Application of the Proper Orthogonal Decomposition in Steady State Inverse Problems

Ryszard A. Białecki1 bialecki@itc.ise.polsl.gliwice.pi; Alain J. Kassab2 kassab@mail.ucf.edu; Ziemowit Ostrowski1 ostry@itc.ise.polsl.gliwice.pi    1 Institute of Thermal Technology, Silesian University of Technology, Gliwice, Poland
2 Department of Mechanical Material and Aerospace Engineering University of Central Florida, Orlando, USA

INTRODUCTION

Reduction of the number of the degrees of freedom in an inverse problem is a well known technique of filtering out the higher frequency error. The Proper Orthogonal Decomposition (POD) offers an elegant way to cut down the number of unknowns without loosing the accuracy. The method has been developed about 100 years ago as a tool of processing statistical data [1–4]. It has been also used in signal processing, pattern recognition, control theory, fluid flow and dynamics [5–7]. Another important area of application is in the turbulence where the technique, know also as Karhunen-Loeve method, has been used to detect the spatial large scale organized motions [8]. Some more recent description of POD theory can be found in [9,10].

The present paper presents a technique of applying POD in inverse analysis and, to the best of the knowledge of the authors, it is the first attempt to use POD in this context. The proposed technique uses the Proper Orthogonal Decomposition (POD) as a regularization technique. The idea is to solve a sequence of forward problems within the body under consideration. The solution of each problem is sampled at a predefined set of points. Each sampled field corresponds to a certain set of assumed values of the parameters defining the distribution of the function to be retrieved. POD detects the correlation between the discretized fields leading to a significant reduction of the degrees of freedom necessary to describe the field with a high accuracy.

BASICS OF POD

The fundamental notion of POD is the snapshot being a collection of N sampled values of the field under consideration. The snapshot is stored in a vector Ui,i = 1,2,…, M. A collection of all snapshots is a rectangular N by M matrix U. The snapshots are generated by changing the values of some parameter(s) upon which the field depends on. Time, parameters entering the boundary conditions, internal heat generation or material properties are examples of such parameters. The snapshots may be obtained either from a mathematical model of the phenomenon or from experiments. The aim of POD is to construct a set of vectors (basis) Φj resembling the original matrix U. The basis is stored in another rectangular matrix Φ of the same dimensionality as U. The elements of the basis are defined as

=U⋅V

where V is a modal matrix defined as a nontrivial solution of a problem

⋅V=ΛV

In the above Λ is a diagonal matrix storing the eigenvalues λi of the positive definite covariance matrix C. The entries of the latter are defined as

ij=∑k=1NUkiUkj

the eigenvalues are real, positive and distinct and should be sorted in an descending order. The basis vectors associated with the eigenvalues are orthogonal ie

T⋅Φ=Λ.

It can be shown [10], that the jth eigenvalue is a measure of the kinetic energy transferred within jth basis mode Φj (strictly speaking this is only true, when the field under consideration is the velocity field). Typically, this energy decreases rapidly with the increased number of the mode, which permits discarding the majority of modes. This can be done by deciding which fraction of the energy may be neglected in further calculations. The resulting POD basis ¯ consists of K < N elements.

This basis captures, in an optimal way, the spatial variation of the field. To have a full picture of the field, dependence on additional parameters used in generating the snapshots needs to be built into the approximation formula. This is accomplished by expressing the field represented by an arbitrary snapshot as

≈∑j=1KαjΦ¯j

with the unknown scalars αj depending on the parameters. The αj’s are found by an appropriate procedure, say the least square fit or weighted residuals.

INVERSE PROBLEM

If the points at which the snapshots were evaluated coincide with the location of the sensors, the values of the unknown parameters αj in eq. (5) can be found by least square curve fitting. Note that only few terms would typically be retained in eq. (5), thereby significantly reducing the number of unknowns, which is a very desirable feature in any inverse analysis algorithm.

To make the discussion more concrete, assume that the aim of inverse problem is to retrieve a distribution of a heat flux on a portion of the boundary. Steady state heat conduction with other boundary conditions known is considered. Both heat conductivity and distribution of the internal heat generation are known. The functional form of the unknown heat flux is postulated, say as a polynomial of a given degree. The coefficients of the polynomial (parameters of heat flux distribution) are unknown. Additional information is produced by temperature sensors located at some points of the domain.

To generate the POD basis, a sequence of direct problems is solved using any analytical or numerical technique. For each problem, other combination of values of the unknown parameters is taken. All solutions are sampled at the same set of points. When a numerical technique is used, a natural snapshot is the set of all nodal temperatures. Adding normal heat fluxes at nodes located on the surface where the heat flux distribution is to be retrieved, significantly improves the stability of the algorithm. In the numerical examples discussed hereafter, snapshots gathering both temperatures and heat fluxes have been used.

To solve an inverse problem, some measured values are needed. The simplest way to approach this problem, is to locate the sensors at a set of points being a subset of points used to create the snapshots. Otherwise, some additional interpolation would be necessary.

Solution of each direct problems, ie., each combination of the unknown parameters generates one snapshot. From the collection of all snapshots, the truncated POD bases in eq.(5) can be constructed, based on the user specified fraction of the energy to be neglected. The values of the unknown multipliers in expansion (5) are calculated by a least square fit of the measurements and the model. Once the values of αj are known, the values of temperatures at all snapshot points can be determined. As the snapshot definition encompasses also the sought for heat fluxes, the procedure also yields the values the retrieved heat flux. Two simple numerical example are used to demonstrate the effectiveness of this technique.

NUMERICAL EXAMPLE 1. ANALYTICAL METHOD USED TO SOLVE THE FORWARD PROBLEM

Steady state heat conduction in a unit square 0 < x, y < 1 with unit conductivity is considered. Edges x = 0 and x = 1 are insulated. At y = 0 temperature is zero. The unknown distribution of the normal heat flux q at y = 1 is to be retrieved. It assumed that this distribution can be described by a cubic polynomial

¯=a+bx+cx2+dx3

where a, b, c, d are unknown parameters. The sketch of the problem has been shown in Fig 1. Every of these parameters has been given a value of -1., -1.5,...