Convex Spline Research Articles

Seakeeping Behavior of Axe Bow Patrol Boat with the Variation of Waterline Spline Type and Submerged Bow Depth

As a sovereign maritime country, Indonesia has been supported with patrol boats to maintain, protect, and manage the fishery and marine resources. Therefore, enhancing the patrol boat's operability is essentially needed for improving the resources maintenance and protection system. The implementation of an advanced hull forms might be conducted for the improvement of boat performance. The axe-bow hull form is one of the advanced hull forms that might improve the patrol boat operability, especially in high-speed conditions (Fr<0.60). However, in order to determine a reliable axe bow hull form, the body lines and the submerge bow depth should be defined appropriately. Based on the condition, this study focuses on the investigation of seakeeping performance of the developed axe-bow hull forms with variations of waterline spline type and submerged bow depth (the depth of foot). The waterline spline type configurations, which consist of the concave spline, convex spline, and straight spline, have been developed. At the same time, the submerged bow depths configuration have been proposed as 30%, 40%, and 50% of the boat draught. Both parameters should be recognized for the improvement of the axe bow hull seakeeping performance. The strip theory method has been adopted for the calculation of the seakeeping behavior. The results show that the waterline spline type has influenced the heave motion behavior. Otherwise, the larger depth of root might improve the pitch motion performance. It is indicated that suitable combination of spline type and depth of root might generate a reliable seakeeping performance of the axe-bow patrol boat.

Aerodynamic Performance Enhancement of an Axisymmetric Deflector Applied to Savonius Wind Turbine Using Novel Transient 3D CFD Simulation Techniques

Many recent studies show that the performance of Savonius turbines can be considerably increased by using wind deflectors. Axisymmetric deflectors are particularly interesting; they concentrate the wind flow in all directions. This study aims to aerodynamically optimize the truncated cone deflector shape through transient 3D CFD simulations using sliding mesh techniques. To reduce the mesh size and thus the simulation time, symmetrical boundary conditions were applied to rotating body faces. A mesh grid sensitivity study was conducted to define the optimum mesh size. Additionally, hybrid numerical approaches combining coupled and SIMPLE solvers were particularly influential in reducing computational time. Concave- and convex-arced-shaped faces deflectors were compared to the original truncated cone deflector, showing an increase in the performance for the convex type and a decrease for the concave one. Then, eight cases involving convex spline shape deflectors were simulated. All these deflectors had an equal volume to the original truncated cone deflector. One of the cases showed a 20% average increase in the performance over the original deflector. This result shows the importance of the geometrical shapes in the design of axisymmetric deflectors.

Data-driven, structure-preserving approximations to entropy-based moment closures for kinetic equations

We present a data-driven approach to construct entropy-based closures for the moment system from kinetic equations. The proposed closure learns the entropy function by fitting the map between the moments and the entropy of the moment system, and thus does not depend on the space-time discretization of the moment system and specific problem configurations such as initial and boundary conditions. With convex and $C^2$ approximations, this data-driven closure inherits several structural properties from entropy-based closures, such as entropy dissipation, hyperbolicity, and H-Theorem. We construct convex approximations to the Maxwell-Boltzmann entropy using convex splines and neural networks, test them on the plane source benchmark problem for linear transport in slab geometry, and compare the results to the standard, optimization-based M$_N$ closures. Numerical results indicate that these data-driven closures provide accurate solutions in much less computation time than the M$_N$ closures.

Study on resistance and intact stability behavior of patrol boat using axe bow hull form to support surveillance activities in the Indonesia territorial sea

As a sovereign maritime country, Indonesia has an obligation to protect, maintain, and manage marine and fishery resources. Patrol boats as the primary means of monitoring marine and fishery resources are needed in response to several problems and threats that may occur, such as illegal fishing, destructive fishing, violations of zoning in fisheries conservation areas and others. Apart from having to be adequate on the number of units, the patrol boats must also have suitable technology to support pursuit operations and suppress violations. The main objective of this research is to investigate the total resistance and intact stability behavior of the developed axe bow hull geometry for the patrol boat hull. The preliminary design is started with the determination of the principal dimension of the patrol boat. Then the hull geometry configurations were made with the variation of the depth of bow and the water line spline type. In the case of the depth of bow, the bow with the additional depth of 30%, 40% and 50% of the draft was investigated. Otherwise, the water line spline type was configured as convex spline, concave spline and straight spline. According to the numerical analysis, the axe bow hull with the depth of bow of 30% and the straight spline type was presented the smallest total resistance performance. However, in the case of intact stability performance, all developed axe bow hull presented a similar righting moment lever arm (GZ curve). It is indicated that the total resistance of the axe bow hull is influenced by the depth of bow and water line spline type. Furthermore, those variables have a slight influence on intact stability performance.

