^{1}

^{2}

We consider a wavelet-based solution to the stochastic heat equation with random inputs. Computational methods based on the wavelet transform are analyzed for solving three types of stochastic heat equation. The methods are shown to be very convenient for solving such problems, since the initial and boundary conditions are taken into account automatically. The results reveal that the wavelet algorithms are very accurate and efficient.

Several applications in science and engineering involve stochasticity in input data. This is usually the result of the stochastic nature of the model coefficients, boundary or initial conditions data, the geometry in which the problem is set, and the source term. Uncertainty may also be introduced into an applied problem owing to the intrinsic variability inherent in the system being modelled [

The stochastic heat equation with random inputs (SHERI) is a stochastic partial differential equation (SPDE) that has received considerable attention in recent years. The approach to the solution depends on the type of random input present in the equation. Usually, the SHERI is analyzed and solved for only a random source term or for random coefficients only (see for example [

In this case,

Currently, several numerical methods are available for solving SPDEs. These include the classical and popular Monte Carlo method (MCM), the stochastic Galerkin method (SGM), and the stochastic collocation method (SCM). It is well known that MCMs have very slow convergence rates since they do not exploit the regularity available in the solution of SPDE’s with respect to input stochastic parameters. Stochastic Galerkin methods and SCM’s tend to have faster convergence rates compared to MCM’s. However, often, scientific and engineering problems involve irregular dependencies of the quantity of interest with respect to the random variable. As such, SGM’s and SCM’s become inefficient and may not converge at all [

In order to overcome the pitfall of global approximation, localized methods are used to arrest the inefficiencies inherent in SCM’s and SGM’s. Adaptive wavelet collocation methods are relied upon to remedy this situation. The use of this method has the additional advantage of eliminating the dreaded curse of dimensionality. Moreover, it maintains a better convergence rate in addition to producing optimal approximation, not only for PDE's, but also, for PDE-constrained optimal problems [

Wavelet-based methods for solving differential equations may be classified in two ways, the wavelet collocation methods and the adaptive wavelet schemes. To implement the adaptive wavelet scheme, we consider a second-generation wavelets constructed form the lifting scheme. Wavelets constructed in this form constitute a Riesz basis and have compact support, the desirable properties that guarantee a multiresolution analysis and required approximation.

The rest of the paper is organized as follows: In Section 2 we review the concept of multiresolution analysis in wavelet bases. This is one of the key concepts that will be used in the paper. In addition, the general properties of wavelet solutions to SPDE’s are considered. Section 3 analyzes the solution of the SHE with random coefficients. The stochastic heat equation with random source term is solved in Section 4, while a detail analysis of the full stochastic heat equation with all types of random inputs is solved and analyzed in Section 5. The paper ends with the conclusion in Section 6.

A wavelet is a function

1)

2)

3)

4)

5) Each subspace V_{j} is spanned by integer translates of a single function_{0}.

6) There exists a function_{0}, such that the sequence _{0}. The approximation of a function ^{j} is defined as the orthogonal projection of _{j}. In general a function _{j}. To compute the orthogonal projection requires that there exists a unique function_{j} is then defined by:

where

The goal of multiresolution analysis is to develop representations of a function ^{j}. To achieve this we seek to expand the given function in terms of basis functions

First, we define the Haar wavelet. Let X denote an infinite dimensional Banach space. A set

such that_{i}. The Haar orthogonal system (see for example [

In the space

and where we put

The Haar basis is convenient for

In general Daubechies wavelets depend on an integer

For

where

Solutions of SDE’s may be classified as weak or strong. If there exist a probability space with filtration, Brownian motion

then

A weak solution of the stochastic differential equation above is a triple

holds for all

The solution of a SDE requires the evaluation of an integral of the type:

where

1) Obtain an approximation for fractional noise

2) Apply an appropriate numerical scheme (for example, implicit or explicit Euler scheme) to obtain an approximation of the solution

3) Prove the almost sure convergence of the approximation to the solution.

Let

If

where

The fractional integral of the function f with respect to the function g is defined as:

If

where

See, for example, [

Second-generation wavelets are a generalized form of bi-orthogonal wavelets. Their applications easily fit functions defined on bounded domains. These wavelets form a Riesz basis for certain desirable function spaces. The lifting scheme is a method for constructing second generation wavelets that are no longer translates and dilates of a single scaling function. The lifting scheme is given by:

See, for example, [

The second generation collocation method makes the treatment of nonlinear terms in PDE’s easier to handle. Moreover, the use of wavelets enables the solution of differential equations with localized structures or sharp transitions more amenable. In order to solve such problem more efficiently, the use of computational grids that adapts dynamically in time to reflect local changes in the solution play an effective role.

Wavelet-based numerical algorithms may be classified into two main types namely the wavelet-Garlekin method and the wavelet collocation method. The wavelet-Garlekin algorithm uses gridless wavelet coefficient space while the collocation method relies on dynamically adaptive computational grid [

Let

1)

2)

3) for each

Since

Here, the MRA is not based on the scaling function

Since

Given the scaling function coefficients

where

Second generation wavelet transform may be considered in terms of filter banks, where filters not only act locally but may be potentially different for each coefficient. Now we can set

where

1) Compact support that is zero outside the interval

2)

3) Linear combinations of

4)

5)

Define the detail function as:

Hence

The lifting scheme is applied to infinite or periodic domains for the construction of the first-generation wavelets. The lifting scheme has the following advantages:

1) Faster implementation of the wavelet transform by a factor of 2.

