Nonsmooth Point Research Articles

Numerical differentiation of the piecewise smooth function by using Fourier extension method

Numerical differentiation of the piecewise smooth function is considered in this paper. To avoid the large error of numerical differentiation that may occur near potential non-smooth points, we identify the discontinuity points of the first or second derivative of the function. Then we divide the domain of the function into several sub-domains. For each sub-domain, the approximation is constructed by Fourier extension, and the global approximation of the piecewise smooth function is formed by superposition to improve accuracy. Some numerical experiments are conducted to further verify the efficacy of the method.

On regular but non-smooth integral curves

Let C be a regular geometrically integral curve over an imperfect field K and assume that it admits a non-smooth point p which — seen as a prime of the separable function field K(C)|K — is non-decomposed in the base field extension K‾⊗KK(C)|K‾. In this paper we establish a bound for the number of iterated Frobenius pullbacks needed in order to transform p into a rational point. This provides an algorithm to compute geometric δ-invariants of non-smooth points and a procedure to construct fibrations with moving singularities of prescribed δ-invariants. We show that the bound is sharp in characteristic 2. We further study the geometry of a pencil of plane projective rational quartics in characteristic 2 whose generic fibre attains our bound. On our way, we prove several results on separable and non-decomposed points that might be of independent interest.

Towards optimal sensor placement for inverse problems in spaces of measures

The objective of this work is to quantify the reconstruction error in sparse inverse problems with measures and stochastic noise, motivated by optimal sensor placement. To be useful in this context, the error quantities must be explicit in the sensor configuration and robust with respect to the source, yet relatively easy to compute in practice, compared to a direct evaluation of the error by a large number of samples. In particular, we consider the identification of a measure consisting of an unknown linear combination of point sources from a finite number of measurements contaminated by Gaussian noise. The statistical framework for recovery relies on two main ingredients: first, a convex but non-smooth variational Tikhonov point estimator over the space of Radon measures and, second, a suitable mean-squared error based on its Hellinger–Kantorovich distance to the ground truth. To quantify the error, we employ a non-degenerate source condition as well as careful linearization arguments to derive a computable upper bound. This leads to asymptotically sharp error estimates in expectation that are explicit in the sensor configuration. Thus they can be used to estimate the expected reconstruction error for a given sensor configuration and guide the placement of sensors in sparse inverse problems.

The isoperimetric inequality for convex subsets of the sphere

We give a new proof of an isoperimetric inequality for a family of closed surfaces, which have Gaussian curvature identically equal to one wherever the surface is smooth. These surfaces are formed from a convex, spherical polygon, with each vertex of the polygon leading to a non-smooth point on the surface. For example, the surface formed from a spherical lune is a surface of revolution, with two non-smooth tips. Combined with a straightforward approximation argument, this inequality was first proved by B\'erard, Besson, and Gallot, where they provide a generalization of the L\'evy-Gromov isoperimetric inequality. The inequality implies an isoperimetric inequality for geodesically convex subsets of the sphere, and, using a Faber-Krahn theorem, it also implies a lower bound on the first Dirichlet eigenvalue of a region of a given area on the closed surfaces. Via approximation, we convert this into a lower bound on the first Dirichlet-Neumann eigenvalue of domains contained in geodesically convex subsets of the sphere.

A nonsmooth variational approach to semipositone quasilinear problems in [formula omitted

This paper concerns the existence of a solution for the following class of semipositone quasilinear problems{−Δpu=h(x)(f(u)−a)inRN,u>0inRN, where 1<p<N, a>0, f:[0,+∞)→[0,+∞) is a function with subcritical growth and f(0)=0, while h:RN→(0,+∞) is a continuous function that satisfies some technical conditions. We prove via nonsmooth critical points theory and comparison principle, that a solution exists for a small enough. We also provide a version of Hopf's Lemma and a Liouville-type result for the p-Laplacian in the whole RN.

Analysis of Markov Chain Approximation for Diffusion Models with Nonsmooth Coefficients

In this study, we analyze the convergence of continuous-time Markov chain approximation for one-dimensional diffusions with nonsmooth coefficients. We obtain a sharp estimate of the convergence rate for the value function and its first and second derivatives, which is generally first order. To improve it to second order, we propose two methods: applying the midpoint rule that places all nonsmooth points midway between two neighboring grid points or applying harmonic averaging to smooth the model coefficients. We conduct numerical experiments for various financial applications to confirm the theoretical estimates. We also show that the midpoint rule can be applied to achieve second-order convergence for some jump-diffusion and two-factor short rate models.

Existence and concentration of solutions for a class of biharmonic Kirchhoff equations with discontinuous nonlinearity

This paper concerns the existence and concentration of positive solutions for a class of biharmonic Kirchhoff equations with discontinuous nonlinearity. The proof relies on variational method, truncated methods, and nonsmooth critical points theory. Some related results are improved.