Model-Free Stochastic Collocation for an Arbitrage-Free Implied Volatility, Part II

This paper explores the stochastic collocation technique, applied on a monotonic spline, as an arbitrage-free and model-free interpolation of implied volatilities. We explore various spline formulations, including B-spline representations. We explain how to calibrate the different representations against market option prices, detail how to smooth out the market quotes, and choose a proper initial guess. The technique is then applied to concrete market options and the stability of the different approaches is analyzed. Finally, we consider a challenging example where convex spline interpolations lead to oscillations in the implied volatility and compare the spline collocation results with those obtained through arbitrage-free interpolation technique of Andreasen and Huge.

An Optimized Image Segmentation Approach Based on Boltzmann Machine

ABSTRACTImage segmentation with complex background is a tedious task. In our study, a convex spline is constructed based on Good Features to Track (GF2T) method’s region-based salient feature (i.e., corner) set. For an optimized edge-based segmentation, an ellipse shape prior based on this convex spline is useful in edge regularization procedure with region-based features. This kind of optimization is achieved by Boltzmann machine (BM) to automatically form an elliptical foreground mask of the GrabCut method. We demonstrated our approach’s usability through traveling salesman problem (TSP), thus, we consider that the TSP’s valid tour’s path solved by BM can be taken as an optimized convex spline for edge-based segmentation. In our experiments, proposed BM-based approach has the performance improvement of segmentation to stand-alone GF2T as 29.79% improvement based on bounding boxes evaluation and as 38.67% improvement based on the overlapping pixel regions for a quantitative evaluation via objective metrics.

A new family of convex splines for data interpolation

This paper develops a new family of convexity-preserving splines of order n, hereby entitled the CP n-spline, that preserves convexity when derivatives at the data points satisfy some reasonable conditions. The spline comprises four components: a constant term, a first order term, and two nth order binomials. A slope-averaging-method is proposed for the general implementation of the new spline. Numerical results that allow for an assessment of the new spline are provided. In particular, a comparative analysis of the CP n-spline, the cubic spline, and of the Carnicer '92 spline is performed. By varying two parameters, the spline shape can be controlled at the local level, while other conventional means can be used to control the shape at the global level. The CP n-spline has no singularities in the case where inflection points are present. Additionally, a less general form of the CP n-spline that applies to most practical cases can be implemented with extreme ease.

Surface fitting using convex tensor-product splines

A construction of linear sufficient convexity conditions for polynomial tensor-product spline functions is presented. As the main new feature of this construction, the obtained conditions are asymptotically necessary: increasing the number of linear inequalities in a suitable manner adapts them to any finite set of strongly convex spline surfaces. Based on the linear constraints we formulate least-squares approximation of scattered data by spline surfaces as a quadratic programming problem.

Three-Convex Approximation by Free Knot Splines in C [0, 1

A Jackson-type estimate is obtained for the approximation of 3 -convex functions by 3 -convex splines with free knots. The order of approximation is the same as for the Jackson-type estimate for unconstrained approximation by splines with free knots. Shape-preserving free knot spline approximation of k -convex functions, k > 3 , is also considered.

Staircase algorithm and construction of convex spline interpolants up to the continuity C3

This paper is concerned with the convex interpolation of data sets. Based on the staircase algorithm, several methods are presented which allow the construction of convex spline interpolants up to the continuity C 3 .

Convex Polynomial and Spline Approximation in $L_p, 0

Convex Polynomial and Spline Approximation in $L_p, 0

An efficient algorithm for line and polygon clipping

We present an algorithm for clipping a polygon or a line against a convex polygonal window. The algorithm demonstrates the practicality of various ideas from computational geometry. It spendsO(logp) time on each edge of the clipped polygon, wherep is the number of window edges, while the Sutherland-Hodgman algorithm spendsO(p) time per edge. Theoretical and experimental analyses show that the constants involved are small enough to make the algorithm competitive even for windows with four edges. The algorithm enables image-space clipping against windows whose boundaries are convex spline curves. The paper contains detailed pseudo-code implementation of the algorithm and an adaptation of the simulation of simplicity method for handling degenerate cases.

