#### DMCA

## Adaptive finite volume methods for distributed non-smooth parameter identification (2007)

Citations: | 13 - 7 self |

### Citations

3326 | Numerical Optimization.
- Nocedal, Wright
- 1999
(Show Context)
Citation Context ...ectively use QuasiNewton techniques without the necessity of excessive storage. In our application we often need to deal with a very large number of sources. We therefore use the reduced space L-BFGS =-=[34]-=- approach for the solution of the problem. 9s4 Adaptive multilevel refinement The cost of the optimization process is directly impacted by the size of the problem and our initial guess for the solutio... |

415 |
Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Society for Industrial and Applied Mathematics:
- Ascher, Matheij, et al.
- 1995
(Show Context)
Citation Context ...2.2). Moreover, differentiating a piecewise constant function in the context of ODE mesh 6sm4 (mx1)1 cellj m1 (mx1)2 (mx1)3 Figure 3: Discretization of ∇m refinement has been known and used for years =-=[6]-=-. But here the resulting approximation appears in the objective function and the grid is adaptively refined based on computational quantities, so we opt to be more careful. A better pointwise approxim... |

396 |
Computational Methods for Inverse Problems
- Vogel
- 2002
(Show Context)
Citation Context ... G(Pkm, uk) ⊤ pk + β GRAD ⊤ Σ(m) GRAD m = 0, (3.12c) G(w, uj) = ∂(A(w)uj) , ∂w and Σ(m) is an approximation to the Hessian of the regularization term. Here we use the lagged diffusivity approach, see =-=[42, 5]-=-. We can then use an inexact Newton method for the solution of the system. For details on the linear systems and preconditioning of the Newton iteration, see [21, 20]. When facing a very large number ... |

372 |
Inverse Problems in Partial Differential Equations.
- Isakov
- 1998
(Show Context)
Citation Context ...tions are prescribed for these PDEs, but other boundary conditions could be easily incorporated. The model problem has multiple right hand sides, which is typical to many other inverse problems (e.g. =-=[25]-=-). Even for the relatively simple forward problem defined by (1.2) the computational cost can be substantial. This is because each PDE can be time consuming to solve in 3D, especially when the coeffic... |

210 | Simulating Water and Smoke with an Octree Data Structure,”
- Losasso, Gibou, et al.
- 2004
(Show Context)
Citation Context ...ay only be refined into two equal halves, and this principle of local refinement is extended to 2sgrids in higher dimensions, resulting in a QuadTree in 2D or an OcTree data structure in 3D (see e.g. =-=[24, 31]-=-, or just google ’octree’ for much, much more). Such an approach has the advantage that gridding is almost trivial, and it yields re-usable sparse matrices. Also, it has been demonstrated that such an... |

164 | Electrical impedance tomography,”
- Cheney, Isaacson, et al.
- 1999
(Show Context)
Citation Context ...s with functions ρ(·) that do not over-penalize rapid profile changes in the model; see e.g. [14, 5]. Numerical optimization problems of this kind are often solved in geophysics [23], medical physics =-=[13]-=-, computer vision [11] and other fields which involve data fitting. To be more ∗ Department of Mathematics and Computer Science, Emory University, Atlanta, GA. † Department of Computer Science, Univer... |

140 |
Electromagnetic theory for geophysical applications, in
- Ward
- 1967
(Show Context)
Citation Context ...sh The grid for u and p is set by choosing a uniform 16 3 grid as the coarsest mesh and refining twice close to the sources. When comparing the data obtained on this first grid with a half space data =-=[43]-=- we see that the agreement is roughly 0.1%, which is within the noise levels. To initialize the grid for m we use an 8 3 uniform grid. We then solve the problem on a sequence of grids and use the refi... |

137 |
Robust modeling with erratic data:
- Claebout, Muir
- 1973
(Show Context)
Citation Context ...unctional, where β is a regularization parameter. We are particularly interested in regularization operators with functions ρ(·) that do not over-penalize rapid profile changes in the model; see e.g. =-=[14, 5]-=-. Numerical optimization problems of this kind are often solved in geophysics [23], medical physics [13], computer vision [11] and other fields which involve data fitting. To be more ∗ Department of M... |

