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In this paper, an inverse problem of determining the space-dependent volatility from the observed market prices of options with different strikes is studied. Being different from other inverse volatility problem with classical parabolic equations, we apply the linearization method and introduce some variable substitutions to convert the original problem into an inverse source problem in a degenerate parabolic equation in a bounded area, from which an unknown volatility can be recovered and deficiencies caused by artificial truncation can be solved. Based on the optimal control framework, the problem is transformed into an optimization problem and the existence of the minimizer is established. After the necessary conditions are deduced, the uniqueness and stability of the minimizer are proved. Then, the Landweber iterative method is used to obtain a stable numerical solution of the inverse problem and some numerical experiments are also performed. The numerical results show that the algorithm which we proposed is robust and the unknown coefficient is recovered quite well. [ABSTRACT FROM AUTHOR]

In this paper, the split feasibility problem is studied in real Banach spaces. Through the W-mapping, a new iterative algorithm with the inertial technique for solving the split feasibility problem is proposed, which the step size is self-adaptive and no prior estimation of operator norm is required. We prove that the proposed algorithm converges weakly to a solution of the split feasibility problem under some mild conditions. Finally, the effectiveness of the proposed algorithm is indicated by numerical experiments. Our results are innovative and can enrich recently announced related results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new split inverse problem called the split equality generalized equilibrium problem which is more general than the split feasibility problem, the split equilibrium problem, and the split equality equilibrium problem. We develop an iterative algorithm for approximating a common solution of this problem and the split equality fixed point problem for Bregman relatively nonexpansive semigroups in p-uniformly convex and uniformly smooth Banach spaces. Using our iterative algorithm, we prove a strong convergence theorem and investigate a split equality convex optimization problem as an application. Finally, we present some numerical experiments to demonstrate the applicability of our proposed method. [ABSTRACT FROM AUTHOR]

This paper introduces a computational framework to incorporate exible regularization techniques in ensemble Kalman methods, generalizing the iterative alternating scheme to nonlinear inverse problems. The proposed methodology approximates the maximum a posteriori (MAP) estimate of a hierarchical Bayesian model characterized by a conditionally Gaussian prior and generalized gamma hyperpriors. Suitable choices of hyperparameters yield sparsity-promoting regularization. We propose an iterative algorithm for MAP estimation, which alternates between updating the unknown with an ensemble Kalman method and updating the hyperparameters in the regularization to promote sparsity. The effectiveness of our methodology is demonstrated in several computed examples, including compressed sensing and subsurface ow inverse problems. [ABSTRACT FROM AUTHOR]

A fast and efficient Newton-Shultz-type iterative method is presented to compute the inverse of an invertible tensor. Analysis of the convergence error shows that the proposed method has the sixth order convergence. It is shown that the proposed algorithm can be used for finding the Moore-Penrose inverse of tensors. Computational complexities of the algorithm is presented to support the theoretical aspects of the paper. Using the new method, we obtain a new preconditioner to solve the multilinear system A *N X = B. The effectiveness and accuracy of this method are re-verified by several numerical examples. Finally, some conclusions are given. [ABSTRACT FROM AUTHOR]