scholarly journals KERNEL CONVERGENCE ESTIMATES FOR DIFFUSIONS WITH CONTINUOUS COEFFICIENTS

2011 ◽  
Vol 14 (07) ◽  
pp. 979-1004
Author(s):  
CLAUDIO ALBANESE
Keyword(s):  
Convergence Rate ◽  
Convergence Rates ◽  
Fourier Transforms ◽  
Uniform Continuity ◽  
Time Step ◽  
Uniform Bounds ◽  
Kernel Convergence ◽  
Discretization Schemes ◽  
A New Technique ◽  
Derivatives Of

Bidirectional valuation models are based on numerical methods to obtain kernels of parabolic equations. Here we address the problem of robustness of kernel calculations vis a vis floating point errors from a theoretical standpoint. We are interested in kernels of one-dimensional diffusion equations with continuous coefficients as evaluated by means of explicit discretization schemes of uniform step h > 0 in the limit as h → 0. We consider both semidiscrete triangulations with continuous time and explicit Euler schemes with time step so small that the Courant condition is satisfied. We find uniform bounds for the convergence rate as a function of the degree of smoothness. We conjecture these bounds are indeed sharp. The bounds also apply to the time derivatives of the kernel and its first two space derivatives. The proof is constructive and is based on a new technique of path conditioning for Markov chains and a renormalization group argument. We make the simplifying assumption of time-independence and use longitudinal Fourier transforms in the time direction. Convergence rates depend on the degree of smoothness and Hölder differentiability of the coefficients. We find that the fastest convergence rate is of order O(h2) and is achieved if the coefficients have a bounded second derivative. Otherwise, explicit schemes still converge for any degree of Hölder differentiability except that the convergence rate is slower. Hölder continuity itself is not strictly necessary and can be relaxed by an hypothesis of uniform continuity.

Stability Analysis of the Nanjung Water Diversion Twin Tunnels based on Convergence Measurement

2019 ◽  
Vol 1 (1) ◽  
pp. 49-60
Author(s):  
Simon Heru Prassetyo ◽  
Ganda Marihot Simangunsong ◽  
Ridho Kresna Wattimena ◽  
Made Astawa Rai ◽  
Irwandy Arif ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Convergence Rate ◽  
Critical Strain ◽  
Convergence Rates ◽  
Strain Analysis ◽  
Water Diversion ◽  
Tunnel Face ◽  
Tunnel Stability ◽  
The Stability ◽  
Twin Tunnels

This paper focuses on the stability analysis of the Nanjung Water Diversion Twin Tunnels using convergence measurement. The Nanjung Tunnel is horseshoe-shaped in cross-section, 10.2 m x 9.2 m in dimension, and 230 m in length. The location of the tunnel is in Curug Jompong, Margaasih Subdistrict, Bandung. Convergence monitoring was done for 144 days between February 18 and July 11, 2019. The results of the convergence measurement were recorded and plotted into the curves of convergence vs. day and convergence vs. distance from tunnel face. From these plots, the continuity of the convergence and the convergence rate in the tunnel roof and wall were then analyzed. The convergence rates from each tunnel were also compared to empirical values to determine the level of tunnel stability. In general, the trend of convergence rate shows that the Nanjung Tunnel is stable without any indication of instability. Although there was a spike in the convergence rate at several STA in the measured span, that spike was not replicated by the convergence rate in the other measured spans and it was not continuous. The stability of the Nanjung Tunnel is also confirmed from the critical strain analysis, in which most of the STA measured have strain magnitudes located below the critical strain line and are less than 1%.

