On the Role of Natural Level Functions to Achieve Global Convergence for Damped Newton Methods

Bock, Hans Georg; Kostina, Ekaterina; Schlöder, Johannes P.

doi:10.1007/978-0-387-35514-6_3

Cited by 24 publications

(29 citation statements)

References 10 publications

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“…We do not regard our step size selection in the damped Newton iteration as a settled matter, and plan to improve our strategy, for example using the idea of restric-tive monotonicity test discussed in Bock et al [2000]. The idea of giving the solution process a kick by taking the full Newton step rather than the smallest allowed step size if the Newton iteration appears to be failing to converge, described in Section 4.1, seems to work well in practice.…”

Section: Limitations and Future Directionsmentioning

confidence: 99%

Automatic Fréchet Differentiation for the Numerical Solution of Boundary-Value Problems

Birkisson

Driscoll

2012

ACM Trans. Math. Softw.

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A new solver for nonlinear boundary-value problems (BVPs) in MATLAB is presented, based on the Chebfun software system for representing functions and operators automatically as numerical objects. The solver implements Newton's method in function space, where instead of the usual Jacobian matrices, the derivatives involved are Fréchet derivatives. A major novelty of this approach is the application of automatic differentiation (AD) techniques to compute the operator-valued Fréchet derivatives in the continuous context. Other novelties include the use of anonymous functions and numbering of each variable to enable a recursive, delayed evaluation of derivatives with forward mode AD. The AD techniques are applied within a new Chebfun class called chebop which allows users to set up and solve nonlinear BVPs, both scalar and systems of coupled equations, in a few lines of code, using the "nonlinear backslash" operator (\). This framework enables one to study the behaviour of Newton's method in function space.

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Section: Limitations and Future Directionsmentioning

confidence: 99%

Automatic Fréchet Differentiation for the Numerical Solution of Boundary-Value Problems

Birkisson

Driscoll

2012

ACM Trans. Math. Softw.

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“…An alternative to this approach is to generalize the restrictive monotonicity test according to Bock et al [2].…”

Section: Robust Estimatorsmentioning

confidence: 99%

“…At each iteration of the generalized Gauss-Newton method we have to solve linearized problems (2), which are equivalent to linear programming problems (LAD estimator) or quadratic programming problems (Huber estimator) with special structures in constraints (see the transformations discussed in the previous section). One may apply any standard method to solve (2).…”

Section: Robust Estimatorsmentioning

confidence: 99%

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Approximate Gauss‐Newton Method for Robust Parameter Estimation

Binder

Kostina

2010

Proc Appl Math and Mech

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We consider robust parameter estimation problems in which either the l 1 norm or the Huber function of a measurement error vector used as cost functionals. In order to avoid high computational effort of computing exact derivatives needed for the solution of these problems with the Gauss-Newton method, we suggest to use approximations of the derivatives in the occurring linearized subproblems. We show how the error introduced by using only approximated derivatives can be compensated by adding a correction term to the objective function of the linearized problems. Robust estimatorsOne of the most important tasks in model validation and calibration is the identification of unknown parameters from sets of data measured during experiments with the real process. This is accomplished by solving parameter estimation problems. However, large measurement errors can strongly affect the estimation results and thus make the calibrated model meaningless for simulation. Thus, robust parameter estimation is needed. The term robustness here means "insensitivity to deviations from the assumptions" as used by Huber [5]. We consider in this paper two robust estimators, namely the least absolute deviations (LAD) estimator and Huber's M -estimator. The objective function of the LAD estimator is the l 1 norm of model deviationIt is commonly known that a characteristic of this estimator is its feature of interpolating then "best" measurements, wheren denotes the number of degrees of freedom in the minimization problem. We can reformulate this estimator as a nonlinear optimization problem with equality and inequality constraints and a smooth objective functionThe Huber criterion for data fitting is a combination of least squares and LAD estimation. As an objective function, the Huber function (s i +t i ), s.t. M (x) − η − v = s − t, v ∈ R m , s ∈ R m , t ∈ R m s, t ≥ 0.

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“…For solving linearized trust region subproblems we develop a structure exploiting method combined with exploitation of known initial conditions, thus reducing the number of necessary directional derivatives of initial value problem to a minimum. The trust region control is based on Restictive Monotonicity Test, which has shown very good performance in practice, see [2].…”

Section: Parameter Estimation Problem For Nonlinear Exchange Rate Dynmentioning

confidence: 99%

Parameter estimation for forward Kolmogorov equation with application to nonlinear exchange rate dynamics

Jäger

Kostina

2005

Proc Appl Math and Mech

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Diffusion processes are widely used for mathematical modeling in finance e.g. in modeling foreign exchange rates. Stochastic differential equations describing diffusion processes are linked directly to the forward Kolmogorov equations. In order to calibrate the models, efficient algorithms identifying the system parameters are in demand. Taking into account nonlinear effects in volatility and drift and dependence on observed economical data, which are not directly modeled, one obtains problems which cannot be treated by standard numerical methods. The coefficients are rapidly oscillatory and strong instabilities may arise. To handle these problem we develop special numerical methods, which are used to simulate the nonlinear dynamics of exchange rates depending on economic data. Nonlinear Price DynamicsIt is common practice in fi nancial modeling that the price dynamics S is modeled by an Itô stochastic differential equation:( 1) Here Z(t) are external, such as economic or political effects and W is a standard Wiener process with the property that dW is distributed as N (0, dt), and µ and σ satisfy Lipschitz and growth conditions suffi cient for the existence of a continuous solution to (1). In case that all necessary coeffi cients of the model equation are known, solutions to (1) can be computed using available algorithms. However, in reality the drift term µ(·) and the volatility σ(·) are unknown and need to be determined by modeling and/or by solving a parameter estimation problem. Parameter Estimation Problem for Nonlinear Exchange Rate DynamicsWe consider a stochastic price process S t , t ∈ [t 0 , T ] with the distribution F t (s) = P (S t ≤ s), t ∈ [t 0 , T ] and s ∈ R. It is assumed that the drift term and the volatility function of the stochastic differential equation (1) depend on the spatial variable s and unknown parameter vector θ: µ = µ(t, s; θ), σ 2 = σ 2 (t, s; θ). Using the fact that the transitional price distribution f (t, s) = dFt(s) ds of the stochastic process S t at each point of time satisfi es the forward Kolmogorov equation, see e.g.[5], we estimate the unknown parameters by solving the following optimization problemHere, the least squares functional (2) can be interpreted as a weighted norm of the difference between the real values ξ j of the random variable s at time points t j , j = 1, .., k, and their expected values. The constraints in the problem are the forward Kolmogorov equation (3) for the density function f (t, s), boundary conditions (4) and initial conditions (5) with two additional parameters a and s 0 to be estimated, η is a normalizing constant.

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On the Role of Natural Level Functions to Achieve Global Convergence for Damped Newton Methods

Cited by 24 publications

References 10 publications

Automatic Fréchet Differentiation for the Numerical Solution of Boundary-Value Problems

Automatic Fréchet Differentiation for the Numerical Solution of Boundary-Value Problems

Approximate Gauss‐Newton Method for Robust Parameter Estimation

Parameter estimation for forward Kolmogorov equation with application to nonlinear exchange rate dynamics

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