89 | A Simple Mesh Generator in MATLAB.
- Persson, Strang
- 2004
(Show Context)
Citation Context ... parameter m is changing through the optimization problem, re-gridding is often needed. However, re-gridding for large scale 3D problems is a computationally intensive task by itself (see for example =-=[35]-=-). Moreover, the stiffness matrix (i.e. the assembled discrete version A of A(m)) must be re-evaluated when m changes. This is also a time consuming process, especially when comparing with the standar... |

76 |
Perspectives in flow control and optimization,
- Gunzburger
- 2003
(Show Context)
Citation Context ...agrange multiplier. In this case the discrete Euler-Lagrange equations do not evolve from a discrete Lagrangian and therefore the discrete reduced gradient is not a discrete gradient of anything (see =-=[18]-=-). This implies that no discrete objective function is decreased and even simple unconstrained optimization algorithms may fail. We therefore keep the grids for pj and uj identical. • If the grid for ... |

73 | Spatially adaptive techniques for level set methods and incompressible flow, Comput. Fluids 35
- Losasso, Fedkiw, et al.
- 2006
(Show Context)
Citation Context ...f cell centered variables is naturally defined on cell faces. One simple way to define the gradient is to use a short difference. This approximation has been investigated by [17] and recently used in =-=[32]-=-. Consider the arrangement in Figure 3 and let x0 be the grid location where (mx1)2 is discretized. With h the width of the smaller cell the approximation is ∂m ∂x1 (x0) ≈ m2 − m1 . (2.6a) h For subdo... |

65 |
On optimization techniques for solving nonlinear inverse problems
- Haber, Ascher, et al.
(Show Context)
Citation Context ...(2.10b) v ⊤ uj = 0, j = 1, . . . , n. (2.10c) 3 Solving the optimization problem In this Section we quickly review solution techniques for the optimization problem. For a more in-depth discussion see =-=[22]-=-. We use variants of Reduced Hessian Sequential Quadratic Programming to solve the optimization problem. The Lagrangian can be written as L = 1 n� �Qjuj − dj� 2 2 + βv ⊤ ρ + j=1 n� j=1 p ⊤ j diag(v)(A... |

62 | Mimetic Finite Difference Methods for Diffusion Equations,
- Hyman, Morel, et al.
- 2002
(Show Context)
Citation Context ...proach has the advantage that gridding is almost trivial, and it yields re-usable sparse matrices. Also, it has been demonstrated that such an approach can deal with adaptivity in the forward problem =-=[17, 16, 29]-=-. While adaptive grid refinement is certainly not a new field, little such work has been done in the general context of inverse problems, and even less attention has been given to recovering rapidly v... |

61 | R.,Adaptive finite element methods for optimal control of partial differential equations: basic concepts
- Becker, Kapp, et al.
- 2000
(Show Context)
Citation Context ...ecovering rapidly varying coefficient functions. A framework for adaptive finite elements for inverse problems has been presented in [7]. Similar work in the context of finite elements is reported in =-=[8, 9, 2, 27, 26]-=-. Some work using finite difference is reported in [10]. See also [30, 44] for a different approach. The above work assumes quadratic regularization operators and obtains smooth model solutions. No pr... |

55 | A survey on level set methods for inverse problems and optimal design,
- Burger, Osher
- 2005
(Show Context)
Citation Context ...ble. Additional restrictions to the solution space can be in the form that m is allowed to take at each spatial location only one of two values, which turns the problem into one of shape optimization =-=[19, 40, 12]-=-. More generally, however, no such information may be available, and we do not assume it in the present article. If m varies rapidly but smoothly then we can seek to resolve regions of such rapid vari... |

45 |
Lipschitz stability for the inverse conductivity problem
- Alessandrini, Vessella
(Show Context)
Citation Context ..., especially when the coefficient function m varies rapidly. In the case that m is outright discontinuous, it is not always clear that this highly ill-posed inverse problem can be meaningfully solved =-=[5, 1]-=-. In such a case it is particularly advantageous to use additional a priori information if possible. Additional restrictions to the solution space can be in the form that m is allowed to take at each ... |

