scholarly journals Inverse Problems via the “Generalized Collage Theorem” for Vector-Valued Lax-Milgram-Based Variational Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
H. Kunze ◽  
D. La Torre ◽  
K. Levere ◽  
M. Ruiz Galán
Keyword(s):  
Inverse Problems ◽  
Variational Problems ◽  
Extended Version ◽  
Numerical Examples ◽  
Vector Valued ◽  
Collage Theorem

We present an extended version of the Generalized Collage Theorem to deal with inverse problems for vector-valued Lax-Milgram systems. Numerical examples show how the method works in practical cases.

scholarly journals Lipschitz Bounds and Nonautonomous Integrals

2021 ◽  
Author(s):  
Cristiana De Filippis ◽  
Giuseppe Mingione
Keyword(s):  
Variational Problems ◽  
Growth Conditions ◽  
Regularity Criteria ◽  
Regularity Of Solutions ◽  
Optimal Regularity ◽  
Lipschitz Regularity ◽  
Elliptic Case ◽  
Different Types ◽  
Vector Valued

AbstractWe provide a general approach to Lipschitz regularity of solutions for a large class of vector-valued, nonautonomous variational problems exhibiting nonuniform ellipticity. The functionals considered here range from those with unbalanced polynomial growth conditions to those with fast, exponential type growth. The results obtained are sharp with respect to all the data considered and also yield new, optimal regularity criteria in the classical uniformly elliptic case. We give a classification of different types of nonuniform ellipticity, accordingly identifying suitable conditions to get regularity theorems.

Solving Inverse Laplace Equation with Singularity by Weighted Reproducing Kernel Collocation Method

2017 ◽  
Vol 09 (05) ◽  
pp. 1750065 ◽  
Author(s):  
Judy P. Yang ◽  
Pai-Chen Guan ◽  
Chia-Ming Fan
Keyword(s):  
Inverse Problems ◽  
Collocation Method ◽  
Reproducing Kernel ◽  
Cauchy Problems ◽  
Overdetermined System ◽  
Numerical Examples ◽  
Kernel Approximation ◽  
Laplace Equations ◽  
Laplace Type ◽  
Least Squares Minimization

This work introduces the weighted collocation method with reproducing kernel approximation to solve the inverse Laplace equations. As the inverse problems in consideration are equipped with over-specified boundary conditions, the resulting equations yield an overdetermined system. Following our previous work, the weighted collocation method using a least-squares minimization has shown to solve the inverse Cauchy problems efficiently without using techniques such as iteration and regularization. In this work, we further consider solving the inverse problems of Laplace type and introduce the Shepard functions to deal with singularity. Numerical examples are provided to demonstrate the validity of the method.

ITERATED FUNCTION SYSTEMS ON FUNCTIONS OF BOUNDED VARIATION

Fractals ◽  
2016 ◽  
Vol 24 (02) ◽  
pp. 1650019 ◽  
Author(s):  
DAVIDE LA TORRE ◽  
FRANKLIN MENDIVIL ◽  
EDWARD R. VRSCAY
Keyword(s):  
Inverse Problems ◽  
Bounded Variation ◽  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Functions Of Bounded Variation ◽  
Complete Space ◽  
Iterated Function ◽  
Collage Theorem ◽  
Contraction Maps

We show that under certain hypotheses, an iterated function system on mappings (IFSM) is a contraction on the complete space of functions of bounded variation (BV). It then possesses a unique attractor of BV. Some BV-based inverse problems based on the Collage Theorem for contraction maps are considered.

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