On Monotone and Convex Spline Interpolation

This paper is concerned with the problem of existence of monotone and/or convex splines, having degree n and order of continuity k, which interpolate to a set of data at the knots. The interpolating splines are obtained by using Bernstein polynomials of suitable continuous piecewise linear functions; they satisfy the inequality $k \leqslant n - k$. The theorems presented here are useful in developing algorithms for the construction of shape-preserving splines interpolating arbitrary sets of data points. Earlier results of McAllister, Passow and Roulier can be deduced from those given in this paper.

On monotone and convex spline interpolation

This paper is concerned with the problem of existence of monotone and/or convex splines, having degree n and order of continuity k, which interpolate to a set of data at the knots. The interpolating splines are obtained by using Bernstein polynomials of suitable continuous piecewise linear functions; they satisfy the inequality k ⩽ n − k k \leqslant n - k . The theorems presented here are useful in developing algorithms for the construction of shape-preserving splines interpolating arbitrary sets of data points. Earlier results of McAllister, Passow and Roulier can be deduced from those given in this paper.

Optimal Interpolation with Convex Splines of Second Degree

Estimation of a convex function interpolating its known values and satisfying certain smooth-ness properties is needed in some applications and has been investigated in many studies from various perspectives. Without the convexity assumption, Karlin’s theorem characterizes the solution to the problem studied here while under convexity, the analogous result is due to Smith and Ward. The aim of this paper is to give a convex programming characterization that can be used to calculate an optimal spline which solves a given problem of degree 2. Specifically, our aim is to determine a smooth (f, $f^{(1)} $ absolutely continuous) convex spline f of second degree, which interpolates $(r + 2)$ given points in $[a,b]$, and minimizes the Tchebycheff norm $\| {f^{(2)} } \|_\infty $, with $f^{(2)} $ essentially bounded in $[a,b]$. The convexity of the formulation enables us to calculate an optimal spline using widely available computer routines for nonlinear optimization. The approach is illustrated by providing the convex programming formulations and the computer-obtained optimal solutions for two numerical examples.

Convex spline interpolants with minimal curvature

The problem of finding convex spline interpolants with minimal mean curvature leads to a quadratic optimization problem of special structure. In the present note a corresponding dual problem without constraints is derived. Its objective function is piecewise quadratic and therefore admits an effective numerical treatment.

Convex Approximation by Splines

Jackson type estimates are obtained for the approximation of convex functions by convex splines with equally spaced knots. The results are of the same order as the Jackson type estimates for unconstrained approximation by splines with equally spaced knots.

Isotonic, Convex and Related Splines

In this paper, we consider the estimation of isotonic, convex or related functions by means of splines. It is shown that certain classes of isotone or convex functions can be represented as a positive cone embedded in a Hilbert space. Using this representation, we give an existence and characterization theorem for isotonic or convex splines. Two special cases are examined showing the existence of a globally monotone cubic smoothing spline and a globally convex quintic smoothing spline. Finally, we examine a regression problem and show that the isotonic-type of spline provides a strongly consistent solution. We also point out several other statistical applications.

Markov-type inequalities and the degree of convex spline interpolation

It has been shown by Passow and Roulier [S] and by McAllister and Roulier [6] that it is impossible to interpolate convex data by a smooth, convex, piecewise polynomial with lixed knots and bounded degree, inde- pendent of the data and knots. Indeed, a lower bound for this degree is given in [6] and [8] which shows that a suitable choice of the j.-coordinates of the data points can make the degree as large as desired. In this paper. we show that the lower bound mentioned above is best possible. This is accomplished by observing that the classical Markov inequality cannot be improved by restricting consideration to convex in- creasing polynomials.

Monotone and Convex Spline Interpolation

For a set of monotone (and/or convex) data, we consider the possibility of finding a spline interpolant, of pre-determined smoothness, which is monotone (and/or convex). The investigation is carried out by constructing an auxiliary set of points and using the well-known monotonicity and convexity preserving properties of Bernstein polynomials. In § 3 we consider the problem of piecewise monotone interpolation.