2,159 research outputs found
Fully discrete finite element data assimilation method for the heat equation
We consider a finite element discretization for the reconstruction of the
final state of the heat equation, when the initial data is unknown, but
additional data is given in a sub domain in the space time. For the
discretization in space we consider standard continuous affine finite element
approximation, and the time derivative is discretized using a backward
differentiation. We regularize the discrete system by adding a penalty of the
-semi-norm of the initial data, scaled with the mesh-parameter. The
analysis of the method uses techniques developed in E. Burman and L. Oksanen,
Data assimilation for the heat equation using stabilized finite element
methods, arXiv, 2016, combining discrete stability of the numerical method with
sharp Carleman estimates for the physical problem, to derive optimal error
estimates for the approximate solution. For the natural space time energy norm,
away from , the convergence is the same as for the classical problem with
known initial data, but contrary to the classical case, we do not obtain faster
convergence for the -norm at the final time
Inverse Problems of Determining Coefficients of the Fractional Partial Differential Equations
When considering fractional diffusion equation as model equation in analyzing
anomalous diffusion processes, some important parameters in the model, for
example, the orders of the fractional derivative or the source term, are often
unknown, which requires one to discuss inverse problems to identify these
physical quantities from some additional information that can be observed or
measured practically. This chapter investigates several kinds of inverse
coefficient problems for the fractional diffusion equation
Identification of Chemotaxis Models with Volume Filling
Chemotaxis refers to the directed movement of cells in response to a chemical
signal called chemoattractant. A crucial point in the mathematical modeling of
chemotactic processes is the correct description of the chemotactic sensitivity
and of the production rate of the chemoattractant. In this paper, we
investigate the identification of these non-linear parameter functions in a
chemotaxis model with volume filling. We also discuss the numerical realization
of Tikhonov regularization for the stable solution of the inverse problem. Our
theoretical findings are supported by numerical tests.Comment: Added bibfile missing in v2, no changes on conten
A convex analysis approach to optimal controls with switching structure for partial differential equations
Optimal control problems involving hybrid binary-continuous control costs are
challenging due to their lack of convexity and weak lower semicontinuity.
Replacing such costs with their convex relaxation leads to a primal-dual
optimality system that allows an explicit pointwise characterization and whose
Moreau-Yosida regularization is amenable to a semismooth Newton method in
function space. This approach is especially suited for computing switching
controls for partial differential equations. In this case, the optimality gap
between the original functional and its relaxation can be estimated and shown
to be zero for controls with switching structure. Numerical examples illustrate
the effectiveness of this approach
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