized Kaiser–Bessel Window Function

The generalized Kaiser–Bessel window function is defined via the modified Bessel function of the first kind and arises frequently in tomographic image reconstruction. In this paper, we study in details the properties of the Kaiser–Bessel distribution, which we define via the symmetric form of the generalized Kaiser–Bessel window function. The Kaiser–Bessel distribution resembles to the Bessel distribution of McKay of the first type, it is a platykurtic or sub-Gaussian distribution, it is not infinitely divisible in the classical sense and it is an extension of the Wigner’s semicircle, parabolic and n-sphere distributions, as well as of the ultra-spherical (or hyper-spherical) and power semicircle distributions. We deduce the moments and absolute moments of this distribution and we find its characteristic and moment generating function in two different ways. In addition, we find its cumulative distribution function in three different ways and we deduce a recurrence relation for the moments and absolute moments. Moreover, by using a formula of Ismail and May on quotient of modified Bessel functions of the first kind, we deduce a closed-form expression for the differential entropy. We also prove that the Kaiser–Bessel distribution belongs to the family of log-concave and geometrically concave distributions, and we study in details the monotonicity and convexity properties of the probability density function with respect to the argument and each of the parameters. In the study of the monotonicity with respect to one of the parameters we complement a known result of Gronwall concerning the logarithmic derivative of modified Bessel functions of the first kind. Finally, we also present a modified method of moments to estimate the parameters of the Kaiser–Bessel distribution, and by using the classical rejection method we present two algorithms for sampling independent continuous random variables of Kaiser–Bessel distribution. The paper is closed with conclusions and proposals for future works.

Remarks on rank‐one convexity and quasiconvexity for planar functions with an additive volumetric–isochoric split

AbstractWe study convexity properties of isotropic energy functions in planar nonlinear elasticity in the context of Morrey's conjecture, which states that rank‐one convexity does not imply quasiconvexity in the two‐dimensional case. Recently, it has been shown that for the special case of isochoric energy functions on GL+(2) = {F ∈ ℝ2×2 | det F > 0}, i.e. for any isotropic function W : GL+(2) → ℝ with W(aF) = W(F) for all a > 0, these two notions of generalized convexity are, in fact, equivalent. Here, we consider the more general case of functions on GL+(2) with an additive volumetric–isochoric split of the form urn:x-wiley:16177061:media:PAMM202200185:pamm202200185-math-0001 with an isochoric function Wiso on GL+(2) and a function Wvol on (0, ∞). In particular, we investigate the importance of the function urn:x-wiley:16177061:media:PAMM202200185:pamm202200185-math-0002 and its convexity properties; here, λmax ≥ λmin > 0 are the ordered singular values of the deformation gradient F ∈ GL+(2). This function arises naturally as an “extremal” case in the class of volumetric–isochorically split energies with respect to rank‐one convexity.

On eigenvalues of symmetric matrices with PSD principal submatrices

We investigate convexity properties of the set of eigenvalue tuples of n×n real symmetric matrices, all of whose k×k (where k≤n is fixed) minors are positive semidefinite. We prove that the set λ(Sn,k) of eigenvalue vectors of all such matrices is star-shaped with respect to the nonnegative orthant R≥0n and not convex already when (n,k)=(4,2). We also show that k is the smallest integer such that Sn,k is a linear projection of a set described by linear matrix inequalities of size k.

Waveform inversion via reduced order modeling

We introduce a novel approach to waveform inversion based on a data-driven reduced order model (ROM) of the wave operator. The presentation is for the acoustic wave equation, but the approach can be extended to elastic or electromagnetic waves. The data are time resolved measurements of the pressure wave gathered by an acquisition system that probes the unknown medium with pulses and measures the generated waves. We propose to solve the inverse problem of velocity estimation by minimizing the square misfit between the ROM computed from the recorded data and the ROM computed from the modeled data, at the current guess of the velocity. We give a step by step computation of the ROM, which depends nonlinearly on the data and yet can be obtained from them in a noniterative fashion, using efficient methods from linear algebra. We also explain how to make the ROM robust to data inaccuracy. The ROM computation requires the full array response matrix gathered with colocated sources and receivers. However, we find that the computation can deal with an approximation of this matrix, obtained from towed-streamer data using interpolation and reciprocity on-the-fly. Although the full-waveform inversion approach of nonlinear least-squares data fitting is challenging without low-frequency information, due to multiple minima of the data fit objective function, we find that the ROM misfit objective function has better behavior, even for a poor initial guess. We also find by explicit computation of the objective functions in a simple setting that the ROM misfit objective function has convexity properties, whereas the least-squares data fit objective function displays multiple local minima.