A NEW TECHNIQUE ESTIMATING LOCATION OF MEAN JOINT CENTERS OF ROTATION WITH ASSOCIATED DISPERSIONS AND ASSESSING TO FUNCTIONAL COORDINATE SYSTEM OF MOVING LIMB SEGMENTS

2006 ◽  
Vol 06 (04) ◽  
pp. 373-384
Author(s):  
ERIC BERTHONNAUD ◽  
JOANNÈS DIMNET
Keyword(s):  
Numerical Technique ◽  
New Technique ◽  
Helical Axis ◽  
Joint Movement ◽  
Time Step ◽  
Data Set ◽  
Minimization Procedure ◽  
A New Technique ◽  
Joint Center

Joint centers are obtained from data treatment of a set of markers placed on the skin of moving limb segments. Finite helical axis (FHA) parameters are calculated between time step increments. Artifacts associated with nonrigid body movements of markers entail ill-determination of FHA parameters. Mean centers of rotation may be calculated over the whole movement, when human articulations are likened to spherical joints. They are obtained using numerical technique, defining point with minimal amplitude, during joint movement. A new technique is presented. Hip, knee, and ankle mean centers of rotation are calculated. Their locations depend on the application of two constraints. The joint center must be located next to the estimated geometric joint center. The geometric joint center may migrate inside a cube of possible location. This cube of error is located with respect to the marker coordinate systems of the two limb segments adjacent to the joint. Its position depends on the joint and the patient height, and is obtained from a stereoradiographic study with specimen. The mean position of joint center and corresponding dispersion are obtained through a minimization procedure. The location of mean joint center is compared with the position of FHA calculated between different sequential steps: time sequential step, and rotation sequential step where a minimal rotation amplitude is imposed between two joint positions. Sticks are drawn connecting adjacent mean centers. The animation of stick diagrams allows clinical users to estimate the displacements of long bones (femur and tibia) from the whole data set.

scholarly journals Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen–Cahn equation

2018 ◽  
Vol 39 (4) ◽  
pp. 2096-2134 ◽  
Author(s):  
Charles-Edouard Bréhier ◽  
Jianbo Cui ◽  
Jialin Hong
Keyword(s):  
Convergence Rate ◽  
Strong Convergence ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
A Priori Estimates ◽  
A Priori ◽  
Auxiliary Problem ◽  
Splitting Scheme ◽  
Cahn Equation ◽  
Strong Convergence Rate

Abstract This article analyses an explicit temporal splitting numerical scheme for the stochastic Allen–Cahn equation driven by additive noise in a bounded spatial domain with smooth boundary in dimension $d\leqslant 3$. The splitting strategy is combined with an exponential Euler scheme of an auxiliary problem. When $d=1$ and the driving noise is a space–time white noise we first show some a priori estimates of this splitting scheme. Using the monotonicity of the drift nonlinearity we then prove that under very mild assumptions on the initial data this scheme achieves the optimal strong convergence rate $\mathcal{O}(\delta t^{\frac 14})$. When $d\leqslant 3$ and the driving noise possesses some regularity in space we study exponential integrability properties of the exact and numerical solutions. Finally, in dimension $d=1$, these properties are used to prove that the splitting scheme has a strong convergence rate $\mathcal{O}(\delta t)$.

A comparison of convergence rates for three models in the theory of dams

1997 ◽  
Vol 34 (01) ◽  
pp. 74-83
Author(s):  
Robert Lund ◽  
Walter Smith
Keyword(s):  
Convergence Rate ◽  
Convergence Rates ◽  
Sample Path ◽  
Jump Process ◽  
Limiting Distribution ◽  
Regenerative Processes ◽  
Input Process ◽  
The Moment ◽  
Finite Dam ◽  
General Conditions

This paper compares the convergence rate properties of three storage models (dams) driven by time-homogeneous jump process input: the infinitely high dam, the finite dam, and the infinitely deep dam. We show that the convergence rate of the infinitely high dam depends on the moment properties of the input process, the finite dam always approaches its limiting distribution exponentially fast, and the infinitely deep dam approaches its limiting distribution exponentially fast under very general conditions. Our methods make use of rate results for regenerative processes and several sample path orderings.

scholarly journals Convergence rates for estimators of geodesic distances and Frèchet expectations