45 | A multigrid method for distributed parameter estimation problems., Electron
- Ascher, Haber
(Show Context)
Citation Context ...ally solving the inverse problem then becomes rather challenging, and advanced computational techniques are needed. One option to achieve major computational savings is by using a multilevel approach =-=[3, 4, 19]-=-. Such an approach is particularly beneficial in this context because it enables us to obtain a good approximation to the regularization parameter, β, as well as the solution m on coarse grids, reduci... |

43 |
Local refinement techniques for elliptic problems on cell-centered grids. I: error analysis,
- Ewing, Lazarov, et al.
- 1991
(Show Context)
Citation Context ...proach has the advantage that gridding is almost trivial, and it yields re-usable sparse matrices. Also, it has been demonstrated that such an approach can deal with adaptivity in the forward problem =-=[17, 16, 29]-=-. While adaptive grid refinement is certainly not a new field, little such work has been done in the general context of inverse problems, and even less attention has been given to recovering rapidly v... |

42 | Inversion of 3D electromagnetic data in frequency and time domain using an inexact all-at-once approach, Geophysics 69
- Haber, Ascher, et al.
- 2004
(Show Context)
Citation Context ...egularization operators with functions ρ(·) that do not over-penalize rapid profile changes in the model; see e.g. [14, 5]. Numerical optimization problems of this kind are often solved in geophysics =-=[23]-=-, medical physics [13], computer vision [11] and other fields which involve data fitting. To be more ∗ Department of Mathematics and Computer Science, Emory University, Atlanta, GA. † Department of Co... |

37 | On effective methods for implicit piecewise smooth surface recovery
- Ascher, Haber, et al.
- 2006
(Show Context)
Citation Context ...unctional, where β is a regularization parameter. We are particularly interested in regularization operators with functions ρ(·) that do not over-penalize rapid profile changes in the model; see e.g. =-=[14, 5]-=-. Numerical optimization problems of this kind are often solved in geophysics [23], medical physics [13], computer vision [11] and other fields which involve data fitting. To be more ∗ Department of M... |

35 |
Grid refinement and scaling for distributed parameter estimation problems. Inverse Problems
- Ascher, Haber
- 2001
(Show Context)
Citation Context ...ally solving the inverse problem then becomes rather challenging, and advanced computational techniques are needed. One option to achieve major computational savings is by using a multilevel approach =-=[3, 4, 19]-=-. Such an approach is particularly beneficial in this context because it enables us to obtain a good approximation to the regularization parameter, β, as well as the solution m on coarse grids, reduci... |

33 | Matrix-dependent multigrid homogenization for diffusion problems
- Knapek
- 1998
(Show Context)
Citation Context ...for the model m is strictly finer than the grid for uj then homogenization is needed in order to accurately evaluate A(m) on the uj grid. It is well known that homogenization is not a trivial process =-=[15, 28]-=-. In fact, some of the better homogenization processes are highly nonlinear with respect to the coefficients. This can generate further complications when solving the optimization problem. We therefor... |

33 |
Two-dimensional DC resistivity inversion for dipole-dipole data
- Smith, Vozoff
- 1984
(Show Context)
Citation Context ...ver. 1sconcrete we motivate this paper using the following 3D model problem which arises in Direct Current resistivity, where one attempts to invert for the log-conductivity m = ln σ (see for example =-=[38, 33, 37]-=- and references therein) the elliptic problems associated with A(m)uj = ∇ · (e m ∇uj) = bj j = 1 . . . n. (1.2) For simplicity, we assume that Neumann boundary conditions are prescribed for these PDEs... |

27 | On level set regularization for highly ill-posed distributed parameter estimation problems
- Doel, Ascher
- 2006
(Show Context)
Citation Context ...ble. Additional restrictions to the solution space can be in the form that m is allowed to take at each spatial location only one of two values, which turns the problem into one of shape optimization =-=[19, 40, 12]-=-. More generally, however, no such information may be available, and we do not assume it in the present article. If m varies rapidly but smoothly then we can seek to resolve regions of such rapid vari... |

