Kronecker-product approximations for some function-related matrices.
The paper focusses on the Kronecker-product approximations for some function-related matrices. The purpose is the formulation and the proof of ``existence theorems'', with estimates on the Kronecker rank and the corresponding approximation error for reasonable wide classes of practically interesting matrices. The first section presents an introduction concerning the approximation of matrices by sums of Kronecker-product of smaller-size matrices. The motivation of the work is given in the second section which begins with some numerical examples of Kronecker-product approximations for the case of uniform and non-uniform respectively grids \(z_1,z_2,\dots, z_n\) of \(n\) nodes in \(\Omega= [0,1]\times [0,1]\) and for a matrix \(A= [a_{ij}]\) associated with a function \(f= f (z',z)\) so that \(a_{ij}= f (z_i,z_j)\), \(i, j =1,\dots,n\) which depends upon the distance \(\rho\) between \(z'\) and \(z\). The third section deals with Kronecker-products and low-rank matrices, stating that a sum of \(r\) Kronecker-products is mapped to a sum of \(r\) rank-one matrices. Section four presents the main theoretical results concerning functions, grids and matrices. One considers the general situation \(f (z', z)= F(u, v)\), \(z'= (x', y')\), \(z= (x, y)\), \(u= x'-x\), \(v= y'-y\) where \(F(u, v)\) is such that any mixed derivative \(D^m F= {\partial^k\partial^l\over(\partial u^k)(\partial v^l)} F\), \(m= k+l\) satisfies the inequality \(| D^m F|\leq cd^m m!\rho^{g- m}\), \(\rho= \sqrt{u^2+ v^2}\) with some constants \(c,d > 0\) and \(g\). \(F\) is called a complete asymptotically smooth function. In section five there are some further results for a formally different case of \(A= [ f (x_i,x_j)]\), \(0\leq i\), \(j\leq n\), \(0< x_1<\cdots< x_n< 1\) where \(f(x',x)= F(x'-x)\) such that \[ \Biggl|{\partial^m F\over\partial x^m}\Biggr|\leq cd^m m!| x|^{g-m}\qquad (\forall)m\in \mathbb{N}. \] Some possible generalizations of the presented results are discussed such as: practically efficient algorithms, a maximal volume like principle for approximations of low Kronecker-rank.
scientific article; zbMATH DE number 2056699