Geomorphology of Lagrangian ridges

We prove an ‘h-principle without pre-conditions’ for the elimination of tangencies of a Lagrangian submanifold with respect to a Lagrangian distribution. The main result states that such tangencies can always be completely removed at the cost of allowing the Lagrangian to develop certain non-smooth points, called Lagrangian ridges, modeled on the corner { p = | q | } ⊂ R 2 $\lbrace p=|q|\rbrace \subset \mathbb {R}^2$ together with its products and stabilizations. This result plays an essential role in the arborealization program.

A robust numerical method for single and multi-asset option pricing

&lt;abstract&gt;&lt;p&gt;In this work, we numerically solve some different single and multi-asset European options with the finite difference method (FDM) and take the advantages of the antithetic variate method in Monte Carlo simulation (AMC) as a variance reduction technique in comparison to the standard Monte Carlo simulation (MC) in the end point of the domain, and the linear boundary condition has been implemented in other boundaries. We also apply the grid stretching transformation to make a non-equidistance discretization with more nodal points around the strike price (K) which is the non-smooth point in the payoff function to reduce the numerical errors around this point and have more accurate results. Superiority of our method (GS&amp;amp;AMC) will be demonstrated by comparison with the finite difference scheme with the equidistance discretization and the linear boundary conditions (Equi&amp;amp;L), the grid stretching discretization around K with linear boundary conditions (GS&amp;amp;L) and also the equidistance discretization with combination of the standard Monte Carlo simulation at the end point of the domain (Equi&amp;amp;MC). Furthermore, the root mean square errors (RMSE) of these four schemes in the whole region and the most interesting region which is around the strike price, have been compared.&lt;/p&gt;&lt;/abstract&gt;

A criterion for the existence of a plane model with two inner Galois points for algebraic curves

A criterion for the existence of a plane model with two non-smooth Galois points for algebraic curves is presented, which is a generalization of Fukasawa’s criterion for two smooth Galois points. Multiplicities and order sequences at Galois points are also described in detail.

On singularities of real algebraic sets and applications to kinematics

Abstract We address the question of identifying non-smooth points in V ℝ ( I ) {{\bf{V}}}_{{\mathbb{R}}}(I) the real part of an affine algebraic variety. Two simple algebraic criteria will be formulated and proven. As an application, we investigate the configuration spaces of the planar four-bar linkage and the delta robot and prove that all singularities are CS-singularities.

Signatures of Topological Branched Covers

Abstract Let $X^4$ and $Y^4$ be smooth manifolds and $f: X\to Y$ a branched cover with branching set $B$. Classically, if $B$ is smoothly embedded in $Y$, the signature $\sigma (X)$ can be computed from data about $Y$, $B$ and the local degrees of $f$. When $f$ is an irregular dihedral cover and $B\subset Y$ smoothly embedded away from a cone singularity whose link is $K$, the second author gave a formula for the contribution $\Xi (K)$ to $\sigma (X)$ resulting from the non-smooth point. We extend the above results to the case where $Y$ is a topological four-manifold and $B$ is locally flat, away from the possible singularity. Owing to the presence of points on $B$ which are not locally flat, $X$ in this setting is a stratified pseudomanifold, and we use the intersection homology signature of $X$, $\sigma _{IH}(X)$. For any knot $K$ whose determinant is not $\pm 1$, a homotopy ribbon obstruction is derived from $\Xi (K)$, providing a new technique to potentially detect slice knots that are not ribbon.

Revisiting nucleation in the phase-field approach to brittle fracture

Twenty years in since their introduction, it is now plain that the regularized formulations dubbed as phase-field of the variational theory of brittle fracture of Francfort and Marigo (1998) provide a powerful macroscopic theory to describe and predict the propagation of cracks in linear elastic brittle materials under arbitrary quasistatic loading conditions. Over the past ten years, the ability of the phase-field approach to also possibly describe and predict crack nucleation has been under intense investigation. The first of two objectives of this paper is to establish that the existing phase-field approach to fracture at large — irrespectively of its particular version — cannot possibly model crack nucleation. This is so because it lacks one essential ingredient: the strength of the material.The second objective is to amend the phase-field theory in a manner such that it can model crack nucleation, be it from large pre-existing cracks, small pre-existing cracks, smooth and non-smooth boundary points, or within the bulk of structures subjected to arbitrary quasistatic loadings, while keeping undisturbed the ability of the standard phase-field formulation to model crack propagation. The central idea is to implicitly account for the presence of the inherent microscopic defects in the material — whose defining macroscopic manifestation is precisely the strength of the material — through the addition of an external driving force in the equation governing the evolution of the phase field. To illustrate the descriptive and predictive capabilities of the proposed theory, the last part of this paper presents sample simulations of experiments spanning the full range of fracture nucleation settings.

Robust approximate optimal solutions for nonlinear semi-infinite programming with uncertainty

In this paper, we deal with robust approximate quasi optimal solutions for a class of nonlinear semi-infinite programming with data uncertainty in both the objective and constraints. By using a new robust-type subdifferential constraint qualification and some generalized convexity assumptions, we first establish approximate optimality conditions for robust approximate quasi optimal solutions of . Then, we introduce a Mixed-type robust approximate dual problem for and investigate robust approximate duality relations between them. Furthermore, we establish a nonsmooth robust approximate saddle point theorem for an uncertain approximate Lagrangian function associated with .