2018 ◽  
Vol 55 (4) ◽  
pp. 1001-1013
Author(s):  
Catherine Aaron ◽  
Olivier Bodart
Keyword(s):  
Probability Distribution ◽  
Convergence Rate ◽  
Compact Manifold ◽  
Convergence Rates ◽  
Geodesic Distance ◽  
Convergence Result ◽  
Regularity Conditions ◽  
Compact Set ◽  
Geodesic Distances ◽  
General Convergence

Abstract Consider a sample 𝒳n={X1,…,Xn} of independent and identically distributed variables drawn with a probability distribution ℙX supported on a compact set M⊂ℝd. In this paper we mainly deal with the study of a natural estimator for the geodesic distance on M. Under rather general geometric assumptions on M, we prove a general convergence result. Assuming M to be a compact manifold of known dimension d′≤d, and under regularity assumptions on ℙX, we give an explicit convergence rate. In the case when M has no boundary, knowledge of the dimension d′ is not needed to obtain this convergence rate. The second part of the work consists in building an estimator for the Fréchet expectations on M, and proving its convergence under regularity conditions, applying the previous results.

Characterization of convergence rates for the approximation of the stationary distribution of infinite monotone stochastic matrices

1996 ◽  
Vol 33 (04) ◽  
pp. 974-985 ◽  
Author(s):  
F. Simonot ◽  
Y. Q. Song
Keyword(s):  
Convergence Rate ◽  
Generating Function ◽  
Convergence Rates ◽  
Transition Probability ◽  
Stochastic Matrix ◽  
Input Sequence ◽  
Transition Probability Matrix ◽  
Stochastic Matrices ◽  
Preceding Result

Let P be an infinite irreducible stochastic matrix, recurrent positive and stochastically monotone and Pn be any n × n stochastic matrix with Pn ≧ Tn , where Tn denotes the n × n northwest corner truncation of P. These assumptions imply the existence of limit distributions π and π n for P and Pn respectively. We show that if the Markov chain with transition probability matrix P meets the further condition of geometric recurrence then the exact convergence rate of π n to π can be expressed in terms of the radius of convergence of the generating function of π. As an application of the preceding result, we deal with the random walk on a half line and prove that the assumption of geometric recurrence can be relaxed. We also show that if the i.i.d. input sequence (A(m)) is such that we can find a real number r 0 > 1 with , then the exact convergence rate of π n to π is characterized by r 0. Moreover, when the generating function of A is not defined for |z| > 1, we derive an upper bound for the distance between π n and π based on the moments of A.

scholarly journals Local Discontinuous Galerkin Method for Nonlinear Time-Space Fractional Subdiffusion/Superdiffusion Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-21
Author(s):  
Meilan Qiu ◽  
Dewang Li ◽  
Yanyun Wu
Keyword(s):  
Convergence Rate ◽  
Fractional Derivative ◽  
Discontinuous Galerkin ◽  
Numerical Schemes ◽  
Step Size ◽  
Time Step ◽  
Local Discontinuous Galerkin ◽  
Time Space ◽  
Backward Euler ◽  
Heterogeneous Rocks

Fractional partial differential equations with time-space fractional derivatives describe some important physical phenomena. For example, the subdiffusion equation (time order 0<α<1) is more suitable to describe the phenomena of charge carrier transport in amorphous semiconductors, nuclear magnetic resonance (NMR) diffusometry in percolative, Rouse, or reptation dynamics in polymeric systems, the diffusion of a scalar tracer in an array of convection rolls, or the dynamics of a bead in a polymeric network, and so on. However, the superdiffusion case (1<α<2) is more accurate to depict the special domains of rotating flows, collective slip diffusion on solid surfaces, layered velocity fields, Richardson turbulent diffusion, bulk-surface exchange controlled dynamics in porous glasses, the transport in micelle systems and heterogeneous rocks, quantum optics, single molecule spectroscopy, the transport in turbulent plasma, bacterial motion, and even for the flight of an albatross (for more physical applications of fractional sub-super diffusion equations, one can see Metzler and Klafter in 2000). In this work, we establish two fully discrete numerical schemes for solving a class of nonlinear time-space fractional subdiffusion/superdiffusion equations by using backward Euler difference 1<α<2 or second-order central difference 1<α<2/local discontinuous Galerkin finite element mixed method. By introducing the mathematical induction method, we show the concrete analysis for the stability and the convergence rate under the L2 norm of the two LDG schemes. In the end, we adopt several numerical experiments to validate the proposed model and demonstrate the features of the two numerical schemes, such as the optimal convergence rate in space direction is close to Ohk+1. The convergence rate in time direction can arrive at Oτ2−α when the fractional derivative is 0<α<1. If the fractional derivative parameter is 1<α<2 and we choose the relationship as h=C′τ (h denotes the space step size, C′ is a constant, and τ is the time step size), then the time convergence rate can reach to Oτ3−α. The experiment results illustrate that the proposed method is effective in solving nonlinear time-space fractional subdiffusion/superdiffusion equations.

