Kernel-based collocation methods for Zakai equations
We examine an application of the kernel-based interpolation to numerical solutions for Zakai equations in nonlinear filtering, and aim to prove its rigorous convergence. To this end, we find the class of kernels and the structure of collocation points explicitly under which the process of iterative interpolation is stable. This result together with standard argument in error estimation shows that the approximation error is bounded by the order of the square root of the time step and the error that comes from a single step interpolation. Our theorem is well consistent with the results of numerical experiments.
Key words: Zakai equations, kernel-based interpolation, stochastic partial differential equations, radial basis functions.
AMS MSC 2010: 60H15, 65M70, 93E11.
We are concerned with numerical methods for Zakai equations, linear stochastic partial differential equations of the form
with initial condition , where the process is an -dimensional standard Wiener processes on a complete probability space . Here, for each , the partial differential operator is given by
where is -valued, is -valued, is -valued, is -valued, and is -valued, all of which are defined on . The conditions for these functions are described in Section 2 below.
It is well known that solving Zakai equations is amount to computing the optimal filter for diffusion processes. We refer to Rozovskii , Kunita , Liptser and Shiryaev , Bensoussan , Bain and Crisan , and the references therein for Zakai equations and its relation with nonlinear filtering. It is also well known that for linear diffusion processes the optimal filters allow for finite dimensional realizations, i.e., they can be represented by some stochastic and deterministic differential equations in finite dimensions. For nonlinear diffusion processes, it is difficult to obtain such realizations except some special cases (see Beneš  and ). Thus one may be led to numerical approach to Zakai equations for computing the optimal filter. Several efforts have been made to obtain approximation methods for the equations during the past several decades. For example, the finite difference method (see Yoo , Gyöngy  and the references therein), the particle method (see Crisan et al. ), a series expansion approach (Lototsky et al. ), Galerkin type approximation (Ahmed and Radaideh  and Frey et al. ) and the splitting up method (Bensoussan et al. ).
In this paper, we examine the approximation of by a collocation method with kernel-based interpolation. Given a points set and a positive definite function , the function
interpolates on . Here, , is the column vector composed of , , and denotes the -th component of for . Thus, with time grid , the function recursively defined by
is a good candidate for an approximate solution of (1.1). The approximation above can be seen as a kernel-based (or meshfree) collocation method for stochastic partial differential equations. The meshfree collocation method is proposed by Kansa , where deterministic partial differential equations are concerned. Since then many studies on numerical experiments and practical applications for this method are generated. As for rigorous convergence, Schaback  and Nakano  study the case of deterministic linear operator equations and fully nonlinear parabolic equations, respectively. However, at least for parabolic equations, there is little known about explicit examples of the grid structure and kernel functions that ensure rigorous convergence. An exception is Hon et.al , where an error bound is obtained for a special heat equation in one dimension. A main difficulty lies in handling the process of the iterative kernel-based interpolation. A straightforward estimates for involves the condition number of the matrix , which in general rapidly diverges to infinity (see Wendland ). Thus we need to take a different route. Our main idea is to introduce a condition on the decay of when becomes large and to choose an appropriate approximation domain whose radius goes to infinity such that the interpolation is still effective. From this together with standard argument in error estimation we find that the approximation error is bounded by the order of the square root of the time step and the error that comes from a single step interpolation. See Lemma 3.7 and Theorem 3.4 below.
The structure of this paper is as follows: Section 2 introduces some notation, and describes the basic results for Zakai equations and the kernel-based interpolation, which are used in this paper. We derive an approximation method for the filter and prove its convergence in Section 3. Numerical experiments are performed in Section 4.
Throughout this paper, we denote by the transpose of a vector or matrix . For we set . For a multiindex of nonnegative integers, the differential operator is defined as usual by
with . For an open set , we denote by the space of continuous real-valued functions on with continuous derivatives up to the order , with the norm
Further, we denote by the space of infinitely differentiable functions on with compact supports. For any and any open set , we denote by the space of all measurable functions such that
For , we write for the space of all measurable functions on such that the generalized derivatives exist for all and that
In addition, for , we write for the space of all measurable functions on such that the generalized derivatives exist for all and that
For we use the notation . By we denote positive constants that may vary from line to line and that are independent of introduced below.