A New Route to Strange Nonchaotic Attractors in an Interval Map

We identify an unusual route to the creation of a strange nonchaotic attractor (SNA) in a quasiperiodically forced interval map. We find that the smooth quasiperiodic torus becomes nonsmooth due to the grazing bifurcation of the torus. The nonsmooth points on the torus increase more and more with the change of control parameter. Finally, the torus gets extremely fractal and becomes a SNA which is termed the grazing bifurcation route to the SNA. We characterize the SNA by maximal Lyapunov exponents and their variance, phase sensitivity exponents and power spectra. We also describe the transition between a torus and a SNA by the recurrence analysis. A remarkable feature of the route to SNAs is that the positive tails decay linearly and the negative tails exhibit recurrent fluctuations in the distribution of the finite-time Lyapunov exponents.

Numerical study on the influence of initial conditions on quasi-periodic oscillation of double pendulum system

The double pendulum system is a common and typical model to investigate nonliner dynamics due to its various and complex dynamical phenomenon. In this paper, the effects of several parameters on the system was studied, such as the length and the initial release angle of the pendulums. Though the Swing length - frequency diagram, the details of ocsillation was researched and a non-smooth inflection point was observed on the curve.

Helmholtz-Kirchhoff integral formula for predicting the acoustic pressure from a radiator including irregular surface

The Helmholtz-Kirchhoff integral (HKI) for the radiation problem from a finite source with a vibrating surface is a method of calculating the acoustic pressure at any position in the three-dimensional space using the velocity conditions of the radiating surface. Previous studies predicted the acoustic pressure from the radiator based on the HKI. However, the acoustic pressure should be evaluated first on the radiation surface in order to calculate the acoustic pressure in the space. Nonetheless, it is difficult to evaluate the acoustic pressure on the radiation surface accurately if there are non-smooth points. Because the exact HKI is not easily found to be applicable for the evaluation, in this study, we investigated the HKI from the beginning. Through the process of deriving the formula, we proposed the form of the HKI that can be used for calculating the acoustic pressure at both smooth and non-smooth points of the radiation surface. The proposed formula is also proved mathematically. As a result, the acoustic pressures calculated by the proposed formula are compared to those using a FEM. The results yield that the overall relative errors are less than 2% even on the radiator surface including vertices. [Work supported by Samsung Display Corporation.]

Characterizations of Approximate Duality and Saddle Point Theorems for Nonsmooth Robust Vector Optimization

This article deals with robust approximate Pareto efficient solutions for a class of nonsmooth vector optimization problems with data uncertainty. By using robust optimization and scalarization methodology, we first obtain some nonsmooth approximate optimality conditions for this uncertain optimization problem. Then, we introduce a Wolfe-type robust approximate dual problem for this uncertain optimization problem and obtain some nonsmooth robust approximate duality properties. Moreover, we investigate nonsmooth approximate saddle point theorems of an uncertain approximate vector Lagrangian function associated with this uncertain optimization problem.

On the acoustic optimality of leading-edge serration profiles

Leading-edge serrations are studied extensively as a way of reducing leading-edge noise and have been shown to be able to reduce leading-edge noise significantly. Previous experiments showed that different serration geometries have different noise reduction capabilities. However, the optimal serration geometry has not been known. Consequently, there are no guides that can be used at the design stage of serrations. In this paper, by performing an asymptotic analysis, we show that in order to achieve greater noise reduction in the high frequency regime (k1h ≫ 1, where k1 denotes the dimensionless streamwise hydrodynamic wavenumber and h half of the dimensionless root-to-tip amplitude of serrations), the serration profile cannot have stationary points. Therefore, piecewise smooth profiles free of stationary points are more desirable. Moreover, we show that greater noise can be achieved in the high frequency regime by using serrations that are sharper around the non-smooth points. The underlying physical mechanisms of these findings are discussed. Based on these findings, a new type of serration profile is proposed, and analytical model evaluations confirm its improved acoustic performance in the frequency range of interest. At low frequencies, a slight deterioration may be expected, but this is often negligible. To verify the conclusion drawn from the analysis, we perform an experimental study to investigate the acoustic performance of this new serration design. The results show that it is indeed superior than conventional sawtooth serrations. For example, a remarkable 7 dB additional noise reduction is observed in the intermediate frequency range with no perceivable noise increase elsewhere. The trends predicted by the analysis are well validated by the experiment. It is expected that these findings can serve as an essential guide for designing serrations, and lead to more acoustically optimized serration geometries.

Stochastic Control of Multidimensional Systems With Relative Optimization

We derive the Tanaka formula for multidimensional semismooth functions with the local time on the semismooth curve along the gradient direction. With this formula, we extend the relative optimization based approach to stochastic control to multidimensional systems. Optimality conditions are derived for systems with semismooth value functions, and no viscosity solution is involved. This approach provides new insights and motivates the research on stochastic control and stochastic calculus, in particular, for problems with nonsmooth features and degenerate points.