25 |
Preconditioned all-at-one methods for large, sparse parameter estimation problems
- Haber, Ascher
- 2001
(Show Context)
Citation Context ...e lagged diffusivity approach, see [42, 5]. We can then use an inexact Newton method for the solution of the system. For details on the linear systems and preconditioning of the Newton iteration, see =-=[21, 20]-=-. When facing a very large number (thousands) of sources and a limited computer memory one may consider working with a reduced space approach rather than the full space approach. The reduced space app... |

22 |
Adaptive finite element methods for the identification of distributed parameters in partial differential equations
- Bangerth
- 2002
(Show Context)
Citation Context ...xt of inverse problems, and even less attention has been given to recovering rapidly varying coefficient functions. A framework for adaptive finite elements for inverse problems has been presented in =-=[7]-=-. Similar work in the context of finite elements is reported in [8, 9, 2, 27, 26]. Some work using finite difference is reported in [10]. See also [30, 44] for a different approach. The above work ass... |

20 | A posteriori error estimation in computational inverse scattering
- Beilina, Johnson
- 2005
(Show Context)
Citation Context ...ecovering rapidly varying coefficient functions. A framework for adaptive finite elements for inverse problems has been presented in [7]. Similar work in the context of finite elements is reported in =-=[8, 9, 2, 27, 26]-=-. Some work using finite difference is reported in [10]. See also [30, 44] for a different approach. The above work assumes quadratic regularization operators and obtains smooth model solutions. No pr... |

18 |
Coarse scale models of two phase flow in heterogeneous reservoirs: Volume averaged equations and their relationship to the existing upscaling techniques
- Durlofsky
(Show Context)
Citation Context ...for the model m is strictly finer than the grid for uj then homogenization is needed in order to accurately evaluate A(m) on the uj grid. It is well known that homogenization is not a trivial process =-=[15, 28]-=-. In fact, some of the better homogenization processes are highly nonlinear with respect to the coefficients. This can generate further complications when solving the optimization problem. We therefor... |

17 |
Optimal finite difference grids for direct and inverse Sturm-Liouville problems
- Borcea, Druskin
(Show Context)
Citation Context ...ptive finite elements for inverse problems has been presented in [7]. Similar work in the context of finite elements is reported in [8, 9, 2, 27, 26]. Some work using finite difference is reported in =-=[10]-=-. See also [30, 44] for a different approach. The above work assumes quadratic regularization operators and obtains smooth model solutions. No previous work on adaptive finite volume methods for inver... |

17 |
A multiresolution method for distributed parameter estimation
- Liu
- 1993
(Show Context)
Citation Context ...ements for inverse problems has been presented in [7]. Similar work in the context of finite elements is reported in [8, 9, 2, 27, 26]. Some work using finite difference is reported in [10]. See also =-=[30, 44]-=- for a different approach. The above work assumes quadratic regularization operators and obtains smooth model solutions. No previous work on adaptive finite volume methods for inverse problems is know... |

16 |
Refinement and coarsening indicators for adaptive parametrization: application to the estimation of the hydraulic transmissivities. Inverse Problems
- Ameur, Chavent, et al.
- 2002
(Show Context)
Citation Context ...ecovering rapidly varying coefficient functions. A framework for adaptive finite elements for inverse problems has been presented in [7]. Similar work in the context of finite elements is reported in =-=[8, 9, 2, 27, 26]-=-. Some work using finite difference is reported in [10]. See also [30, 44] for a different approach. The above work assumes quadratic regularization operators and obtains smooth model solutions. No pr... |

16 | Speeding up construction of quadtrees for spatial indexing
- Hjaltason, Samet
(Show Context)
Citation Context ...ay only be refined into two equal halves, and this principle of local refinement is extended to 2sgrids in higher dimensions, resulting in a QuadTree in 2D or an OcTree data structure in 3D (see e.g. =-=[24, 31]-=-, or just google ’octree’ for much, much more). Such an approach has the advantage that gridding is almost trivial, and it yields re-usable sparse matrices. Also, it has been demonstrated that such an... |

15 | multilevel, level-set method for optimizing eigenvalues in shape design problems
- Haber, A
- 2004
(Show Context)
Citation Context ...ble. Additional restrictions to the solution space can be in the form that m is allowed to take at each spatial location only one of two values, which turns the problem into one of shape optimization =-=[19, 40, 12]-=-. More generally, however, no such information may be available, and we do not assume it in the present article. If m varies rapidly but smoothly then we can seek to resolve regions of such rapid vari... |

