The Facts about the proposed Bloomington-Normal Park District /

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Greater West Park system. West Chicago Park Commission; [Chicago, Illinois]

Green, red and black ink on linen. Map of Chicago with parks, boulevards, municipal kitchen gardens. Signed. 132 cm. x 95 cm. Scale: 1"=1500' [from photographic copy by Lance Burgharrdt]

Planting plan; Columbus Park; Chicago, Illinois

Ink on linen. Type of plantings; notes. At lower left is key to orientation of sections A-D. Unsigned. 91 cm. x 87 cm. Scale: 1"=40' [from photographic copy by Lance Burgharrdt]

Planting plan for Columbus Park; Chicago, Illinois

Ink on linen. Type of plantings; notes. At upper left is key to orientation sections A-D. Unsigned. 87 cm. x 87 cm. Scale 1"=40' [from photographic copy by Lance Burgharrdt]

Section C. Planting plan for Columbus Park; Chicago, Illinois

Ink on linen. Type of plantings. Unsigned. 95 cm. x 85 cm. Scale: 1"=40' [from photographic copy by Lance Burgharrdt]

Preliminary planting plan; Columbus Park; Section D; Chicago, Illinois

Ink on linen. Location, type of plantings; notes. Key to orientation of sections A-D at upper right. Signed. 98 cm. x 75 cm. Scale: 1"=40' [from photographic copy by Lance Burgharrdt]

Survey showing actual location of trees on Columbus Park site; [Chicago]

Unsigned. 140 cm. x 105 cm. Scale: 1"=50' [from photographic copy by Lance Burgharrdt]

Planting plan for the playground area. Columbus Park; Chicago, Illinois

Ink on linen. Location, type of plantings, pools, footpaths, football field; notes. Signed. 172 cm. x 58 cm. Scale: 1"=20' [from photographic copy by Lance Burgharrdt]

Plans and sections for two springs. Columbus Park; Chicago, Illinois

Ink on linen. Elevations, sections of springs. Signed. 88 cm. x 60 cm. Vertical sale: 1"=5'; Horizontal scale: 1"=10' [from photographic copy by Lance Burgharrdt]

Planting plan for player's green. Columbus Park. West Chicago Parks [Chicago, Illinois]

Ink on linen. Location, type of plantings; notes. Unsigned. 58 cm. x 84 cm. Scale: 1"=20' [from photographic copy by Lance Burgharrdt]

Rock work for the wading pool at Columbus Park; Chicago, Illinois

Orange, green, red, black pencil on tracing papter. Location, type of plantings; notes. Unsigned. 58 cm. x 84 cm. Scale: 1"=20' [from photographic copy by Lance Burgharrdt]

Kenilworth, Illinois. Parks. Mahoney Park

Blueprint; unsigned. 59 x 113 cm. Scale: 1"=20' [from photographic copy by Lance Burgharrdt]

Kenilworth, Illinois. Parks. Mahoney Park. A wild flower preserve and bird sanctuary for the Mahoney Parks. Kenilworth, Illinois.

Photocopy; aerial sketch; signed. 43 x 31 cm. Scale : 1"=20' [from photographic copy by Lance Burgharrdt]

Hierarchical Reconstruction Method for Solving Ill-posed Linear Inverse Problems

We present a detailed analysis of the application of a multi-scale Hierarchical Reconstruction method for solving a family of ill-posed linear inverse problems. When the observations on the unknown quantity of interest and the observation operators are known, these inverse problems are concerned with the recovery of the unknown from its observations. Although the observation operators we consider are linear, they are inevitably ill-posed in various ways. We recall in this context the classical Tikhonov regularization method with a stabilizing function which targets the specific ill-posedness from the observation operators and preserves desired features of the unknown. Having studied the mechanism of the Tikhonov regularization, we propose a multi-scale generalization to the Tikhonov regularization method, so-called the Hierarchical Reconstruction (HR) method. First introduction of the HR method can be traced back to the Hierarchical Decomposition method in Image Processing. The HR method successively extracts information from the previous hierarchical residual to the current hierarchical term at a finer hierarchical scale. As the sum of all the hierarchical terms, the hierarchical sum from the HR method provides an reasonable approximate solution to the unknown, when the observation matrix satisfies certain conditions with specific stabilizing functions. When compared to the Tikhonov regularization method on solving the same inverse problems, the HR method is shown to be able to decrease the total number of iterations, reduce the approximation error, and offer self control of the approximation distance between the hierarchical sum and the unknown, thanks to using a ladder of finitely many hierarchical scales. We report numerical experiments supporting our claims on these advantages the HR method has over the Tikhonov regularization method.

Women's Rights National Historical Park, Seneca Falls, New York /

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