scholarly journals Product rule for vector fractional derivatives

2012 ◽  
Vol 15 (3) ◽  
Author(s):  
Diogo Bolster ◽  
Mark Meerschaert ◽  
Alla Sikorskii
Keyword(s):  
Vector Space ◽  
Fractional Derivatives ◽  
Fourier Transforms ◽  
Product Rule ◽  
Finite Dimensional ◽  
Euclidean Vector Space ◽  
Derivatives Of

AbstractThis paper establishes a product rule for fractional derivatives of a realvalued function defined on a finite dimensional Euclidean vector space. The proof uses Fourier transforms.

Geometric renewal convergence rates from hazard rates

2001 ◽  
Vol 38 (01) ◽  
pp. 180-194 ◽  
Author(s):  
Kenneth S. Berenhaut ◽  
Robert Lund
Keyword(s):  
Convergence Rate ◽  
Hazard Rate ◽  
Convergence Rates ◽  
Hazard Rates ◽  
Finite Support ◽  
Good Convergence ◽  
Renewal Sequence ◽  
Geometric Convergence ◽  
New Better Than Used ◽  
Better Than

This paper studies the geometric convergence rate of a discrete renewal sequence to its limit. A general convergence rate is first derived from the hazard rates of the renewal lifetimes. This result is used to extract a good convergence rate when the lifetimes are ordered in the sense of new better than used or increasing hazard rate. A bound for the best possible geometric convergence rate is derived for lifetimes having a finite support. Examples demonstrating the utility and sharpness of the results are presented. Several of the examples study convergence rates for Markov chains.

scholarly journals Efficient Actor-Critic Algorithm with Hierarchical Model Learning and Planning

2016 ◽  
Vol 2016 ◽  
pp. 1-15 ◽  
Author(s):  
Shan Zhong ◽  
Quan Liu ◽  
QiMing Fu
Keyword(s):  
Convergence Rate ◽  
Hierarchical Model ◽  
Function Approximation ◽  
Local Linear Regression ◽  
Benchmark Problems ◽  
Time Step ◽  
Model Learning ◽  
Linear Function Approximation ◽  
Efficient Learning ◽  
The Value Function

To improve the convergence rate and the sample efficiency, two efficient learning methods AC-HMLP and RAC-HMLP (AC-HMLP withl2-regularization) are proposed by combining actor-critic algorithm with hierarchical model learning and planning. The hierarchical models consisting of the local and the global models, which are learned at the same time during learning of the value function and the policy, are approximated by local linear regression (LLR) and linear function approximation (LFA), respectively. Both the local model and the global model are applied to generate samples for planning; the former is used only if the state-prediction error does not surpass the threshold at each time step, while the latter is utilized at the end of each episode. The purpose of taking both models is to improve the sample efficiency and accelerate the convergence rate of the whole algorithm through fully utilizing the local and global information. Experimentally, AC-HMLP and RAC-HMLP are compared with three representative algorithms on two Reinforcement Learning (RL) benchmark problems. The results demonstrate that they perform best in terms of convergence rate and sample efficiency.

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