11 | RESINVM3D: A MATLAB 3-D Resistivity Inversion Package." Geophysics
- Pidlisecky, Haber
- 2007
(Show Context)
Citation Context ...ver. 1sconcrete we motivate this paper using the following 3D model problem which arises in Direct Current resistivity, where one attempts to invert for the log-conductivity m = ln σ (see for example =-=[38, 33, 37]-=- and references therein) the elliptic problems associated with A(m)uj = ∇ · (e m ∇uj) = bj j = 1 . . . n. (1.2) For simplicity, we assume that Neumann boundary conditions are prescribed for these PDEs... |

10 |
Elimination of adaptive grid interface errors in the discrete cell centered pressure equation
- Edwards
- 1996
(Show Context)
Citation Context ...proach has the advantage that gridding is almost trivial, and it yields re-usable sparse matrices. Also, it has been demonstrated that such an approach can deal with adaptivity in the forward problem =-=[17, 16, 29]-=-. While adaptive grid refinement is certainly not a new field, little such work has been done in the general context of inverse problems, and even less attention has been given to recovering rapidly v... |

10 |
Forward modeling and inversion of DC resistivity and MMR data
- McGillivray
- 1992
(Show Context)
Citation Context ...ver. 1sconcrete we motivate this paper using the following 3D model problem which arises in Direct Current resistivity, where one attempts to invert for the log-conductivity m = ln σ (see for example =-=[38, 33, 37]-=- and references therein) the elliptic problems associated with A(m)uj = ∇ · (e m ∇uj) = bj j = 1 . . . n. (1.2) For simplicity, we assume that Neumann boundary conditions are prescribed for these PDEs... |

5 |
Quasi-newton methods methods for large scale electromagnetic inverse problems
- Haber
(Show Context)
Citation Context ...e lagged diffusivity approach, see [42, 5]. We can then use an inexact Newton method for the solution of the system. For details on the linear systems and preconditioning of the Newton iteration, see =-=[21, 20]-=-. When facing a very large number (thousands) of sources and a limited computer memory one may consider working with a reduced space approach rather than the full space approach. The reduced space app... |

5 |
A posteriori error estimates for control problems governed by Stokes’ equation
- Liu, Yan
(Show Context)
Citation Context ...ements for inverse problems has been presented in [7]. Similar work in the context of finite elements is reported in [8, 9, 2, 27, 26]. Some work using finite difference is reported in [10]. See also =-=[30, 44]-=- for a different approach. The above work assumes quadratic regularization operators and obtains smooth model solutions. No previous work on adaptive finite volume methods for inverse problems is know... |

4 | Adaptive local regularization methods for the inverse ECG problem
- Johnson, MacLeod
- 1998
(Show Context)
Citation Context |

4 | Cone-based electrical resistivity tomography
- Pidlisecky
- 2006
(Show Context)
Citation Context ...sing many sources. Of course, this is obtained at the price of a substantial increase in computational cost to solve the problem. Our experimental setting is similar to the field setting presented in =-=[36]-=-, where sources are placed in boreholes and data is measured on the surface of the earth. In our setting we have a grid of 64 2 receivers on the surface of the earth. We decree the existence of 4 bore... |

2 |
Optimal contol formulation for determining optical flow
- Borzi, Kunisch
(Show Context)
Citation Context ...that do not over-penalize rapid profile changes in the model; see e.g. [14, 5]. Numerical optimization problems of this kind are often solved in geophysics [23], medical physics [13], computer vision =-=[11]-=- and other fields which involve data fitting. To be more ∗ Department of Mathematics and Computer Science, Emory University, Atlanta, GA. † Department of Computer Science, University of British Columb... |

1 |
Matrix Iterative Analysis
- Varge
- 1962
(Show Context)
Citation Context ...e (see for example [39]), we take a different approach for the discretization of the forward problem and use the variation principle. This approach has been discussed for finite difference only in 1D =-=[41]-=- and it is commonly used in finite element discretization. The discretization of the inverse problem is then developed in a similar 3smanner. 2.1 OcTree data structure The Quad/OcTree representation i... |