A direct or explicit solution to the inverse design problem requires use of an inverse formulation. Ordinary techniques (e.g., Gauss-Seidel, Gauss elimination or LU decomposition) are likely to either identify non-physical solutions with large amplitude fluctuations and/or complex absolute temperatures, or completely fail to find a solution.

To achieve an accurate and physically reasonable solution, the explicit system of equations may be regularized by modifying the ill-conditioned set to a nearly-equivalent set of well-conditioned equations. The solution is then subject to some error and the level of regularization must be selected so that the accuracy of the solution satisfies the designer’s needs. One regularization technique is truncated singular value decomposition (TSVD).

TSVD is based on modifying the singular value decomposition of A. The solution uses the pseudo-inverse matrix that is formed by filtering or truncating some (or many) of the smallest singular values, thus reducing the condition number of the matrix A; the solution to Ax = C using the p largest singular values from the singular value decomposition of A, ${\mathbf{A = US}}{{\mathbf{V}}^T}$, becomes

${x_n} = \sum\limits_{k = 1}^p {{V_{n,k}}\frac{{\sum\limits_m^N {{C_m}U_{m,k}^T} }}{{{S_{k,k}}}},\quad n = 1,2, \ldots ,N} \qquad \qquad(1)$

where p has a value less than or equal to the rank of A (Hansen, 1998). Retaining different numbers of singular values yields alternative solutions. Those with acceptable accuracy and physically reasonable characteristics present allowable alternatives.

A problem that incorporates many attributes of a practical design process for a radiant furnace was proposed and solved by a diverse team to compare various solution methods for inverse problems (Daun et al., 2006). A three-dimensional geometry was considered, and the problem was to determine the radiant heater settings that provide a prescribed transient but spatially-uniform heating of the design surface.

## References

Daun, K.J. França, F., Larsen, M., Leduc, G., and Howell, J.R., 2006, “Comparison of Methods for Inverse Design of Radiant Enclosures,” J. Heat Transfer, Vol. 128, pp. 269-282.

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Hansen, P.C., 1998, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, SIAM, Philadelphia, PA.