2) No auxiliary memory required. The original signal is replaced with its wavelet transform.

3) Inverse wavelet transform is simply the reversal of the order of operations and switching of addition and operations. The scaling function and mother wavelet have vanishing moments, that is

where D is the domain over which the wavelets are constructed.

Consider the function

where the grid points

[

where

Hence

and the number of significant wavelet coefficients

where the coefficients

The adaptive grid is calculated as follows:

1) Sample

2) Perform the forward wavelet transform to obtain the values of

3) Analyze wavelet coefficients

4) Incorporate into the mask M all grid points associated with the scaling functions at the coarsest level of res- olution.

5) Starting from

The process of grid adaptation for the solution of PDE’s is made up of the following steps [

1) Use the values of the solution

2) Analyze wavelet coefficients

3) Extend the mask M with grid points associated with type I or II adjacent wavelets.

4) Perform the reconstruction check procedure to obtain a complete mask M.

5) Construct the new computational grid

When solutions of differential equations are intermittent in both space and time, methods combining adjustable time step with spatial grid to obtain approximate solutions. However, several problems depend on small spatial scales that are highly localized and as such, using a uniformly fine grid does not necessarily lead to and efficient method of solution. To address this concern, locally adapted grids are appealed to.

Wavelets can be used to used as an efficient tool to develop adaptive numerical methods capable of limiting the global approximation error associated with the numerical scheme. In addition to being fast, such wavelet- based schemes are asymptotically optimal when applied to elliptic differential equations [

The second generation adaptive wavelet can be used to discretize PDE’s as follows:

where

where

In order to construct grid points that adapt to intermittent solution, we consider the collocation points

The second generation wavelet decomposition takes the form:

[

In real thermal environments, the heat transfer coefficient of media surfaces are subject to temporal and spatial variations due to several factors [

with

If κ is random, three possible approaches to the solution are possible. Two of these methods are provided by [

In this case the solution is a complex nonlinear function of the coefficient κ [^{J} is approximately constant [

We assume a stochastic solution of the form:

where W_{0} = 1 and _{min} < κ_{max} < ∞. Here,

To obtain the approximation given by the equation above which yields an optimal wavelet basis by minimizing the total mean square error, we consider the sample space Ω equipped with the

where

and

and the random variables

where

We consider the heat equation with an additional forcing term. The quation now becomes:

A weak solution may be given as

where

which is almost Holder-

The greatest difficulty encountered in solving this problem involves the representation of the source term. [

where u_{k} are deterministic coefficients and

For any intermediate resolution level j (0 ≤ j < J) we have

where

Ususlly,

We consider the partial differential equation with random inputs in the form:

where

where

where

Using polynomials that have the property of diagonal interpolation matrix, leads to the stochastic collocation method. We re-formulate the problem by letting D denote a bounded domain in

Theorem 1. Find

where

The above problem may be solved using Lagrange Interpolation in parameter space. Let

After solving for the finite element approximation of the solution

Instead of using global polynomial interpolating spaces, piecewise polynomial interpolation spaces requiring only a fixed polynomial degree is needed. this method is based on refining the grid used and is suitable for problems having solutions with irregular behavior.

For each parameter dimension

where

and where

where

hence we have:

The hierarchical sparse-grid approximation of L is given by:

where

The approximation spaces

1)

2) Supp

3)

4) There is a constant C, independent of the level L, such that

For example, consider the hat function:

The major disadvantage of this that the linear hierarchical basis does not form a stable multiscale splitting of the approximation scale. The scheme does not ensure efficiency and optimality with respect to complexity as previously claimed.

A multi-resolution wavelet approximation though similar, performs better to achieve optimality since it possesses the additional property:

5) Riesz Property: The basis

By implication, other methods without this property are not

Suppose the wavelet decomposition is truncated at level J, we define the residual of the truncation by

This error is a function of the wavelet thresholding parameter

Wavelets can handle periodic boundary conditions efficiently. Moreover, the use of antiderivatives of wavelet bases as trial functions smoothen singurarities in wavelets. The basic principle is summarized as follows:

1) Represent the geometric region for the bvp in terms of wavelet series.

2) Represent the functions defined on the boundary and on the interior of the region in terms of wavelet series defined on a rectangular region containing the domain.

3) Convert the differential equation to some weak form.

4) Formulate and solve the wavelet Garlerkin problem for the domain and differential equation, using localized wavelets as orthonormal basis.

An important property of this method is that the coding for the solution is independent of the geometry of the boundary [

We have shown that wavelet-based solution to the stochastic heat equation with random inputs is stable. Computational methods based on the wavelet transform are analyzed for every possible type of stochastic heat equation. The methods are shown to be very convenient for solving such problems, since the initial and boundary conditions are taken into account automatically. The results reveal that the wavelet algorithms are very accurate and efficient.

Anthony Y.Aidoo,MatildaWilson, (2015) A Review of Wavelets Solution to Stochastic Heat Equation with Random Inputs. Applied Mathematics,06,2226-2239. doi: 10.4236/am.2015.614196