2.2 Zakai equations
We impose the following conditions for the equation (1.1):
All components of the functions , , , , and are infinitely differentiable with bounded continuous derivatives of any order.
For any ,
for any , where with and for any ;
Here, denots the inner product in , and for each , the partial differential operator is the formal adjoint of . Moreover, satisfies
Further, as in [17, Proposition 3, Section 1.3, Chapter 4], there exists a version , with respect to , of such that for and that for any and ,
In particular, is a solution to the Zakai equation in the strong sense, i.e., satisfies
2.3 Kernel-based interpolation
In this subsection, we recall the basis of the interpolation theory with positive definite functions. We refer to  for a complete account. Let be a radial and positive definite function, i.e., for all and for every , for all pairwise distinct and for all , we have
Let be a finite subset of and put . Then is invertible and thus for any the function
interpolates on . If , then
called the native space, is a real Hilbert space with inner product
and the norm . Here, for , the function is the Fourier transform of , defined as usual by
Moreover, is a reproducing kernel for . If satisfies
for some constants and , then we have from Corollary 10.13 in  that and
with . For example,
where denotes equality up to a positive constant factor.
3 Collocation method for Zakai equations
Let us describe the collocation methods for (2.1). In what follows, we always consider the version of , and thus by abuse of notation, we write for . Moreover, we restrict ourselves to the class of Wendland kernels described in Section 2.3. Suppose that the open rectangle for some is the set of points at which the approximate solution is to be computed. We take a set of grid points That is, we choose a set consisting of pairwise distinct points such that
To construct an approximate solution of Zakai equation, we first take a set of time discretized points such that . The solution of the Zakai equation approximately satisfies
where and . Since , we see
Thus, we define the function , a candidate of an approximate solution parametrized with a parameter , by
With this definition, the -dimensional vector of the collocation points satisfies
Here, we have set with . This follows from
To discuss the error of the approximation above, set and consider the Hausdorff distance between and , and the separation distance defined respectively by
Then suppose that , , , and are functions of .
The parameters , , , and satisfy , , , and as .
There exist and , positive constants independent of , such that
for with , and that
Notice that . Thus the condition implies as .
Suppose that is quasi-uniform in the sense that
hold for some positive constants . In this case, a sufficient condition for which the latter part of Assumption 3.1 (ii) holds is
with , for some positive constants and .
The approximation error for the Zakai equation is estimated as follows:
The rest of this section is devoted to the proof of Theorem 3.4. To this end, for every , put
In what follows, denotes the cardinality of a finite set .
Suppose that . Then there exists such that
Fix . Put for notational simplicity. It follows from the definition of that
Further, is disjoint. Indeed, otherwise, there exists , such that and . This implies , and so . Since we have assumed that ’s are pairwise distinct, we have . Denote by for the Lebesgue measure for . Then we have . Thus, for some that is independent of . ∎
Suppose that Assumption 3.1 (i) and hold. Then, there exists such that for any multi-index with and , we have
This result is reported in [19, Corollary 11.33] for more general domains. However, a simple application of that result leads to an ambiguity of the dependence of the constant on . Here we will confirm that we can take to be independent of .
Let with . Set and , . Then, and
Since as and , we can apply [19, Theorem 11.32] to to obtain
for some . Here, for an open set and ,
with being the smallest integer that exceeds and . It is straightforward to see that
Substituting these relations into (3.2), we have
This and (2.5) yield
Thus the lemma follows. ∎
Observe that for any ,
The following result tells us that the process of iterative kernel-based interpolation is stable, which is a key to our convergence analysis.
Suppose that Assumption 3.1 and hold. Then,
Fix and . Set for simplicity. First consider the set
Then, by Assumption 3.1 (ii),
This together with Lemma 3.5 leads to
Then, by Kergin interpolation (see Kergin ) there exists a polynomial on with degree at most that interpolates at for all . This leads to
for some that is independent of and . In particular, is Lipschitz continuous on with Lipschitz coefficient . Hence the function
is Lipschitz continuous on with the same Lipschitz coefficient and satisfies on . Further, for to be specified later, define the function on by
where is a -function such that for , for , and for for some . It is straightforward to verify that this function satisfies