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We revisit one of the classical search problems in which a diffusing target encounters a stationary searcher. Under the condition that the searcher’s detection region is much smaller than the search region in which the target roams diffusively, we carry out an asymptotic analysis to derive the decay rate of the non-detection probability. We consider two different geometries of the search region: a disk and a square, respectively. We construct a unified asymptotic expression valid for both of these two cases. The unified asymptotic expression shows that the decay rate of the non-detection probability, to the leading order, is proportional to the diffusion constant, is inversely proportional to the search region, and is inversely proportional to the logarithm of the ratio of the search region to the searcher’s detection region. Furthermore, the second term in the unified asymptotic expansion indicates that the decay rate of the non-detection probability for a square region is slightly smaller than that for a disk region of the same area. We also demonstrate that the asymptotic results are in good agreement with numerical solutions.

Searching is a common activity in our everyday life. For example, we look for lost car keys in a big parking lot, search for hidden natural resources such as oil and metals in a vast field, and search for missing people in a national park; the U.S. Coast Guards conduct thousands of open ocean search and rescue missions every year; the police hunts for drug smugglers; the military searches for Improvised Explosive Devices (IEDs) and insur- gents in Iraq and Afghanistan. More examples can be found in Koopman’s classical book [

Historically, the search for enemy submarines during World War II stimulated intensive scientific studies, giving rise to a branch of operations research now known as search theory [

In search theory, we call the object being sought the target. The search region is the region that both the target and the searcher are confined to. Generally speaking, search problems can be loosely divided into two cate- gories: one-sided search and two-sided search. In one-sided search, the searcher tries to detect the target while the target does not know the presence of the searcher, i.e., the target does not try to avoid the searcher in any active way. In two-sided search, the target has some capability of sensing the presence or the approaching of the searcher and may design a way to avoid being detected by the searcher. In one-sided search, the main objective is to maximize the probability of detection in a given time period and/or to minimize the cost or time of the search for a given tolerance of the non-detection probability. One-sided search can be further characterized by the constraints placed on the searcher’s actions and the target motion. This includes stationary target problems and moving target problems. Different approaches have been taken to address these problems. For example, Stone [

One of the classical one-sided search problems involves a lone searcher looking for a single moving target. A classical mathematical problem is to examine the non-detection probability of a diffusing target by a stationary sensor such as fixed acoustic sensors, sonobuoys, or possibly mines. This problem has been investigated by Eagle [

In this paper we revisit this classical problem of detecting a moving target by a stationary sensor or searcher. We first nondimensionalize the problem, then we derive asymptotic approximations to the decay rate of the non-detection probability of a diffusing target by a fixed searcher in a large detection region for two different geometries of the search region: a disk and a square, respectively. For the disk-shaped search region, we show that the asymptotic solution agrees very well with an accurate numerical solution obtained by solving an algebraic equation involving Bessel functions. For a square search region, asymptotic solutions show that the decay rate of the non-detection probability is slightly smaller than that for a disk search region of the same area. Finally, we combine the two cases into a unified form by expressing the decay rate of non-detection probability in terms of a large parameter: the ratio of the search region to the searcher’s detection region. The significance of our results lies in the simple and explicit asymptotic expressions of the decay rate of the non-detection pro- bability. It shows that the decay rate of the non-detection probability, to the leading order, is inversely propor- tional to the logarithm of the ratio of the search region to the searcher’s detection region.

Consider a search region A of “characteristic” radius

We consider the situation where the searcher is fixed at

where

p along the normal vector

The objective of this paper is to seek asymptotic solutions when the detection radius R of the searcher is much smaller than the characteristic radius

The function

After non-dimensionalization, the characteristic radius of the search region is 1 and the detection radius of the searcher is

The initial boundary value problem (2), in principle, can be solved using the method of separation of variables. More specifically, the solution can be expressed as

where

In the expression of

Let

That is, over long time, the decay rate of the survival or non-detection probability is given by the smallest eigenvalue

We consider a disk-shaped search region as illustrated in

In (Muskat, 1934), it is shown that

where

To derive an asymptotic solution, we view the right-hand side of (7) as a source term and recast the equation for

where

When

・ probability flows out at the absorbing boundary and gets added back in as the source term

・ the source term

・ the probability outflow (detection by the searcher) is slow;

・ the relaxation within the region is relatively fast.

From these observations, it follows that as long as the total amount of the source term

We use a delta function at

A general solution of Equation (10) is

The probability out-flow at the absorbing boundary is

In the calculations above, we have ignored terms smaller than

Note that

Specifically, in Equation (9), we set

multiple of an eigenfunction is still an eigenfunction. Equation (9) with boundary conditions becomes

A particular solution of Equation (13) is

Enforcing the boundary conditions

where

In the calculations above, we have neglected terms smaller than

With the new approximation of

constant multiple of an eigenfunction is still an eigenfunction. We set

and we pick

For each term on the right-hand side of (16), we find a corresponding particular solution. A particular solution of

Determining coefficients

We now have 3 approximations for eigenfunction

Based on the most recently updated

probability out-flow at the absorbing boundary and the total amount of the source term. In this new round of improvement iteration, we expect to get the coefficient of

where

asymptotic results for the normalized decay rate of non-detection probability,

Going back to the physical quantities before non-dimensionalization, we conclude that, over long time, the decay rate of non-detection probability for a disk search region has the asymptotic expression:

where

Next we study the case of a square search region as shown in

Even though the search region is not axisymmetric, we will not completely abandon the polar coordinates

For a square of size 2 centered at

We use a similar approach as we did in the case of a disk region. We treat the right-hand side as a source term and view

where

・ probability flows out at the absorbing boundary and gets added back in as the source term

・ the input of probability from the source term

・ in contrast, the relaxation of probability distribution within the region is relatively fast (fast time scale).

