Numerical solution of Helmholtz equation by the modified Hopfield finite difference techniques

Dehghan, Mehdi; Nourian, Mojtaba; Menhaj, Mohammad Bagher

doi:10.1002/num.20366

Cited by 17 publications

(10 citation statements)

References 32 publications

(58 reference statements)

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“…We now point out a selection of other approaches from the scientific literature to numerically approximate solutions of nonlinear parabolic PDEs. Deterministic approximation methods, such as finite element and sparse-grid methods, can be found, for example, in [34,98,99,101]. There are also a number of approximation methods for nonlinear parabolic PDEs in the scientific literature whose derivation is based on probabilistic concepts.…”

Section: Introductionmentioning

confidence: 99%

Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks

Hutzenthaler¹,

Jentzen²,

Wurstemberger³

2020

Electron. J. Probab.

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Parabolic partial differential equations (PDEs) are widely used in the mathematical modeling of natural phenomena and man-made complex systems. In particular, parabolic PDEs are a fundamental tool to approximately determine fair prices of financial derivatives in the financial engineering industry. The PDEs appearing in financial engineering applications are often nonlinear (e.g., in PDE models which take into account the possibility of a defaulting counterparty) and high-dimensional since the dimension typically corresponds to the number of considered financial assets. A major issue in the scientific literature is that most approximation methods for nonlinear PDEs suffer from the so-called curse of dimensionality in the sense that the computational effort to compute an approximation with a prescribed accuracy grows exponentially in the dimension of the PDE or in the reciprocal of the prescribed approximation accuracy and nearly all approximation methods for nonlinear PDEs in the scientific literature have not been shown not to suffer from the curse of dimensionality. Recently, a new class of approximation schemes for semilinear parabolic PDEs, termed full history recursive multilevel Picard (MLP) algorithms, were introduced and it was proven that MLP algorithms do overcome the curse of dimensionality for semilinear heat equations. In this paper we extend and generalize those findings to a more general class of semilinear PDEs which includes as special cases the important examples of semilinear Black-Scholes equations used in pricing models for financial derivatives with default risks. In particular, we introduce an MLP algorithm for the approximation of solutions of semilinear Black-Scholes equations and prove, under the assumption that the nonlinearity in the PDE is globally Lipschitz continuous, that the computational effort of the proposed method grows at most polynomially in both the

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Section: Introductionmentioning

confidence: 99%

Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks

Hutzenthaler¹,

Jentzen²,

Wurstemberger³

2020

Electron. J. Probab.

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“…Therefore, the use of the ML algorithms can be advantageous because they can process nonlinear and noisy data, and predict the performance of the biological systems. ANN is one of the commonly used models and it is inspired by the basic function of a biological neuron (Dehghan et al 2009). Its use was reported in several phytoextraction studies such as the phytoextraction of basic red 46 using watercress (Torbati et al 2014) and Lemna minor (Movafeghi et al 2013).…”

Section: Introductionmentioning

confidence: 99%

Phytoextraction capability of Azolla pinnata in the removal of rhodamine B from aqueous solution: artificial neural network and random forests approaches

et al. 2019

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This study used live Azolla pinnata (AP) to remediate rhodamine B (RB) from aqueous solutions via the phytoextraction method, and machine learning algorithms such as artificial neural networks and random forests were used as predictive models. The pH was found to have a major influence on the phytoextraction process, and the AP dosage can change the pH of the aqueous solution. The optimum condition for the phytoextraction of RB (initial dye concentration at 10 ppm) is at pH 3.0 with a plant dosage of 0.4 g, resulting in removal efficiency as high as 76%. The growth estimation (relative frond number) indicates that AP can tolerate RB concentration of as high as 20 mg L −1 , and the estimations from the pigment studies showed that the exposure of AP to RB causes AP to produce higher concentrations of plant pigments than the control, which hinted the possibility of AP using RB and its intermediates for growth.

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“…The existence of the solution of this kind of problem is not always guaranteed, but when the conditions on the accessible part of the boundary are compatible, then the existence is assured , and the uniqueness results can be found in . In recent years, there are many special numerical methods to deal with this problem, such as the Boundary element method , the method of Fundamental solution , the Conjugate gradient method , the Landweber method , Quasi‐reversibility and truncation methods , Quasi‐boundary and Tikhonov‐type regularization methods , the Fourier regularization method , and so on . However, most of the mentioned numerical methods choosing the regularization parameter is based on the a priori rule.…”

Section: Introductionmentioning

confidence: 99%

A regularization method for the cauchy problem of the modified Helmholtz equation

Cheng¹,

Ping

Gao

2014

Math Methods in App Sciences

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In the present paper, an iteration regularization method for solving the Cauchy problem of the modified Helmholtz equation is proposed. The a priori and a posteriori rule for choosing regularization parameters with corresponding error estimates between the exact solution and its approximation are also given. The numerical example shows the effectiveness of this method. Copyright © 2014 John Wiley & Sons, Ltd.

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Numerical solution of Helmholtz equation by the modified Hopfield finite difference techniques

Cited by 17 publications

References 32 publications

Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks

Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks

Phytoextraction capability of Azolla pinnata in the removal of rhodamine B from aqueous solution: artificial neural network and random forests approaches

A regularization method for the cauchy problem of the modified Helmholtz equation

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