| Project ID |
D16
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| Application Area |
D
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| Project Title |
Adapted linear algebra for TR1 updates
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| Description |
The Twosided Rank-One Update (TR1) is well suited for the succssive approximation of not
necessarily symmetric Jacobians of arbitrary vector functions. It is essentially based on the possibility to compute products of transposed Jacobians with weight vectors as adjoints. Here there is no difference whether the adjoint equations have been formulated and solved explicitly or if the backward mode of automated differentiation is used. In both cases such products can be computed with a complexity which is in the range of the evaluation of the underlying vector function. If this function is a gradient and therefore has a symmetric Jacobian, the TR1 formula reduces to the symmetric rank one update (SR1) which is well known in optimization literature.
The projects focusses on two cases based on low rank update problems and which are challenging for the numerical linear algebra. Here the objective is to avoid cubic complexity with respect to the matrix dimension in each update step, at least in the normal case.
1. QR and LU factorization for nonsquare matrices
2. Structure-preserving factorization of symmetric indefinite matrices
In the first case existing QR or LU factorizations are being updated after a low rank correction. Within project C12 currently NLP solvers are being developed. These solvers would be greatly enhanced if an efficient and stable implementation of the more general problem of rank-1-modification could be provided.
The second case discusses symmetric not necessarily postive definite matrices which typically occur as Hessians in nonlinear optimizations problems.The NLP solver which subject to C12 requires to keep the projected Hessian in a suitable factored form. Beside the SR1 update it is also necessary to modify the system with respect to certain symmetric rank-2 modifications which are caused by the the change of the nullspace when updating the Jacobian.
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| Duration |
06/05-05/06
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| Status |
completed
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| Members |
| No assigned Project Members |
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| Heads |
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| Guests |
| No assigned Project Guests |
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| Publications |
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| Preprints |
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| Website |
http://www.math.tu-berlin.de/~stange/d16.html
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