1. A method for constructing a multi-level solver for a linear system, the method comprising the steps of:
a. providing a processor to execute the following steps of the method;
b. decomposing a planar graph A of the linear system into a plurality of pieces, wherein each of the pieces has a plurality of edges and a plurality of interface nodes that are fewer in number than the edges;
c. producing a local preconditioner for each of the pieces of the planar graph A;
d. producing a planar global preconditioner graph B of the planar graph A by aggregating the local preconditioners;
e. generating a planar graph C based on the planar global preconditioner B of the planar graph A, wherein the planar graph C is smaller than the planar graph A;
f. substituting the planar graph A with the planar graph C;
g. iterating between steps b-f until the planar graph C reaches a predetermined size to generate a sequence of graph A, global preconditioner graph B, and graph C to form a preconditioning hierarchy, wherein the planar graph A has an iteration count, wherein the global preconditioner graph B has an iteration count, wherein the iteration count of the planar graph A is the same as the iteration count of the global preconditioner graph B, wherein the planar graph C is smaller than the planar graph A, wherein the planar graph C has an iteration count; and
h. applying the preconditioning hierarchy to a recursive preconditioned iterative method to solve the linear system.
2. The method of claim 1, wherein the step of decomposing the planar graph A includes:
constructing a set of independent neighborhoods; and
decomposing the planar graph A into edge-connected Voronoi regions, wherein each of the edge-connected Voronoi regions corresponds to one of the independent neighborhoods.
3. The method of claim 2, wherein the step of decomposing the planar graph A includes decomposing the edge-connected Voronoi regions into Voronoi-Pair regions.
4. The method of claim 2, wherein the step of decomposing the planar graph A into the set of independent neighborhoods includes using parallel and sequential algorithms working locally to produce the set of independent neighborhoods of the planar graph A.
5. The method of claim 1, further comprising the step of eliminating at least one node in the pieces of the planar graph A before producing the planar global preconditioner graph B of the planar graph A.
6. The method of claim 5, wherein the step of eliminating includes eliminating nodes having a degree greater than two.
7. The method of claim 1, wherein the step of producing the local preconditioner includes independently and in parallel constructing local preconditioners for each of the pieces of the planar graph A.
8. The method of claim 7, wherein the step of independently and in parallel constructing the local preconditioners for each of the pieces of the planar graph A includes testing quality of the first local preconditioners.
9. The method of claim 8, further comprising the step of substituting different local preconditioners in response to the step of testing the quality of the local preconditioners.
10. The method of claim 9, further comprising the step of adjusting parameters of the recursive preconditioned iterative method in response to testing the quality of the local preconditioners.
11. The method of claim 1, wherein the step of generating the planar graph C includes eliminating nodes of degree one and degree two in the planar global preconditioner graph B of the planar graph A.
12. A method for constructing a multi-level solver for a linear system, the method comprising the steps of:
a. providing a processor to execute the following steps of the method;
b. providing a planar graph A of the system, wherein the planar graph A has an iteration count;
c. decomposing the planar graph A into a plurality of subgraphs;
d. producing a planar graph B for each of the plurality of subgraphs, wherein the planar graph B has an iteration count, wherein the iteration count of the planar graph A is the same as the iteration count of the planar graph B;
e. aggregating the planar graphs B of the plurality of subgraphs to form a global planar graph B;
f. generating a planar graph C based on the global planar graph B, wherein the planar graph C is smaller than the planar graph A, wherein the planar graph C has an iteration count,
g. forming a next planar graph A, wherein the next planar graph A is one iteration count higher than the iteration count of the planar graph A;
h. equating the next planar graph A to the planar graph C;
i. repeating steps c-h substituting the next planar graph A of step h for the planar graph A of step b until the planar graph C reaches a determined size;
j. forming a hierarchy of paired planar graphs A and B, wherein a paired planar graph A is paired with a paired planar graph B of a lower iteration count than the paired planar graph A; and
k. applying the hierarchy of paired planar graphs A and B to a recursive preconditioned iterative solver to solve the linear system.
13. The method of claim 12, wherein the planar graph B is a preconditioner planar graph B.
14. The method of claim 12, wherein the global planar graph B is a global preconditioner planar graph B.
The claims below are in addition to those above.
All refrences to claim(s) which appear below refer to the numbering after this setence.
1. A hearing aid, comprising:
a main body having a storage space;
a revolving shaft disposed in the main body;
a battery holder configured to be opened and closed with respect to the main body by revolving around the revolving shaft, and being configured to be installed in the storage space;
a lock bar disposed in the main body and being configured and arranged to pass through the storage space;
a protruding portion that disposed in the main body and protruding into the storage space; and
a latching portion disposed on the battery holder and having a slit forming a cut-out space, the slit having a first concave portion disposed on an open end side and a second concave portion disposed farther inside the slit relative to the first concave portion, the latching portion moving relative to the protruding portion and the lock bar on the main body when the battery holder is opened or closed with respect to the main body, the first concave portion being configured to latch to the protruding portion and the lock bar, and the second concave portion being configured to latch to the lock bar when the protruding portion is latched in the first concave portion.
2. The hearing aid according to claim 1, wherein
the battery holder is configured so as to be capable of being opened by a specific angle with respect to the main body in a first latched state in which the lock bar is latched in the first concave portion, and is configured so as to be completely closed with respect to the main body in a second latched state in which the protruding portion is latched in the first concave portion and the lock bar is latched in the second concave portion.
3. The hearing aid according to claim 2, wherein
in the second latched state, power is on, and in the first latched state, the power is off.
4. The hearing aid according to claim 1, wherein,
a diameter of the protruding portion is R1, and a diameter of the lock bar is R2, and the following relation is satisfied:
R1>R2.
5. The hearing aid according to claim 1, wherein,
a size of the gap of the slit in the first concave portion is r1, a size of the gap of the slit in the second concave portion is r2, a diameter of the protruding portion is R1, and a diameter of the lock bar is R2, and the following relation is satisfied:
r1\u2212R1>r2\u2212R2,
6. The hearing aid according to claim 1, wherein,
a size of the gap of the slit in the first concave portion is r1, a size of the gap of the slit in the second concave portion is r2, a size of the gap of the slit at the portion other than the first and second concave portions is r3, a diameter of the protruding portion is R1, and a diameter of the lock bar is R2, and the following relation is satisfied:
r1\u2248r2>R1>R2>r3,
7. The hearing aid according to claim 1, wherein,
the latching portion is configured and arranged such that first the lock bar and then the protruding portion are latched in the first and second concave portions, respectively, when the battery holder is closed.
8. The hearing aid according to claim 1, wherein
the lock bar metal.
9. The hearing aid according to claim 1, wherein
the main body is a combination of a first housing and a second housing, and
the lock bar links the first housing and the second housing.