scholarly journals Graph-based prior and forward models for inverse problems on manifolds with boundaries

2021 ◽  
Author(s):  
John Harlim ◽  
Shixiao Willing Jiang ◽  
Hwanwoo Kim ◽  
Daniel Sanz-Alonso
Keyword(s):  
Inverse Problems ◽  
Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Gaussian Field ◽  
Forward Models ◽  
Bayesian Inverse Problems ◽  
Flexible Modeling ◽  
Covariance Models ◽  
Classical Boundary ◽  
Second Order Elliptic Operators

Abstract This paper develops manifold learning techniques for the numerical solution of PDE-constrained Bayesian inverse problems on manifolds with boundaries. We introduce graphical Matérn-type Gaussian field priors that enable flexible modeling near the boundaries, representing boundary values by superposition of harmonic functions with appropriate Dirichlet boundary conditions. We also investigate the graph-based approximation of forward models from PDE parameters to observed quantities. In the construction of graph-based prior and forward models, we leverage the ghost point diffusion map algorithm to approximate second-order elliptic operators with classical boundary conditions. Numerical results validate our graph-based approach and demonstrate the need to design prior covariance models that account for boundary conditions.

Inverse problems on a graph with loops

2017 ◽  
Vol 25 (3) ◽  
Author(s):  
Chuan-Fu Yang ◽  
Feng Wang
Keyword(s):  
Inverse Problems ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Internal Vertex ◽  
Dirichlet Boundary Conditions ◽  
Matching Conditions ◽  
Nodal Points

AbstractA graph with loops, endowed with standard matching conditions in the internal vertex and with Dirichlet boundary conditions at the boundary vertices, is considered. We show that the potential on a graph with loops can be constructed by the dense nodal points on the interval considered. Moreover, we investigate the so-called incomplete inverse problems of recovering the potential on a fixed edge from a subset of nodal points situated only on a part of the edge.

scholarly journals Random Forward Models and Log-Likelihoods in Bayesian Inverse Problems

2018 ◽  
Vol 6 (4) ◽  
pp. 1600-1629 ◽  
Author(s):  
H. C. Lie ◽  
T. J. Sullivan ◽  
A. L. Teckentrup
Keyword(s):  
Inverse Problems ◽  
Forward Models ◽  
Bayesian Inverse Problems

Investigation of Multiply Connected Inverse Cauchy Problems by Efficient Weighted Collocation Method

2020 ◽  
Vol 12 (01) ◽  
pp. 2050012 ◽  
Author(s):  
Judy P. Yang ◽  
Qizheng Lin
Keyword(s):  
Inverse Problems ◽  
Boundary Conditions ◽  
Collocation Method ◽  
Reproducing Kernel ◽  
Dirichlet Boundary Conditions ◽  
Benchmark Problems ◽  
Cauchy Problems ◽  
Multiply Connected ◽  
Simply Connected ◽  
Gradient Approach

This work introduces an efficient weighted collocation method to solve inverse Cauchy problems. As it is known that the reproducing kernel approximation takes time to compute the second-order derivatives in the meshfree strong form method, the gradient approach alleviates such a drawback by approximating the first-order derivatives in a similar way to the primary unknown. In view of the overdetermined system derived from inverse Cauchy problems with incomplete boundary conditions, the weighted gradient reproducing kernel collocation method (G-RKCM) is further introduced in the analysis. The convergence of the method is first demonstrated by the simply connected inverse problems, in which the same set of source points and collocation points is adopted. Then, the multiply connected inverse problems are investigated to show that high accuracy of approximation can be reached. The sensitivity and stability of the method is tested through the disturbance added on both Neumann and Dirichlet boundary conditions. From the investigation of four benchmark problems, it is concluded that the weighted gradient reproducing kernel collocation method is more efficient than the reproducing kernel collocation method.

scholarly journals The Singular Perturbation Problem for a Class of Generalized Logistic Equations Under Non-classical Mixed Boundary Conditions

2019 ◽  
Vol 19 (1) ◽  
pp. 1-27 ◽  
Author(s):  
Sergio Fernández-Rincón ◽  
Julián López-Gómez
Keyword(s):  
Singular Perturbation ◽  
Boundary Conditions ◽  
Distance Function ◽  
Dirichlet Boundary ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Logistic Equations ◽  
Perturbation Problem ◽  
Second Order Elliptic Operators

Abstract This paper studies a singular perturbation result for a class of generalized diffusive logistic equations, {d\mathcal{L}u=uh(u,x)} , under non-classical mixed boundary conditions, {\mathcal{B}u=0} on {\partial\Omega} . Most of the precursors of this result dealt with Dirichlet boundary conditions and self-adjoint second order elliptic operators. To overcome the new technical difficulties originated by the generality of the new setting, we have characterized the regularity of {\partial\Omega} through the regularity of the associated conormal projections and conormal distances. This seems to be a new result of a huge relevance on its own. It actually complements some classical findings of Serrin, [39], Gilbarg and Trudinger, [21], Krantz and Parks, [27], Foote, [18] and Li and Nirenberg [28] concerning the regularity of the inner distance function to the boundary.

scholarly journals Inverse problems for Sturm-Liouville difference equations

Filomat ◽  
2016 ◽  
Vol 30 (5) ◽  
pp. 1297-1304 ◽  
Author(s):  
Martin Bohner ◽  
Hikmet Koyunbakan
Keyword(s):  
Inverse Problems ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Difference Equations ◽  
Dirichlet Boundary ◽  
Liouville Problem ◽  
Dirichlet Boundary Conditions ◽  
Sturm Liouville

We consider a discrete Sturm-Liouville problem with Dirichlet boundary conditions. We show that the specification of the eigenvalues and weight numbers uniquely determines the potential. Moreover, we also show that if the potential is symmetric, then it is uniquely determined by the specification of the eigenvalues. These are discrete versions of well-known results for corresponding differential equations.

scholarly journals Bayesian inverse problems with Monte Carlo forward models

2013 ◽  
Vol 7 (1) ◽  
pp. 81-105 ◽  
Author(s):  
Guillaume Bal ◽  
◽  
Ian Langmore ◽  
Youssef Marzouk ◽  
Keyword(s):  
Monte Carlo ◽  
Inverse Problems ◽  
Forward Models ◽  
Bayesian Inverse Problems

Identification of the string boundary conditions using natural frequencies

2016 ◽  
Vol 11 (1) ◽  
pp. 38-52
Author(s):  
I.M. Utyashev ◽  
A.M. Akhtyamov
Keyword(s):  
Inverse Problems ◽  
Power Series ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Boundary Value ◽  
External Environment ◽  
Direct And Inverse Problems ◽  
Symmetric Potential ◽  
Modified Method

The paper discusses direct and inverse problems of oscillations of the string taking into account symmetrical characteristics of the external environment. In particular, we propose a modified method of finding natural frequencies using power series, and also the problem of identification of the boundary conditions type and parameters for the boundary value problem describing the vibrations of a string is solved. It is shown that to identify the form and parameters of the boundary conditions the two natural frequencies is enough in the case of a symmetric potential q(x). The estimation of the convergence of the proposed methods is done.

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