Due to the separation of slow and fast time scales, the exact distribution of the source term, to the leading order, will not affect solution

The exact solution of (23) is given by

The probability out-flow at the absorbing boundary is

where function

Geometrically, when we look at the intersection of the circle of radius r and the square,

Equating the out-flow and the source term, we have an expression of

where

Note that

approximate differential equation for

we choose

Both the differential equation and the absorbing boundary are still axisymmetric. When the reflecting condition at the square boundary is put away, the system allows axisymmetric solutions. We first solve for an axisymmetric solution of this system and then we use superposition to take care of the reflecting condition at the

boundary of the square. A particular axisymmetric solution of (29) is

general axisymmetric solution of (29) with the absorbing boundary condition can be written as

where

Note that in the general solution, both coefficients

is the coefficient of

For each of

of an integer multiple of

is zero if and only if the integral over one side of the square is zero. Specifically, in step 1 for

Substituting (31) into (33), we obtain

In step 2 for

The last condition

This property plays an important role in the calculation of

This quantity is also important in the calculation of

Next, we enforce the reflecting boundary condition for solution

Using Equation (38) to determine coefficient

In step 2 for

Similar to the situation for

The exact expression of

Putting all components together, the solution of (29) with both the absorbing boundary condition and the reflecting boundary condition is approximately

where functions

As before,

source term

where the coefficients

The 2-term asymptotic expansion of

where

We like to find the accuracy of the asymptotic solutions we derived above for a square search region. Unfortunately, for a square search region, no analytical or semi-analytical solution is known yet. A very accurate numerical solution is also difficult to compute. The circular cookie-cutter detector is incompatible with the square search region in a numerical discretization. It is difficult to design a numerical grid to accommodate both the square outer boundary and the circular inner boundary. Instead, we use Monte Carlo simulations to compute the decay rate of non-detection probability density.

To gauge the accuracy of Monte Carlo simulations, we first carry out Monte Carlo simulations in the case of a disk search region for which a very accurate numerical solution is known.

In the case of a square search region, we use the Monte Carlo solution as the accurate solution and we compare it with the two asymptotic expansions.

In terms of the physical quantities before non-dimensionalization, we conclude that over long time, the decay rate of non-detection probability for a square search region has the asymptotic expression:

where

Finally, we compare the results of the two cases that we have analyzed so far. We define the large parameter as the ratio of the area of search region A (disk or square) to the searcher’s detection area:

Since we only have a two-term expansion for a square search region, in the comparison we will use only two terms from the asymptotic expansion for a disk search region. We write the results of these two cases in a unified form:

where the shape factor

Note that in the unified form (50), the normalized decay rate is defined slightly differently from the one used in the discussion of a disk or a square search region. In the discussion of case 1 and case 2, we used

In summary, for both a disk search region and a square search region, the decay rate of the survival probability, to the leading order, is proportional to the diffusion constant, is inversely proportional to the search area, and is inversely proportional to the logarithm of the ratio of the search area to the searcher’s detection area. Furthermore, the second term in the asymptotic expansion implies that the decay rate of the non-detection probability for a square region is slightly smaller than that for a disk region of the same area.

We have derived asymptotic expressions for the decay rate of non-detection probability of a diffusing target in the presence of a stationary searcher in a large disk or square search region. The asymptotic expansions agree well with a very accurate numerical result obtained by solving an algebraic equation involving Bessel functions in the case of a disk search region. In the case of a square search region, the asymptotic expansions are validated by comparing them with an accurate result computed in large scale Monte Carlo simulations. Based on the asymptotic expansions obtained, respectively, for a disk and for a square, we write out a unified asymptotic expression valid for both of these two cases, in which the effect of the area of the search region is separated from the effect of the shape of the search region. The unified asymptotic expression shows that the decay rate of non-detection probability, to the leading order, is proportional to the diffusion constant, is inversely proportional to the search area, and is inversely proportional to the logarithm of the ratio of the search area to the searcher’s detection area. The leading term is not affected by the shape of the search region provided that the area of the search region is kept unchanged. The second term in the unified asymptotic expansion is affected by the shape of the search region. It indicates that the decay rate of non-detection probability for a square region is slightly smaller than that for a disk region of the same area.

Hong Zhou would like to thank Professor James Eagle, Professor Jim Scrofani and Professor Sivaguru Sritharan for helpful discussions.

HongyunWang,HongZhou, (2016) Non-Detection Probability of a Diffusing Target by a Stationary Searcher in a Large Region. Applied Mathematics,07,250-266. doi: 10.4236/am.2016.73023