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Parameter Identification in PDEs Using Monotone Inclusion Problems

Core Concepts
The authors propose a novel approach to parameter identification in PDEs using monotone inclusion problems, demonstrating well-posedness and convergence of the regularization method.
Parameter identification in partial differential equations (PDEs) is addressed through a total variation based regularization method. The inverse problem of reconstructing the source term from noisy data is discussed, emphasizing the need for regularization due to ill-posedness. Various regularization approaches like Tikhonov and iterative methods are compared, with a focus on Lavrentiev regularization for monotone problems. The study highlights numerical algorithms and inertial techniques for solving inclusion problems efficiently. Primal-dual splitting algorithms with inertial effects are explored, showcasing advancements in solving complex monotone inclusion problems.
A solution algorithm for the numerical solution of inclusion problems is discussed. Regularization parameters are chosen appropriately to ensure convergence to the true solution. Total variation and Sobolev norms are combined in the regularization method. The subdifferential calculus results are applied to analyze the proximal operator. The convergence speed of algorithms is enhanced by incorporating inertial terms.

Deeper Inquiries

How does Lavrentiev regularization compare to other methods in handling ill-posed problems

Lavrentiev regularization offers a unique approach to handling ill-posed problems compared to other methods like Tikhonov regularization. While Tikhonov regularization focuses on finding a stable solution by balancing data fidelity and regularity through the minimization of a cost function, Lavrentiev regularization tackles the problem as a monotone inclusion equation. By formulating the problem in this way, Lavrentiev regularization ensures that there is a unique solution under certain coercivity conditions on the operator and regularizer. This uniqueness leads to stable regularization results even in non-linear cases where traditional methods may struggle with local minima or non-optimal solutions.

What implications do nested primal-dual algorithms have on computational efficiency

Nested primal-dual algorithms have significant implications for computational efficiency in solving complex structured monotone inclusion problems. By incorporating inertial effects into the algorithm, these methods can leverage information from previous iterations to determine the next step more efficiently. This "warm-start" strategy allows for faster convergence rates as it uses momentum from previous iterations to guide the optimization process towards the optimal solution. Additionally, by iteratively approximating proximal operators during each outer iteration through inner iterative algorithms applied to dual problems, nested primal-dual algorithms strike a balance between accuracy and computational cost.

How can inertial effects improve convergence rates beyond traditional methods

Inertial effects play a crucial role in improving convergence rates beyond traditional methods by introducing momentum into the optimization process. By discretizing second-order differential equations proposed by Polyak into an inertial term, these algorithms use information from two preceding terms to determine the next iteration's direction effectively speeding up convergence rates significantly. The presence of an inertial term enables leveraging historical information about gradients and steps taken previously leading to faster convergence towards optimal solutions especially when dealing with large-scale optimization problems or highly non-linear functions where traditional methods might struggle with slow convergence rates or getting stuck in local optima.