One-Dimensional Failure Modes for Bodies with Non-convex Plastic Energies

In this paper, a complete picture of the different plastic failure modes that can be predicted by the strain gradient plasticity model proposed in Del Piero et al. (J. Mech. Mater. Struct. 8:109–151, 2013) is drawn. The evolution problem of the elasto-plastic strain is formulated in Del Piero et al. (J. Mech. Mater. Struct. 8:109–151, 2013) as an incremental minimization problem acting on an energy functional which includes a local plastic term and a non-local gradient contribution. Here, an approximate analytical solution of the evolution problem is determined in the one-dimensional case of a tensile bar. Different solutions are found describing specific plastic strain processes, and correlations between the different evolution modes and the convexity/concavity properties of the plastic energy density are established.The variety of solutions demonstrates the large versatility of the model in describing many failure mechanisms, ranging from brittle to ductile. Indeed, for a convex plastic energy, the plastic strain diffuses in the body, while, for a concave plastic energy, it localizes in regions whose amplitude depends on the internal length parameter included into the non-local energy term, and, depending on the convexity properties of the first derivative of the plastic energy, the localization band expands or contracts. Complex failure processes combining different modes can be reproduced by assuming plastic energy functionals with specific convex and concave branches.The quasi-brittle failure of geomaterials in simple tension tests was reproduced by assuming a convex-concave plastic energy, and the accuracy of the analytical predictions was checked by comparing them with the numerical results of finite element simulations.

Upper Tail Bounds for Stationary KPZ Models

We present a proof of an upper tail bound of the correct order (up to a constant factor in the exponent) in two classes of stationary models in the KPZ universality class. The proof is based on an exponential identity due to Rains in the case of last passage percolation with exponential weights, and recently re-derived by Emrah–Janjigian–Seppäiläinen (EJS). Our proof follows very similar lines for the two classes of models we consider, using only general monotonocity and convexity properties, and can thus be expected to apply to many other stationary models.

Practical Applications of Primer on Agency Mortgage-Backed Securities Specified Pools and Their Convexity Profiles

In <b>Primer on Agency Mortgage-Backed Securities Specified Pools and Their Convexity Profiles</b> from the Spring 2022 issue of <b><i>The Journal of Fixed Income</i></b>, authors <b>Glenn M. Schultz</b> (of <b>MUFG Securities</b>) and <b>Frank J. Fabozzi</b> (of <b>Johns Hopkins University</b> and <b>EDHEC Risk Institute</b>) describe the various sectors of the market for agency mortgage-backed securities (MBS) specified pools. While most MBS are traded on a generic basis—without identifying the underlying pool of mortgage loans—some trades specify the particular underlying pool of the security that must be delivered to clear the trade. The first type of trade is called a to-be-announced (TBA) trade, and the second type is called a specified-pool trade. When market participants trade agency MBS on a specified pool basis, they generally do so because they expect the loans in the underlying pools to display atypical prepayment behavior, which gives the related MBS atypical convexity properties. Specified pools are categorized into sectors, based on loan balance, loan-to-value (LTV) ratio, and geographical location, plus whether they consist of refinanced mortgages or mortgages for investment properties, relocating employees, or people with suboptimal credit scores. The authors describe each specified-pool sector and its convexity profile. Most have positive convexity relative to generic pools and trade at a premium, but jumbo and relocation mortgage pools have negative convexity relative to generic pools and trade at a concession. Investors can manage risk by allocating assets to customize the convexity profiles of their MBS portfolios.

Characterization of certain fractional-type set-valued functions

In this paper, we give some characterizations of a special class of fractional-type set-valued functions in terms of convexity-preserving properties of sets by direct and inverse images. We begin by generalizing the so-called ratios of affine functions, initially introduced by Rothblum, to set-valued functions by using an affinity concept introduced in the literature by Gorokhovik. Next, we investigate some convexity properties for general fractional-type set-valued functions and provide a series of convexity-preserving results of sets under set-valued ratios of affine functions.

On Generalization Bounds for Deep Networks Based on Loss Surface Implicit Regularization

The classical statistical learning theory implies that fitting too many parameters leads to overfitting and poor performance. That modern deep neural networks generalize well despite a large number of parameters contradicts this finding and constitutes a major unsolved problem towards explaining the success of deep learning. While previous work focuses on the implicit regularization induced by stochastic gradient descent (SGD), we study here how the local geometry of the energy landscape around local minima affects the statistical properties of SGD with Gaussian gradient noise. We argue that under reasonable assumptions, the local geometry forces SGD to stay close to a low dimensional subspace and that this induces another form of implicit regularization and results in tighter bounds on the generalization error for deep neural networks. To derive generalization error bounds for neural networks, we first introduce a notion of stagnation sets around the local minima and impose a local essential convexity property of the population risk. Under these conditions, lower bounds for SGD to remain in these stagnation sets are derived. If stagnation occurs, we derive a bound on the generalization error of deep neural networks involving the spectral norms of the weight matrices but not the number of network parameters. Technically, our proofs are based on controlling the change of parameter values in the SGD iterates and local uniform convergence of the empirical loss functions based on the entropy of suitable neighborhoods around local minima.