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On stable numerical differentiation
 Mathem. of Computation
, 1968
"... Abstract. A new approach to the construction of finitedifference methods is presented. It is shown how the multipoint differentiators can generate regularizing algorithms with a stepsize h being a regularization parameter. The explicitly computable estimation constants are given. Also an iterative ..."
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Cited by 39 (22 self)
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Abstract. A new approach to the construction of finitedifference methods is presented. It is shown how the multipoint differentiators can generate regularizing algorithms with a stepsize h being a regularization parameter. The explicitly computable estimation constants are given. Also an iteratively regularized scheme for solving the numerical differentiation problem in the form of Volterra integral equation is developed. 1.
A Mollification Framework For Improperly Posed Problems
 Numer. Math
, 1996
"... this paper is to show how, using the classical semigroup theory of Hille and Phillips [8], a quite general theory can be constructed for the mollification of operator equations ..."
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Cited by 5 (0 self)
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this paper is to show how, using the classical semigroup theory of Hille and Phillips [8], a quite general theory can be constructed for the mollification of operator equations
For Numerical Differentiation, Dimensionality Can Be A Blessing!
, 1997
"... . Finite difference methods, like the midpoint rule, have been applied successfully to the numerical solution of ordinary and partial differential equations. If such formulas are applied to observational data, in order to determine derivatives, the results can be disastrous. The reason for this is ..."
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Cited by 4 (0 self)
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. Finite difference methods, like the midpoint rule, have been applied successfully to the numerical solution of ordinary and partial differential equations. If such formulas are applied to observational data, in order to determine derivatives, the results can be disastrous. The reason for this is that measurement errors, and even rounding errors in computer approximations, are strongly amplified in the differentiation process, especially if small stepsizes are chosen and higher derivatives are required. A number of authors have examined the use of various forms of averaging which allows the stable computation of low order derivatives from observational data. The size of the averaging set acts like a regularization parameter and has to be chosen as a function of the grid size h. In this paper, it is initially shown how first (and higher) order singlevariate numerical differentiation of higher dimensional observational data can be stabilized with a reduced loss of accuracy than occu...
OMAE200979051 NUMERICAL SIMULATION OF SLOSHING IN A TANK, CFD CALCULATIONS AGAINST MODEL TESTS
, 2009
"... Simulation of liquid dynamics in an LNG tank is studied numerically. The applied CFD code solves NavierStokes equations and uses an improved Volume of Fluid (iVOF) method to track movement of fluid’s free surface. Relative advantages of using two different fluid models, singlephase (liquid+void) a ..."
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Simulation of liquid dynamics in an LNG tank is studied numerically. The applied CFD code solves NavierStokes equations and uses an improved Volume of Fluid (iVOF) method to track movement of fluid’s free surface. Relative advantages of using two different fluid models, singlephase (liquid+void) and twophase (liquid+compressible gas) are discussed, the latter model being capable of simulating bubbles and gas entrapped in liquid. Furthermore, the 1 st and 2 nd order upwind differencing schemes are used with both physical models leading to a total of four possible approaches to solve the problem. Numerical results are verified against experimental data from large scale (1:10) sloshing experiments of 2D section of an LNG carrier. The CFD vs. experiment comparison is shown for tank filling rates of practical interest, ranging from 10 % to 95%, and includes both fluid height and fluid pressure exerted on tank walls. A visual comparison in form of computer animation frames, synchronised with cameramade movies taken during the experiments is included as well. Finally, an exhaustive computational grid convergence study is presented for lower filling rates of the tank.
Adaptive Parameter Choice for OneSided Finite Difference Schemes and its Application in Diabetes Technology
"... In this paper we discuss the problem of an adaptive parameter choice in onesided finite difference schemes for the numerical differentiation in case when noisyvaluesofthefunctiontobedifferentiatedareavailableonlyatthegiven points. This problem is motivated by diabetes therapy management, where it is ..."
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In this paper we discuss the problem of an adaptive parameter choice in onesided finite difference schemes for the numerical differentiation in case when noisyvaluesofthefunctiontobedifferentiatedareavailableonlyatthegiven points. This problem is motivated by diabetes therapy management, where it is important to provide estimations of the future blood glucose trend from current and past measurements. Here we show, how the proposed approach can be used for this purpose and demonstrate some illustrative tests, as well as the results of numerical experiments with simulated clinical data. 1. Problem formulation In this paper we consider the problem of approximation of a derivative y ′ (B) at the boundary point of some interval [b,B] under the condition that at the given points B = tN> tN−1> ·· ·> t1 ≥ b (1) only noisy values yδ(tj) of y(tj) are available such that
ON COMPLEXVALUED 2D EIKONALS. PART FOUR: CONTINUATION PAST A CAUSTIC
, 905
"... Abstract. Theories of monochromatic highfrequency electromagnetic fields have been designed by Felsen, Kravtsov, Ludwig and others with a view to portraying features that are ignored by geometrical optics. These theories have recourse to eikonals that encode information on both phase and amplitude ..."
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Abstract. Theories of monochromatic highfrequency electromagnetic fields have been designed by Felsen, Kravtsov, Ludwig and others with a view to portraying features that are ignored by geometrical optics. These theories have recourse to eikonals that encode information on both phase and amplitude — in other words, are complexvalued. The following mathematical principle is ultimately behind the scenes: any geometric optical eikonal, which conventional rays engender in some light region, can be consistently continued in the shadow region beyond the relevant caustic, provided an alternative eikonal, endowed with a nonzero imaginary part, comes on stage. In the present paper we explore such a principle in dimension 2. We investigate a partial differential system that governs the real and the imaginary parts of complexvalued twodimensional eikonals, and an initial value problem germane to it. In physical terms, the problem in hand amounts to detecting waves that rise beside, but on the dark side of, a given caustic. In mathematical terms, such a problem shows two main peculiarities: on the one hand, degeneracy near the initial curve; on the other hand, illposedness in the sense of Hadamard. We benefit from using a number of technical devices: hodograph transforms, artificial viscosity, and a suitable discretization. Approximate differentiation and a parody of the quasireversibility method are also involved. We offer an algorithm that restrains instability and produces effective approximate solutions. 1.